Lionel Eyraud-Dubois

Loris Marchal

Oliver Sinnen

Parallel scheduling of task trees with limited memory

Keywords: Approximation algorithms, memory usage, multi-criteria optimization, pebblegame, scheduling, task graphs

This paper investigates the execution of tree-shaped task graphs using multiple processors. Each edge of such a tree represents some large data. A task can only be executed if all input and output data fit into memory, and a data can only be removed from memory after the completion of the task that uses it as an input data. Such trees arise, for instance, in the multifrontal method of sparse matrix factorization. The peak memory needed for the processing of the entire tree depends on the execution order of the tasks. With one processor the objective of the tree traversal is to minimize the required memory. This problem was well studied and optimal polynomial algorithms were proposed. Here, we extend the problem by considering multiple processors, which is of obvious interest in the application area of matrix factorization. With multiple processors comes the additional objective to minimize the time needed to traverse the tree, i.e., to minimize the makespan. Not surprisingly, this problem proves to be much harder than the sequential one. We study the computational complexity of this problem and provide inapproximability results even for unit weight trees. We design a series of practical heuristics achieving different trade-offs between the minimization of peak memory usage and makespan. Some of these heuristics are able to process a tree while keeping the memory usage under a given memory limit. The different heuristics are evaluated in an extensive experimental evaluation using realistic trees.

Ordonnancement parallèle d'arbres de tâches avec mémoire limitée

Résumé : Dans ce rapport, nous nous intéressons au traitement d'arbres de tâches par plusieurs processeurs. Chaque arête d'un tel arbre représente un gros fichier d'entrée/sortie. Une tâche peut être traitée seulement si l'ensemble de ses fichiers d'entrée et de sortie peut résider en mémoire, et un fichier ne peut être retiré de la mémoire que lorsqu'il a été traité. De tels arbres surviennent, par exemple, lors de la factorisation de matrices creuses par des méthodes multifrontales. La quantité de mémoire nécessaire dépend de l'ordre de traitement des tâches. Avec un seul processeur, l'objectif est naturellement de minimiser la quantité de mémoire requise. Ce problème a déjà été étudié et des algorithmes polynomiaux ont été proposés. Nous étendons ce problème en considérant plusieurs processeurs, ce qui est d'un intérêt évident pour le problème de la factorisation de grandes matrices. Avec plusieurs processeurs se pose également le problème de la minimisation du temps nécessaire pour traiter l'arbre. Nous montrons que comme attendu, ce problème est bien plus compliqué que dans le cas séquentiel. Nous étudions la complexité de ce problème et nous fournissons des résultats d'inaproximabilité, même dans le cas de poids unitaires. Nous proposons plusieurs heuristiques qui obtiennent différents compromis entre mémoire et temps d'exécution. Certaines d'entre elles sont capables de traiter l'arbre tout en gardant la consommation mémoire inférieure à une limite donnée. Nous analysons les performances de toutes ces heuristiques par une large campagne de simulations utilisant des arbres réalistes.

Mots-clés : Algorithmes d'approximation, consommation mémoire, optimisation multi-critères, ordonnancement, graphe de tâches

Introduction

Parallel workloads are often modeled as task graphs, where nodes represent tasks and edges represent the dependencies between tasks. There is an abundant literature on task graph scheduling when the objective is to minimize the total completion time, or makespan. However, with the increase of the size of the data to be processed, the memory footprint of the application can have a dramatic impact on the algorithm execution time, and thus needs to be optimized. This is best exemplified with an application which, depending on the way it is scheduled, will either fit in the memory, or will require the use of swap mechanisms or out-of-core techniques. There are very few existing studies on the minimization of the memory footprint when scheduling task graphs, and even fewer of them targeting parallel systems.

We consider the following memory-aware parallel scheduling problem for rooted trees. The nodes of the tree correspond to tasks, and the edges correspond to the dependencies among the tasks. The dependencies are in the form of input and output files1 : each node takes as input several large files, one for each of its children, and it produces a single large file, and the different files may have different sizes. Furthermore, the execution of any node requires its execution file to be present; the execution file models the program and/or the temporary data of the task. We are to execute such a set of tasks on a parallel system made of p identical processing resources sharing the same memory. The execution scheme corresponds to a schedule of the tree where processing a node of the tree translates into reading the associated input files and producing the output file. How can the tree be scheduled so as to optimize the memory usage?

Modern computing platforms exhibit a complex memory hierarchy ranging from caches to RAM and disks and even sometimes tape storage, with the classical property that the smaller the memory, the quicker. Thus, to avoid large running times, one usually wants to avoid the use of memory devices whose IO bandwidth is below a given threshold: even if out-of-core execution (when large data are unloaded to disks) is possible, this requires special care when programming the application and one usually wants to stay in the main memory (RAM). This is why in this paper, we are interested in the question of minimizing the amount of main memory needed to completely process an application.

Throughout the paper, we consider in-trees where a task can be executed only if all its children have already been executed (This is absolutely equivalent to considering out-trees as a solution for an in-tree can be transformed into a solution for the corresponding out-tree by just reversing the arrow of time, as outlined in [START_REF] Jacquelin | On optimal tree traversals for sparse matrix factorization[END_REF]). A task can be processed only if all its files (input, output, and execution) fit in currently available memory. At a given time, many files may be stored in the memory, and at most p tasks may be processed by the p processors. This is obviously possible only if all tasks and execution files fit in memory. When a task finishes, the memory needed for its execution file and its input files is released. Clearly, the schedule which determines the processing times of each task plays a key role in determining which amount of main memory is needed for a successful execution of the entire tree.

The first motivation for this work comes from numerical linear algebra. Tree workflows (assembly or elimination trees) arise during the factorization of sparse matrices, and the huge size of the files involved makes it absolutely necessary to reduce the memory requirement of the factorization. The sequential version of this problem (i.e., with p = 1 processor) has already been studied. Liu [START_REF] Joseph | On the storage requirement in the out-of-core multifrontal method for sparse factorization[END_REF] discusses how to find a memory-minimizing traversal when the traversal is required to correspond to a postorder traversal of the tree. A follow-up study [START_REF] Joseph | An application of generalized tree pebbling to sparse matrix factorization[END_REF] presents an optimal algorithm to solve the general problem, without the postorder constraint on the traversal. Postorder traversals are known to be arbitrarily worse than optimal traversals for memory minimization [START_REF] Jacquelin | On optimal tree traversals for sparse matrix factorization[END_REF]. However, they are very natural and straightforward solutions to this problem, as they allow to fully process one subtree before starting a new one. Therefore, they are thus widely used in sparse matrix software like MUMPS [START_REF] Amestoy | A fully asynchronous multifrontal solver using distributed dynamic scheduling[END_REF][START_REF] Amestoy | Hybrid scheduling for the parallel solution of linear systems[END_REF], and in practice, they achieve close to optimal performance on actual elimination trees [START_REF] Jacquelin | On optimal tree traversals for sparse matrix factorization[END_REF].

The parallel version of this problem is a natural continuation of these studies: when processing large elimination trees, it is very meaningful to take advantage of parallel processing resources. However, to the best of our knowledge, no theoretical study exists for this problem. A preliminary version of this work, with fewer complexity results and proposed heuristics, was presented at IPDPS 2013 [START_REF] Marchal | Scheduling tree-shaped task graphs to minimize memory and makespan[END_REF]. The key contributions of this work are:

• A new proof that the parallel variant of the pebble game problem is NP-complete (simpler than in [START_REF] Marchal | Scheduling tree-shaped task graphs to minimize memory and makespan[END_REF]). This shows that the introduction of memory constraints, in the simplest cases, suffices to make the problem NP-hard (Theorem 1). • The proof that no schedule can simultaneously achieve a constant-ratio approximation for the memory minimization and for the makespan minimization (Theorem 2); bounds on the achievable approximation ratios for makespan and memory when the number of processors is fixed (Theorems 3 and 4). • A series of practical heuristics achieving different trade-offs between the minimization of peak memory usage and makespan; some of these heuristics are guaranteed to keep the memory under a given memory limit. • An exhaustive set of simulations using realistic tree-shaped task graphs corresponding to elimination trees of actual matrices; the simulations assess the relative and absolute performance of the heuristics. The rest of this paper is organized as follows. Section 2 reviews related studies. The notation and formalization of the problem are introduced in Section 3. Complexity results are presented in Section 4 while Section 5 proposes different heuristics to solve the problem, which are evaluated in Section 6.

2 Background and Related Work

Sparse matrix factorization

As mentioned above, determining a memory-efficient tree traversal is very important in sparse numerical linear algebra. The elimination tree is a graph theoretical model that represents the storage requirements, and computational dependencies and requirements, in the Cholesky and LU factorization of sparse matrices. In a previous study [START_REF] Jacquelin | On optimal tree traversals for sparse matrix factorization[END_REF], we have described how such trees are built, and how the multifrontal method [START_REF] Liu | The multifrontal method for sparse matrix solution: Theory and practice[END_REF] organizes the computations along the tree. This is the context of the founding studies of Liu [START_REF] Joseph | On the storage requirement in the out-of-core multifrontal method for sparse factorization[END_REF][START_REF] Joseph | An application of generalized tree pebbling to sparse matrix factorization[END_REF] on memory minimization for postorder or general tree traversals presented in the previous section. Memory minimization is still a concern in modern multifrontal solvers when dealing with large matrices. Among other, efforts have been made to design dynamic schedulers that take into account dynamic pivoting (which impacts the weights of edges and nodes) when scheduling elimination trees with strong memory constraints [START_REF] Guermouche | Memory-based scheduling for a parallel multifrontal solver[END_REF], or to consider both task and tree parallelism with memory constraints [START_REF] Agullo | Robust memory-aware mappings for parallel multifrontal factorizations[END_REF]. While these studies try to optimize memory management in existing parallel solvers, we aim at designing a simple model to study the fundamental underlying scheduling problem.

Scientific workflows

The problem of scheduling a task graph under memory constraints also appears in the processing of scientific workflows whose tasks require large I/O files. Such workflows arise in many scientific fields, such as image processing, genomics or geophysical simulations. The problem of task graphs handling large data has been identified in [START_REF] Ramakrishnan | Scheduling dataintensiveworkflows onto storage-constrained distributed resources[END_REF] which proposes some simple heuristic solutions. Surprisingly, in the context of quantum chemistry computations, Lam et al. [START_REF] Lam | Memory-optimal evaluation of expression trees involving large objects[END_REF] have recently rediscovered the algorithm published in 1987 in [START_REF] Joseph | An application of generalized tree pebbling to sparse matrix factorization[END_REF].

Pebble game and its variants

On the more theoretical side, this work builds upon the many papers that have addressed the pebble game and its variants. Scheduling a graph on one processor with the minimal amount of memory amounts to revisiting the I/O pebble game with pebbles of arbitrary sizes that must be loaded into main memory before firing (executing) the task. The pioneering work of Sethi and Ullman [START_REF] Sethi | The generation of optimal code for arithmetic expressions[END_REF] deals with a variant of the pebble game that translates into the simplest instance of our problem when all input/output files have weight 1 and all execution files have weight 0. The concern in [START_REF] Sethi | The generation of optimal code for arithmetic expressions[END_REF] was to minimize the number of registers that are needed to compute an arithmetic expression. The problem of determining whether a general DAG can be executed with a given number of pebbles has been shown NP-hard by Sethi [START_REF] Sethi | Complete register allocation problems[END_REF] if no vertex is pebbled more than once (the general problem allowing recomputation, that is, re-pebbling a vertex which has been pebbled before, has been proven Pspace complete [START_REF] Gilbert | The pebbling problem is complete in polynomial space[END_REF]). However, this problem has a polynomial complexity for tree-shaped graphs [START_REF] Sethi | The generation of optimal code for arithmetic expressions[END_REF].

To the best of our knowledge, there have been no attempts to extend these results to parallel machines, with the objective of minimizing both memory and total execution time. We present such an extension in Section 4.

Model and objectives

Application model

We consider in this paper a tree-shaped task-graph T composed of n nodes, or tasks, numbered from 1 to n. Nodes in the tree have an output file, an execution file (or program), and several input files (one per child). More precisely:

• Each node i in the tree has an execution file of size n i and its processing on a processor takes time w i .

• Each node i has an output file of size f i . If i is not the root, its output file is used as input by its parent parent(i); if i is the root, its output file can be of size zero, or contain outputs to the outside world.

• Each non-leaf node i in the tree has one input file per child. We denote by Children(i) the set of the children of i. For each child j ∈ Children(i), task j produces a file of size f j for i.

If i is a leaf-node, then Children(i) = ∅ and i has no input file: we consider that the initial data of the task either resides in its execution file or is read from disk (or received from the outside word) during the execution of the task.

During the processing of a task i, the memory must contain its input files, the execution file, and the output file. The memory needed for this processing is thus:

  j∈Children(i) f j   + n i + f i
After i has been processed, its input files and execution file (program) are discarded, while its output file is kept in memory until the processing of its parent.

Platform model and objectives

In this paper, our goal is to design a simple platform model which allows to study memory minimization on a parallel platform. We thus consider p identical processors sharing a single memory.

Any sequential optimal schedule for memory minimization is obviously an optimal schedule for memory minimization on a platform with any number p of processors. Therefore, memory minimization on parallel platforms is only meaningful in the scope of multi-criteria approaches that consider trade-offs between the following two objectives: root

N 1 L 1 1 L 1 2 . . . L 1 3m×a1 N 2 L 2 1 L 2 2 . . . L 2 3m×a2 . . . N 3m L 3m 1 L 3m 2 . . . L 3m 3m×a3m
Figure 1: Tree used for the NP-completeness proof

• Makespan: the classical makespan, or total execution time, which corresponds to the timespan between the beginning of the execution of the first leaf task and the end of the processing of the root task.

• Memory: the amount of memory needed for the computation. At each time step, some files are stored in the memory and some task computations occur, inducing a memory usage. The peak memory is the maximum usage of the memory over the whole schedule, hence the memory that needs to be available, which we aim to minimize.

Complexity results in the pebble game model

Since there are two objectives, the decision version of our problem can be stated as follows.

Definition 1 (BiObjectiveParallelTreeScheduling). Given a tree-shaped task graph T with file sizes and task execution times, p processors, and two bounds B Cmax and B mem , is there a schedule of the task graph on the processors whose makespan is not larger than B Cmax and whose peak memory is not larger than B mem ?

This problem is obviously NP-complete. Indeed, when there are no memory constraints (B mem = ∞) and when the task tree does not contain any inner node, that is, when all tasks are either leaves or the root, then our problem is equivalent to scheduling independent tasks on a parallel platform which is an NP-complete problem as soon as tasks have different execution times [START_REF] Lenstra | Complexity of machine scheduling problems[END_REF]. Conversely, minimizing the makespan for a tree of same-size tasks can be solved in polynomial-time when there are no memory constraints [START_REF] Hu | Parallel sequencing and assembly line problems[END_REF]. In this section, we consider the simplest variant of the problem. We assume that all input files have the same size (∀i, f i = 1) and no extra memory is needed for computation (∀i, n i = 0). Furthermore, we assume that the processing of each node takes unit time: ∀i, w i = 1. We call this variant of the problem the Pebble Game model since it perfectly corresponds to the pebble game problems introduced above: the weight f i = 1 corresponds to the pebble one must put on node i to process it; this pebble must remain there until the parent of node i has been completed, because the parent of node i uses as input the output of node i. Processing a node is done in unit time.

In this section, we first show that, even in this simple variant, the introduction of memory constraints (a limit on the number of pebbles) makes the problem NP-hard (Section 4.1). Then, we show that when trying to minimize both memory and makespan, it is not possible to get a solution with a constant approximation ratio for both objectives, and we provide tighter ratios when the number of processors is fixed (Section 4.2).

NP-completeness

Theorem 1. The BiObjectiveParallelTreeScheduling problem is NP-complete in the Pebble Game model (i.e., with ∀i,

f i = w i = 1, n i = 0).
Proof. First, it is straightforward to check that the problem is in NP: given a schedule, it is easy to compute its peak memory and makespan.

To prove the problem NP-hard, we perform a reduction from 3-Partition, which is known to be NP-complete in the strong sense [START_REF] Garey | Computers and Intractability, A Guide to the Theory of NP-Completeness[END_REF]. We consider the following instance I 1 of the 3-Partition problem: let a i be 3m integers and B an integer such that a i = mB. We consider the variant of the problem, also NP-complete, where ∀i, B/4 < a i < B/2. To solve I 1 , we need to solve the following question: does there exist a partition of the a i 's in m subsets S 1 , . . . , S m , each containing exactly 3 elements, such that, for each S k , i∈S k a i = B? We build the following instance I 2 of our problem, illustrated in Figure 1. The tree contains a root r with 3m children, the N i 's, each one corresponding to a value a i . Each node N i has 3m × a i children, L i 1 , ..., L i 3m×ai , which are leaf nodes. The question is to find a schedule of this tree on p = 3mB processors, whose peak memory is not larger than B mem = 3mB + 3m and whose makespan is not larger than B Cmax = 2m + 1.

Assume first that there exists a solution to I 1 , i.e., that there are m subsets S k of 3 elements with i∈S k a i = B. In this case, we build the following schedule:

• At step 1, we process all the nodes L i1

x , L j1 y , and L k1 z with S 1 = {a i1 , a j1 , a k1 }. There are 3mB = p such nodes, and the amount of memory needed is also 3mB.

• At step 2, we process the nodes N i1 , N j1 , N k1 . The memory needed is 3mB + 3.

• At step 2n + 1, with 1 ≤ n ≤ m -1, we process the 3mB = p nodes L in x , L jn y , L kn z with S n = {a in , a jn , a kn }. The amount of memory needed is 3mB + 3n (counting the memory for the output files of the N t nodes previously processed).

• At step 2n + 2, with 1 ≤ n ≤ m -1, we process the nodes N in , N jn , N kn . The memory needed for this step is 3mB + 3(n + 1).

• At step 2m + 1, we process the root node and the memory needed is 3m + 1.

Thus, the peak memory of this schedule is B mem and its makespan B Cmax . Reciprocally, assume that there exists a solution to problem I 2 , that is, there exists a schedule of makespan at most B Cmax = 2m + 1. Without loss of generality, we assume that the makespan is exactly 2m+1. We start by proving that at any step of the algorithm, at most three of the N i nodes are being processed. By contradiction, assume that four (or more) such nodes N is , N js , N ks , N ls are processed during a certain step s. We recall that a i > B/4 so that a is + a js + a ks + a ls > B and thus a is + a js + a ks + a ls ≥ B + 1. The memory needed at this step is thus at least (B + 1)3m for the children of the nodes N is , N js , N ks , and N ls and 4 for the nodes themselves, hence a total of at least (B + 1)3m + 4, which is more than the prescribed bound B mem . Thus, at most three of N i nodes are processed at any step. In the considered schedule, the root node is processed at step 2m + 1. Then, at step 2m, some of the N i nodes are processed, and at most three of them from what precedes. The a i 's corresponding to those nodes make the first subset S 1 . Then all the nodes L j

x such that a j ∈ S 1 must have been processed at the latest at step 2m -1, and they occupy a memory footprint of 3m aj ∈S1 a j at steps 2m -1 and 2m. Let us assume that a node N k is processed at step 2m -1. For the memory bound B mem to be satisfied we must have a k + aj ∈S1 a j ≤ B. (Otherwise, we would need a memory of at least 3m(B + 1) for the involved L j

x nodes plus 1 for the node N k ). Therefore, node N k can as well be processed at step 2m instead of step 2m -1. We then modify the schedule so as to schedule N k at step 2m and thus we add k to S 1 . We can therefore assume, without loss of generality, that no N i node is processed at step 2m -1. Then, at step 2m -1 only the children of the N j nodes with a j ∈ S 1 are processed, and all of them are. So, none of them have any memory footprint before step 2m -1. We then generalize this analysis: at step 2i, for 1 ≤ i ≤ m -1, only some N j nodes are processed and they define a subset S i ; at step 2i -1, for 1 ≤ i ≤ m -1, are processed exactly the nodes L k

x that are children of the nodes N j such that a j ∈ S i .

Because of the memory constraint, each of the m subsets of a i 's built above sum to at most B. Since they contain all a i 's, their sum is mB. Thus, each subset S k sums to B and we have built a solution for I 1 .

Joint minimization of both objectives

As our problem is NP-complete, it is natural to wonder whether approximation algorithms can be designed. In this section, we prove that there does not exist any scheduling algorithm which approximates both the minimum makespan and the minimum peak memory with constant factors. This is equivalent to saying that there is no Zenith (also called simultaneous) approximation. We first state a lemma, valid for any tree-shaped task graph, which provides lower bounds for the makespan of any schedule.

Lemma 1. For any schedule S on p processors with a peak memory M , we have the two following lower bounds on the makespan C max :

C max ≥ 1 p n i=1 w i M × C max ≥ n i=1   n i + f i + j∈Children(i) f j   w i
In the pebble game model, these equations can be written as:

C max ≥ n/p M × C max ≥ 2n -1
Proof. The first inequality is a classical bound that states that all tasks must be processed before C max . Similarly, each task i uses a memory of n i + f i + j∈Children(i) f j during a time w i . Hence, the total memory usage (i.e., the sum over all time instants t of the memory used by S at time t) needs to be at least equal to n i=1 n i + f i + j∈Children(i) f j w i . Because S uses a memory that is not larger than M at any time, the total memory usage is upper bounded by M × C max . This gives us the second inequality. In the pebble game model, the right-hand-side term of the second inequality can be simplified:

n i=1   n i + f i + j∈Children(i) f j   w i = n i=1 f i +   n i=1 j∈Children(i) f j   = n + (n -1)
In the next theorem, we show that it is not possible to design an algorithm with constant approximation ratios for both makespan and maximum memory, i.e. approximation ratios independent of the number of processors p. In Theorem 3, we will provide a refined version which analyzes the dependence on p. Theorem 2. For any given constants α and β, there does not exist any algorithm for the pebble game model that is both an α-approximation for makespan minimization and a β-approximation for peak memory minimization when scheduling in-tree task graphs.

Proof. We consider in this proof the tree depicted in Figure 2. The root of this tree has m children a 1 , . . . , a m . Any of these children, a i , has m children b i,1 , . . . , b i,m . Therefore, overall this tree contains n = 1 + m + m × m nodes. On the one hand, with a large number of processors (namely, m 2 ), this tree can be processed in C * max = 3 time steps: all the leaves are processed in the first step, all the a i nodes in the second step, and finally the root in the third and last step. On the other hand, the minimum memory required to process the tree is M * = 2m. This is achieved by processing the tree with a single processor. The subtrees rooted at the a i 's are processed one at a time. The processing of the subtree rooted at node a i requires a memory of m + 1 (for the root processing of its root a i once the m leaves have been processed). Once such a subtree is processed there is a unit file that remains in memory. Hence, the peak memory usage when processing the j-th of these subtrees is (j -1) + (m + 1) = j + m. The overall peak M * = 2m is thus reached when processing the root of the last of these subtrees.

a 1 b 1,1 b 1,2 ... b 1,m a 2 b 2,1 b 2,2 ... b 2,m . . .
Let us assume that there exists a schedule S which is both an α-approximation for the makespan and a β-approximation for the peak memory. Then, for the tree of Figure 2, the makespan C max of S is at most equal to 3α, and its peak memory M is at most equal to 2βm. Because

n = 1+m+m 2 , Lemma 1 implies that M × C max ≥ 2n -1 = 2m 2 + 2m + 1. Therefore M ≥ 2m 2 +2m+1 3α
. For a sufficiently large value of m, this is larger than 2βm, the upper bound on M . This contradicts the hypothesis that S is a β-approximation for peak memory usage.

Theorem 2 only considers approximation algorithms whose approximation ratios are constant. In the next theorem we consider algorithms whose approximations ratios may depend on the number of processors in the platform. Theorem 3. When scheduling in-tree task graphs in the pebble-game model on a platform with p ≥ 2 processors, there does not exist any algorithm that is both an α(p)-approximation for makespan minimization and a β(p)-approximation for peak memory minimization, with

α(p)β(p) < 2p ⌈log(p)⌉ + 2 •
Proof. We establish this result by contradiction. We assume that there exists an algorithm that is an α(p)-approximation for makespan minimization and a β(p)-approximation for peak memory minimization when scheduling in-tree task graphs, with α(p)β(p) = 2p (⌈log(p)⌉+2)ǫ with ǫ > 0. The proof relies on a tree similar to the one depicted in Figure 3 for the case p = 13. The top part of the tree is a complete binary subtree with ⌈ p 2 ⌉ leaves, l 1 , . . . , l ⌈ p 2 ⌉ , and of height m. Therefore, this subtree contains 2⌈ p 2 ⌉ -1 nodes, all of its leaves are at depth either m or m -1, and

m = 1 + ⌈log(⌈ p 2 ⌉)⌉ = ⌈log(p)⌉.
To prove the last equality, we consider whether p is even:

• p is even: ∃l ∈ N, p = 2l. Then, 1 + ⌈log(⌈ p 2 ⌉)⌉ = ⌈log(2⌈ 2l 2 ⌉)⌉ = ⌈log(2l)⌉ = ⌈log(p)⌉. • p is odd: ∃l ∈ N, p = 2l + 1. Then, 1 + ⌈log(⌈ p 2 ⌉)⌉ = ⌈log(2⌈ 2l+1 2 ⌉)⌉ = ⌈log(2l + 2)⌉. Since 2l + 1 is odd, ⌈log(2l + 2)⌉ = ⌈log(2l + 1)⌉ = ⌈log(p)⌉.
Each node l i is the root of a comb subtree of height k (except the last node if p is odd); each comb subtree contains 2k -1 nodes. If p is odd, the last leaf of the binary top subtree, l ⌈ p 2 ⌉ , is the root of a chain subtree with k -1 nodes. Then, the entire tree contains n = pk -1 nodes (be careful not to count twice the roots of the comb subtrees):

• p is even: ∃l ∈ N, p = 2l. Then, • p is odd: ∃l ∈ N, p = 2l + 1. Then,

n = 2 p 2 -1 + p 2 (2k -2) = (2l -1) + l(2k -2) = pk -1. RR n°8606 root l 1 l 2 l 3 l 4 l 5 l 6 l 7 m levels k levels k -1 levels
n = 2 p 2 -1 + p 2 -1 (2k -2) + (k -2) = (2(l + 1) -1) + (l + 1 -1)(2k -2) + (k -2) = (2l + 1)k -1 = pk -1.
With the p processors, it is possible to process all comb subtrees (and the chain subtree if p is odd) in parallel in k steps by using two processors per comb subtree (and one for the chain subtree). Then, m -1 steps are needed to complete the processing of the binary reduction (the l i nodes have already been processed at the last step of the processing of the comb subtrees). Thus, the optimal makespan with p processors is C * max = k + m -1. We now compute the optimal peak memory usage, which is obtained with a sequential processing. Each comb subtree can be processed with 3 units of memory, if we follow any postorder traversal starting from the deepest leaves. We consider the sequential processing of the entire tree that follows a postorder traversal that process each comb subtree as previously described, that process first the leftmost comb subtree, then the second leftmost comb subtree, the parent node of these subtrees, and so on, and finishes with the rightmost comb subtree (or the chain subtree if p is odd). The peak memory is reached when processing the last comb subtree. At that time, either m -2 or m -1 edges of the binary subtree are stored in memory (depending on the value of ⌈ p 2 ⌉). The processing of the last comb subtree itself uses 3 units of memory. Hence, the optimal peak memory is not greater than m + 2:

M * ≤ m + 2.
Let C max denote the makespan achieved by the studied algorithm on the tree, and let M denote its peak memory usage. By definition, the studied algorithm is an α(p)-approximation for the makespan:

C max ≤ α(p)C * max . Thanks to Lemma 1, we know that M × C max ≥ 2n -1 = 2pk -3.
Therefore,

M ≥ 2pk -3 C max ≥ 2pk -3 α(p)(k + m -1)
.

The approximation ratio of the studied algorithm with respect to the peak memory usage is, thus, bounded by:

β(p) ≥ M M * ≥ 2pk -3 α(p)(k + m -1)(m + 2) • Therefore, if we recall that m = ⌈log(p)⌉, α(p)β(p) ≥ 2pk -3 (k + m -1)(m + 2) = 2pk -3 (k + ⌈log(p)⌉ -1)(⌈log(p)⌉ + 2) ----→ k→∞ 2p ⌈log(p)⌉ + 2 • root cp 1 1 d 1 1 a 1,1 δ a 1,1 δ-1 ... a 1,1 2 a 1,1 1 cp 1 2 d 1 2 a 1,2 δ-1 ... a 1,2 2 a 1,2 1 cp 1 δ-1 d 1 δ-1 a 1,δ-1 2 a 1,δ-1 1 c 1 1 c 1 2 c 1 3 c 1 k cp 2 1 d 2 1 a 2,1 δ a 2,1 δ-1 ... a 2,1 2 a 2,1 1 cp 2 2 d 2 2 a 2,2 δ-1 ... a 2,2 2 a 2,2 1 cp 2 δ-1 d 2 δ-1 a 2,δ-1 2 a 2,δ-1 1 c 2 1 c 2 2 c 2 3 c 2 k . . . cp p-1 1 d p-1 1 a p-1,1 δ a p-1,1 δ-1
... Hence, there exists a value k 0 such that, for any k ≥ k 0 ,

a p-1,1 2 a p-1,1 1 cp p-1 2 d p-1 2 a p-1,2 δ-1 ... a p-1,2 2 a p-1,2 1 cp p-1 δ-1 d p-1 δ-1 a p-1,δ-1 2 a p-1,δ-1 1 c p-1 1 c p-1 2 c p-1 3 c p-1 k
α(p)β(p) ≥ 2p ⌈log(p)⌉ + 2 - ǫ 2 •
This contradicts the definition of ǫ and, hence, concludes the proof.

Readers may wonder whether the bound in Theorem 3 is tight. This interrogation is especially relevant because the proof of Theorem 3 uses the average memory usage as a lower bound to the peak memory usage. This technique enables to design a simple proof, which may however be very crude. In fact, in the special case where α(p) = 1, that is, for makespan-optimal algorithms, a stronger result holds. For that case, Theorem 3 states that β(p) ≥ 2p ⌈log(p)⌉+2 . Theorem 4 below states that β(p) ≥ p -1 (which is a stronger bound when p ≥ 4). This result is established through a careful, painstaking analysis of a particular task graph. Using the average memory usage argument on this task graph would not enable to obtain a non-trivial bound.

Theorem 4. There does not exist any algorithm that is both optimal for makespan minimization and that is a (p -1ǫ)-approximation algorithm for the peak memory minimization, where p is the number of processors and ǫ > 0.

Proof. To establish this result, we proceed by contradiction. Let p be the number of processors. We then assume that there exists an algorithm A which is optimal for makespan minimization and which is a β(p)-approximation for peak memory minimization, with β(p) < p -1. So, there exists ǫ > 0 such that β(p) = p -1ǫ.

The tree. Figure 4 presents the tree used to derive a contradiction. This tree is made of p -1 identical subtrees whose roots are the children of the tree root. The value of δ will be fixed later on.

Optimal peak memory. A memory-optimal sequential schedule processes each subtree rooted at cp i 1 sequentially. Each of these subtrees can be processed with a memory of δ + 1 by processing first the subtree rooted at d i 1 , then the one rooted at d i 2 , etc., until the one rooted at d i δ-1 , and then the chain of c i j nodes, and the remaining cp i j nodes. The peak memory, reached when processing the last subtree rooted at cp i 1 , is thus δ + p -1. Optimal execution time. The optimal execution time with p processors is at least equal to the length of the critical path. The critical path has a length of δ + k, which is the length of the path from the root to any c i k node, with 1 ≤ i ≤ p -1. We now define k for this lower bound to be an achievable makespan, with an overall schedule as follows:

• Each of the first p -1 processors processes one of the critical paths from end to end (except obviously for the root node that will only be processed by one of them). • The last processor processes all the other nodes. We define k so that this processor finishes processing all the nodes it is allocated at time k + δ -2. This allows, the other processors to process all p -1 nodes cp 1 1 through cp p-1

1 from time k + δ -2 to time k + δ -1.
In order to find such a value for k, we need to compute the number of nodes allocated to the last processor. In the subtree rooted in cp i 1 , the last processor is in charge of processing the δ -

1 nodes d i 1 through d i δ-1
, and the descendants of the d i j nodes, for 1 ≤ j ≤ δ -1. As node d i j has δj + 1 descendants, the number of nodes in the subtree rooted in cp i 1 that are allocated to the last processor is equal to:

(δ -1) + δ-1 j=1 (δ -j + 1) = δ -2 + δ(δ + 1) 2 = δ 2 + 3δ -4 2 = (δ + 4)(δ -1) 2 .
All together, the last processor is in charge of the processing of (p -1) (δ+4)(δ-1) 2

nodes. As we have stated above, we want this processor to be busy from time 0 to time k + δ -2. This gives the value of k:

k + δ -2 = (p -1) (δ + 4)(δ -1) 2 ⇔ k = (p -1)δ 2 + (3p -5)δ + 4(2 -p) 2 .
Remark, by looking at the first equality, that the expression on the right-hand side of the second equality is always an integer; therefore, k is well defined.

To conclude that the optimal makespan with p processors is k + δ -1 we just need to provide an explicit schedule for the last processor. This processor processes all its allocated nodes, except node d 1 1 , and nodes a 1,1 1 through a 1,1 δ-3 , in any order between the time 0 and k. Then, between time k and k + δ -2, it processes the remaining a 1,1 j nodes and then node d 1 1 . Lower bound on the peak memory usage. We first remark that, by construction, under any makespan-optimal algorithm the p-1 nodes c i j are processed during the time interval [k-j, kj + 1]. Similarly, the p -1 nodes cp i j are processed during the time interval [k + δj -1, k + δj]. Without loss of generality, we can assume that processor P i , for 1 ≤ i ≤ p -1, processes nodes c i k through c i 1 and then nodes cp i δ-1 through cp i 1 from time 0 to time k + δ -1. The processor P p processes the other nodes. The only freedom an algorithm has is in the order processor P p is processing its allocated nodes.

To establish a lower bound, we consider the class of schedules which are makespan-optimal for the studied tree, whose peak memory usage is minimum among makespan-optimal schedules, and which satisfy the additional properties we just defined. We first show that, without loss of generality, we can further restrict our study to schedules which, once one child of a node d i j has started being processed, process all the other children of that node and then the node d i j itself before processing any children of any other node d i ′ j ′ . We also establish this property by contradiction by assuming that there is no schedule (in the considered class) that satisfies the last property. Then, for any schedule B, let t(B) be the first date at which B starts the processing of a node a i,j m while some node a i ′ ,j ′ m ′ has already been processed, but node d i ′ j ′ has not been processed yet. We consider a schedule B which maximizes t(B) (note that t can only take values no greater than δ + k -1 and that the maximum and B are thus well defined). We then build from B another schedule B ′ whose peak memory is not greater and that does not overlap the processing of nodes d i j and d i ′ j ′ . This schedule is defined as follows. It is identical to B except for the time slots at which node d i j , node d i ′ j ′ or any of their children was processed. If, under B node d i j was processed before node

d i ′ j ′ (respectively, node d i ′ j ′
was processed before node d i j ), then under the new schedule, in these time slots, all the children of d i j are processed first, then d i j , then all the children of d i ′ j ′ and finally d i ′ j ′ (resp. all the children of

d i ′ j ′ are processed first, then d i ′ j ′
, then all the children of d i j and finally d i j ). The peak memory due to the processing of nodes d i j and d i ′ j ′ and of their children is now max{δj + 2, δj ′ + 3} (resp. max{δj ′ + 2, δj + 3}). In the original schedule it was no smaller than max{δj + 3, δj ′ + 3} because at least the output of the processing of one of the children of node d i ′ j ′ (resp. d i j ) was in memory while node d i j (resp. d i ′ j ′ ) was processed. Hence the peak memory of the new schedule B ′ is no greater than the one of the original schedule. The new schedule satisfies the desired property at least until time t + 1. Hence t(B ′ ) is larger than t(B), which contradicts the maximality of B with respect to the value t.

From the above, in order to establish a lower bound on the peak memory of any makespanoptimal schedule, it is sufficient to consider only those schedules that do not overlap the processing of different nodes d i j (and of their children). Let B be such a schedule. We know that, under schedule B, processor P p processes nodes without interruption from time 0 until time k + δ -2. Therefore, in the time interval [k + δ -3, k + δ -2] processor P p processes a node d i 1 , say d 1 1 . Then we have shown that we can assume without loss of generality that processor P p exactly processes the δ children of

d 1 1 in the time interval [k -3, k + δ -3]
. Therefore, at time k, processor P p has processed all nodes allocated to it except node d 1 1 and δ -3 of its children. Therefore, during the time interval [k, k + 1] the memory must contain:

1. The output of the processing of all d i j nodes, for 1 ≤ i ≤ p-1 and 1 ≤ j ≤ δ-1, except for the output of node d 1 1 (which has not yet been processed). This corresponds to (p -1)(δ -1) -1 elements. 2. The output of the processing of 3 children of node d 1 1 and a additional unit to store the result of a fourth one. This corresponds to 4 elements. 3. The result of the processing of the c i 1 nodes, for 1 ≤ i ≤ p -1 and room to store the results of the cp i δ-1 nodes, for 1 ≤ i ≤ p -1. This corresponds to 2(p -1) elements. Overall, during this time interval the memory must contain:

((p -1)(δ -1) -1) + 4 + 2(p -1) = (p -1)δ + p + 2
elements. As the optimal peak memory is δ + p -1 this gives us a lower bound on the ratio ρ of the memory used by B with respect to the optimal peak memory usage:

ρ ≥ (p -1)δ + p + 2 δ + p -1 ---→ δ→∞ p -1.
Therefore, there exists a value δ 0 such that

(p -1)δ 0 + p + 2 δ 0 + p -1 > p -1 + 1 2 ǫ.
As algorithm A cannot have a strictly lower peak memory than algorithm B by definition of B, this proves that the ratio for A is at least equal to p -1 + 1 2 ǫ, which contradicts the definition of ǫ.

Furthermore, a similar result can also be derived in the general model (with arbitrary execution times and file sizes), but without the restriction that α(p) = 1. This is done in the next lemma. Lemma 2. When scheduling in-tree task graphs in the general model on a platform with p ≥ 2 processors, there does not exist any algorithm that is both an α(p)-approximation for makespan minimization and a β(p)-approximation for peak memory minimization, with α(p)β(p) < p.

Proof. Consider the tree drawn in Figure 5. This tree can be scheduled in time C * max = 1 on p processors if all non-root nodes are processed simultaneously (by using a peak memory of p), or sequentially in time p by using only M * = 1 memory. Lemma 1 states that for any schedule with makespan C max and peak memory M , we have M C max ≥ p. This immediately implies that any algorithm with approximation ratio α(p) for makespan and β(p) for peak memory minimization must verify α(p)β(p) ≥ p. This bound is tight because, in this model, any memory-optimal sequential schedule is an approximation algorithm with β(p) = 1 and α(p) = p.

n0 = 0, w0 = 0, f0 = 0 n1 = 1, w1 = 1, f1 = 0 n2 = 1, w2 = 1, f2 = 0 . . . np = 1, wp = 1, fp = 0

Heuristics

Given the complexity of optimizing the makespan and memory at the same time, we have investigated heuristics and we propose six algorithms. The intention is that the proposed algorithms cover a range of use cases, where the optimization focus wanders between the makespan and the required memory. The first heuristic, ParSubtrees (Section 5.1), employs a memory-optimizing sequential algorithm for each of its subtrees, the different subtrees being processed in parallel. Hence, its focus is more on the memory side. In contrast, ParInnerFirst and ParSubtrees are two list scheduling based algorithms (Section 5.2), which should be stronger in the makespan objective. Nevertheless, the objective of ParInnerFirst is to approximate a postorder in parallel, which is good for memory in sequential. The focus of ParDeepestFirst is fully on the makespan. Then, we move to memory-bounded heuristics (Section 5.3). Initially, we adapt the two list-scheduling heuristics to obtain bounds on their memory consumption. Finally, we design a heuristic, MemBookingInnerFirst (Section 5.3.3), that proposes a parallel execution of a sequential postorder while satisfying a memory bound given as input.

Parallel execution of subtrees

The most natural idea to process a tree T in parallel is arguably to split it into subtrees, to process each of these subtrees with a sequentially memory-optimal algorithm [START_REF] Jacquelin | On optimal tree traversals for sparse matrix factorization[END_REF][START_REF] Joseph | An application of generalized tree pebbling to sparse matrix factorization[END_REF], and to have these sequential processings happen in parallel. The underlying idea is to assign to each processor a whole subtree in order to enable as much parallelism as there are processors, while allowing to use a single-processor memory-optimal traversal on each subtree. Algorithm 1 outlines such an algorithm, using Algorithm 2 for splitting T into subtrees. The makespan obtained using ParSubtrees is denoted by C ParSubtrees max . Algorithm 1: ParSubtrees (T , p)

1 Split tree T into q subtrees (q ≤ p) and a set of remaining nodes, using SplitSubtrees (T , p). 2 Concurrently process the q subtrees, each using a memory minimizing algorithm, e.g., [START_REF] Jacquelin | On optimal tree traversals for sparse matrix factorization[END_REF]. 3 Sequentially process the set of remaining nodes, using a memory minimizing algorithm.

In this approach, q subtrees of T , q ≤ p, are processed in parallel. Each of these subtrees is a maximal subtree of T . In other words, each of these subtrees includes all the descendants (in T ) of its root. The nodes not belonging to the q subtrees are processed sequentially. These are the nodes where the q subtrees merge, the nodes included in subtrees that were produced in excess (if more than p subtrees were created), and the ancestors of these nodes. An alternative approach, as discussed below, is to process all produced subtrees in parallel, assigning more than one subtree to each processor when q > p. The advantage of Algorithm 1 is that we can construct a splitting into subtrees that minimizes its makespan, established shortly in Lemma 3.

As w i is the computation weight of node i, W i denotes the total computation weight (i.e., sum of weights) of all nodes in the subtree rooted in i, including i. SplitSubtrees uses a node priority queue PQ in which the nodes are sorted by non-increasing W i , and ties are broken according to non-increasing w i . head(PQ) returns the first node of PQ, while popHead(PQ) also removes it. PQ[i] denotes the i-th element in the queue.

SplitSubtrees starts with the root of the entire tree and continues splitting the largest subtree (in terms of the total computation weight W) until this subtree is a leaf node (W head(PQ) = w head(PQ) ). The execution time of Step 2 of ParSubtrees is that of the largest of the q subtrees of the splitting, hence W head(PQ) for the solution found by SplitSubtrees. Splitting subtrees that are smaller than the largest leaf (W j < max i∈T w i ) cannot decrease the parallel time, but only increase the sequential time. More generally, given any splitting s of T into subtrees, the best execution time for s with ParSubtrees is achieved by choosing the p largest subtrees for the parallel Step 2. This can be easily derived, as swapping a large tree included in the sequential part with a smaller tree included in the parallel part cannot increase the total execution time. Hence, the value C ParSubtrees max (s) computed in Step 10 is the makespan that would be obtained by ParSubtrees on the splitting computed so far. At the end of algorithm SplitSubtrees (Step 12), the splitting which yields the smallest makespan is selected. Proof. The proof is by contradiction. Let S be the splitting into subtrees selected by Split-Subtrees. Assume now that there is a different splitting S opt which results in a strictly shorter processing with ParSubtrees.

Because of the termination condition of the while-loop, SplitSubtrees splits any subtree that is heavier than the heaviest leaf. Therefore, any such tree will be at one time at the head of the priority queue. Let r be the root node of a heaviest subtree in S opt . From what precedes, there always exists a step t which is the first step in SplitSubtrees where a node, say r t , of weight W r , is the head of PQ at the end of the step (r t is not necessarily equal to r, as there can be more than one subtree of weight W r ). Let S t be the solution built by SplitSubtrees at the end of step t. By definition of r, there cannot be any leaf node in the entire tree that is heavier than W r . The cost of the solution S t is equal to the execution time of the parallel processing of the min{q, p} subtrees plus the execution time of the sequential processing of the remaining nodes. Therefore C ParSubtrees max (t) = W r + Seq(t), where Seq(t) is the total weight of the sequential set seqSet(t) plus the total weight of the surplus subtrees (that is, of all the subtrees in PQ except the p subtrees of largest weights). The cost of S opt is C * max = W r + Seq(S opt ), given that r is the root of a heaviest subtree of S opt by definition.

SplitSubtrees splits subtrees by non-increasing weights. Furthermore, by definition of step t, all subtrees split by SplitSubtrees, up to step t included, were subtrees whose weights were strictly greater than W r . Therefore, because r is the weight of the heaviest subtree in S opt , all the subtrees split by SplitSubtrees up to step t included must have been split to obtain the solution S opt . This has several consequences. Firstly, seqSet(t) is a subset of seqSet(S opt ), because, for any solution S, seqSet(S) is the set of all nodes that are roots of subtrees split to obtain the solution S. Secondly, either a subtree of S t belongs to S opt or this subtree has been split to obtain S opt . Therefore, the sequential processing of the max{qp, 0} exceeding subtrees is no smaller in S opt than in the solution built at step t. It directly follows from the two above consequences that Seq(t) ≤ Seq(S opt ). However, S opt and S t have the same execution time for the parallel phase The complexity of the sequential traversal algorithms used in Steps 2 and 3 of ParSubtrees is at most O(n 2 ), e.g., [START_REF] Jacquelin | On optimal tree traversals for sparse matrix factorization[END_REF][START_REF] Joseph | An application of generalized tree pebbling to sparse matrix factorization[END_REF], or O(n log n) if the optimal postorder is sufficient. Thus the total complexity of ParSubtrees is O(n 2 ) or O(n(log n + p)), depending on the chosen sequential algorithm.

Memory

Lemma 4. ParSubtrees is a p-approximation algorithm for peak memory minimization: the peak memory, M , verifies M ≤ pM seq , where M seq is the memory required for the complete sequential execution.

Proof. We first remark that during the parallel part of ParSubtrees, the total memory used, M p , is not more than p times M seq . Indeed, each of the p processors executes a maximal subtree and the processing of any subtree does not use, obviously, more memory (if done optimally) than the processing of the whole tree. Thus, M p ≤ p • M seq .

During the sequential part of ParSubtrees, the memory used, M S , is bounded by M seq + i∈Q f i , where the second term is for the output files produced by the root nodes of the q ≤ p subtrees processed in parallel (Q is the set of the root nodes of the q trees processed in parallel). We now claim that at least two of those subtrees have a common parent. More specifically, let us denote by X the node that was split last (i.e., it was split in the step S which is selected at the end of SplitSubtrees). Our claim is that at least two children of X are processed in the parallel part. Before X was split (in step S -1), the makespan as computed in Step 10 of SplitSubtrees is C max (S -1) = W X + Seq(S -1), where Seq(S -1) is the work computed in sequential ( i∈seqSet w i + |PQ| i=PQ[p+1] W i ). Let D denote the set of children of X which are not executed in parallel, then the total weight of their subtrees is W D = i∈D W i . We now show that if at most one child of X is processed in the parallel part, X was not the node that was split last:

• If exactly one child C of X is processed in the parallel part, then C max (S) = W X ′ + Seq(S - 1) + w X + W D
, where X ′ is the new head of the queue, and thus verifies

W X ′ ≥ W C . And since W X = w X + W C + W D , we can conclude that C max (S) ≥ C max (S -1). • If no child of X is processed in the parallel part, then C max (S) = W X ′ + Seq(S -1) -W Y + w X + W D
, where X ′ is the new head of the queue and Y is the newly inserted node in the p largest subtrees in the queue. Since W X ′ ≥ W Y and W X = w X + W D , we obtain once again C max (S) ≥ C max (S -1).

In Furthermore, given that up to p processors work in parallel, each on its own subtree, it is easy to see that this bound is tight if the sequential peak memory can be reached in each subtree.

Makespan ParSubtrees delivers a p-approximation algorithm for makespan minimization, and this bound is tight. Because at least one processor is working at any time under ParSubtrees, ParSubtrees delivers, in the worst case, a p-approximation for makespan minimization. To prove that this bound is tight, we consider a tree of height 1 with p • k leaves (a fork), where all execution times are equal to 1 (∀i ∈ T , w i = 1), and where k is a large integer (this tree is depicted in Figure 6). The optimal makespan for such a tree is C * max = kp/p + 1 = k + 1 (the leaves are processed in parallels, in batches of size p, and then the root is processed). With ParSubtrees p leaves are processed in parallel, and then the remaining nodes are processed sequentially. The makespan is thus C max = (1 + pkp) + 1 = p(k -1) + 2. When k tends to +∞ the ratio between the makespans tends to p.

Given the just observed worst case for the makespan, a makespan optimization for ParSubtrees is to allocate all produced subtrees to the p processors instead of only p subtrees. This can be done by ordering the subtrees by non-increasing total weight and allocating each subtree in turn to the processor with the lowest total weight. Each of the parallel processors executes its subtrees sequentially. This optimized form of the algorithm is named ParSubtreesOptim. Note that this optimization should improve the makespan, but it will likely worsen the peak memory usage.

List scheduling algorithms

ParSubtrees is a high-level algorithm employing sequential memory-optimized algorithms. An alternative, explored in this section, is to design algorithms that directly work on the tree in parallel. We first present two such algorithms that are event-based list scheduling algorithms [START_REF] Hwang | Scheduling precedence graphs in systems with interprocessor communication times[END_REF]. One of the strong points of list scheduling algorithms is that they are (2 -1 p )-approximation algorithms for makespan minimization [START_REF] Graham | Bounds for certain multiprocessing anomalies[END_REF].

Algorithm 3 outlines a generic list scheduling, driven by node finish time events. At each event at least one node has finished so at least one processor is available for processing nodes. Each available processor is given the respective head node of the priority queue. The priority of nodes is given by the total order O, a parameter to Algorithm 3.

Heuristic ParInnerFirst

From the study of the sequential case, one knows that a postorder traversal, while not optimal for all instances, provides good results [START_REF] Jacquelin | On optimal tree traversals for sparse matrix factorization[END_REF]. Our intention is to extend the principle of postorder traversal to the parallel processing. For the first heuristic, called ParInnerFirst, the priority queue uses the following ordering O: 1) inner nodes, in an arbitrary order; 2) leaf nodes ordered according to a given postorder traversal. It makes heuristic sense that the chosen postorder is an optimal sequential postorder, so that memory consumption can be minimized. We do not further define the order of inner nodes because it has absolutely no impact. Indeed, because we target Assign node to proc 12 eventSet ← eventSet ∪ finishTime(node) the processing of tree-shaped task-graphs, the processing of a node makes at most one new inner node available, and the processing of this new inner node can start right away on the processor that freed it by completing the processing of its last un-processed child.

1 1 • • • • • • p -1 • • • • • • k -1 1 • • • • • • p -1 k 2k -2
Complexity The complexity of ParInnerFirst is that of determining the input order O and that of the list scheduling. Computing the optimal sequential postorder is O(n log n) [START_REF] Joseph | On the storage requirement in the out-of-core multifrontal method for sparse factorization[END_REF]. In the list scheduling algorithm there are O(n) events and n nodes are inserted and retrieved from PQ. An insertion into PQ costs O(log n), so the list scheduling complexity is O(n log n). Hence, the total complexity is also O(n log n).

Memory ParInnerFirst is not an approximation algorithm with respect to peak memory usage. This is derived considering the tree in Figure 7. All output files have size 1 and the execution files have size 0 (∀i ∈ T : f i = 1, n i = 0). Under an optimal sequential processing, leaves are processed in a deepest first order. The resulting optimal memory requirement is M seq = p + 1, reached when processing a join node. With p processors, all leaves have been processed at the time the first join node (k -1) can be executed. (The longest chain has length 2k -2.) At that time there are (k -1) • (p -1) + 1 files in memory. When k tends to +∞ the ratio between the memory requirements also tends to +∞.

Heuristic ParDeepestFirst

The previous heuristic, ParInnerFirst, tries to take advantage of the memory performance of optimal sequential postorders. Going in the opposite direction, another heuristic objective can be the minimization of the makespan. For trees, an inner node depends on all the nodes in the subtree it defines. Therefore, it makes heuristic sense to try to process the deepest nodes first to try to reduce any possible waiting time. For the parallel processing of a tree, the most meaningful definition of the depth of a node i is the w-weighted length of the path from i to the root of the tree, including w i (therefore, the depth of node i is equal to its top-level plus w i [START_REF] Casanova | Parallel Algorithms[END_REF]). A deepest node in the tree is a deepest node in a critical path of the tree.

ParDeepestFirst is our proposed list-scheduling deepest-first heuristic. ParDeepestFirst is defined by Algorithm 3 called with the following node ordering O: nodes are ordered according to their depths and, in case of ties, inner nodes have priority over leaf nodes, and remaining ties are broken according to an optimal sequential postorder.

Complexity

The complexity is the same as for ParInnerFirst, namely O(n log n). See ParIn-nerFirst's complexity analysis.

Memory

The memory required by ParDeepestFirst is unbounded with respect to the optimal sequential memory M seq . Consider the tree in Figure 8 with many long chains, assuming the Pebble Game model (i.e., ∀i ∈ T : f i = 1, n i = 0, w i = 1). The optimal sequential memory requirement is 3. The memory usage of ParDeepestFirst will be proportional to the number of leaves, because they are all at the same depth, the deepest one. As we can build a tree like the one of Figure 8 for any predefined number of chains, the ratio between the memory required by ParDeepestFirst and the optimal one is unbounded.

Memory bounded heuristics

From the analysis of the three algorithms presented so far we have seen that only ParSubtrees gives a guaranteed bound on the required peak memory. The memory behavior of the two other algorithms, ParInnerFirst and ParDeepestFirst, will be analyzed in the experimental evaluation presented in Section 6. In a practical setting it might be very desirable to have a strictly bounded memory consumption, so as to be certain that the algorithm can be executed with the available memory. In fact, a guaranteed upper limit might be more important than a good average behavior as the system needs to be equipped with sufficient memory for the worst case. ParSubtrees's guarantee of at most p times the optimal sequential memory seems high, and thus an obvious goal would be to have a heuristic that minimizes the makespan while keeping the peak memory usage below a given bound. In order to approach this goal, we first study how to limit the memory consumption of ParInnerFirst and ParDeepestFirst. Our study relies on some reduction property on trees and we show how to transform general trees into reduction trees in Section 5.3.1. We then develop memory bounded versions of ParInnerFirst and ParDeepestFirst (Section 5.3.2). The memory bounds achieved by these new variants are rather lax. Therefore, we design our last heuristic MemBookingInnerFirst, with stronger memory properties (Section 5.3.3). In the experimental section (Section 6), we will show that these three heuristics achieve different trade-offs between makespan and memory usage.

Simplifying tree properties

To design our memory-bounded heuristics, we make two simplifying assumptions. First, the considered trees do not have any execution files. In other words, we assume that, for any task i, n i = 0.

Eliminating execution files

To still be able to deal with general trees, we can transform any tree T with execution files into a strictly equivalent tree T ′ where all execution files have a null size. Let i be any node of T . We add to i a new leaf child i ′ whose execution time is null (w i ′ = 0), whose execution file is of null size (n i ′ = 0), and whose output file has size n i (f i ′ = n i ). Then we set n i to 0. Any schedule S for the original tree T can be easily transformed into a schedule S ′ for the new tree T ′ with the exact same memory and execution-time characteristics: S ′ schedules a node from T at the same time than S, and a node i from T ′ \ T at the same time than the father of i is scheduled by S (because i has a null execution time).

The second simplifying assumption is that all considered trees are reduction trees: Definition 2 (reduction tree). A task tree is a reduction tree if the size of the output file of any inner node i is not more than the sum of its input files:

f i ≤ j∈Children(i) f j .
(

) 1
This reduction property is very useful, because it implies that executing an inner node does not increase the amount of memory needed (this will be used for instance in Theorem 5).

For convenience, we sometimes use the following notation to denote the sum of the sizes of the input files of an inner node i:

inputs(i) = j∈Children(i) f j .
We now show how general trees can be transformed into reduction trees.

Turning trees into reduction trees We can transform any tree T that does not satisfy the reduction property stated by Equation ( 1) into a tree where each (inner) node satisfies it. Let i be any inner node of T . We add to i a new leaf child i ′ whose execution time is null (w i ′ = 0), whose execution file is of null size (n i ′ = 0), and whose output file has size:

f i ′ = max    0, f i -   j∈Children(i) f j      = max{0, f i -inputs(i)}.
The new tree is not equivalent to the original one. Let us consider an inner node i that did not satisfy the reduction property. Then, f i ′ = f iinputs(i) > 0. The memory used to execute node i in the tree T is: inputs(i) + n i + f i . In the new tree, the memory needed to execute this node is:

(inputs(i) + (f i -inputs(i)) + n i + f i > inputs(i) + n i + f i .
Any schedule of the original tree can be transformed into a schedule of the new tree with an increase of the memory usage bounded by:

p × max i {0, f i -inputs(i)}.

RR n°8606

Obviously, a more clever approach is to transform a tree first into a tree without execution files, and then to transform the new tree into a tree with the reduction property. Under this approach, the increase of the memory usage is bounded by:

p × max i {0, f i -inputs(i) -n i }.
Transforming schedules The algorithms proposed in the following subsections produce schedules for reduction trees without execution files, which might have been created from general trees which do not possess our simplifying properties. The schedule S ′ produced by an algorithm for a reduction tree without execution files T ′ can be readily transformed into a schedule S for the original tree T . To create schedule S, we simply remove all (leaf) nodes from the schedule S ′ that were introduced in the simplification transform (i ′ ∈ T ′ \ T ). Because those nodes have zero processing time (∀i ′ ∈ T ′ \ T : w i ′ = 0) there is no impact on the ordering and on the starting time of the other nodes of T . In terms of memory consumption, the peak memory for schedule S is never higher than that for schedule S ′ . A leaf i ′ that was added to eliminate an execution file might use memory earlier in S ′ than the execution file n i in S, but it is the same amount and freed at the same time. In terms of leaf nodes introduced to enforce the reduction property, they might only increase the memory needed for tree T ′ (as discussed above); hence, removing these nodes can not increase the peak memory needed for schedule S. In summary, the schedule S for tree T has the same makespan as S ′ and a peak memory that is not greater than that of S ′ .

Memory-bounded ParInnerFirst and ParDeepestFirst

Both ParInnerFirst and ParDeepestFirst are based on the list scheduling approach presented in Algorithm 3. To achieve a memory bounded version of these algorithms for reduction trees, we modify Algorithm 3 to obtain Algorithm 4. The code common to both algorithms is shown in gray in Algorithm 4 and the new code is printed in black.

We use the same event concept as previously. However, we only start processing a node if i) it is an inner node; or ii) it is a leaf node and the current memory consumption plus the leaf's output file (f c ) is less than the amount M of available memory. Once a node is encountered that fulfills neither of these conditions, the node assignment is stopped (P ← ∅) until the next event. Therefore, Algorithm 4 may deliberately keep some processors idle when there are available tasks, and thus does not necessarily produce a list schedule (hence, the name of "pseudo" list schedules). Subsequently, the only approximation guarantee on the makespan produced by these heuristics is that they are p-approximations, the worst case for heuristics that always use at least one processor at any time before the entire processing completes.

The algorithms based on this memory-bounded scheduling approach are called ParInner-FirstMemLimit and ParDeepestFirstMemLimit.

Theorem 5. The peak memory requirement of ParInnerFirstMemLimit and ParDeepest-FirstMemLimit for a reduction tree without execution files processed with a memory bound M and a node order O is at most 2M , if M ≥ M seq , where M seq is the peak memory usage of the corresponding sequential algorithm with the same node order O.

Proof. We first show that the required memory never exceeds 2M and then we show that the algorithms completely process the considered tree T .

We analyze the memory usage at the time a new candidate node c is considered for execution (line 11 of Algorithm 4). The amount of currently used memory is then M used = In in + Out in + Out lf + InIdle, where:

• In in is the size of the input files of the currently processed inner nodes;

• Out in is the size of the output files of the currently processed inner nodes;

• Out lf is the size of the output files of the currently processed leaves;

• InIdle is the size of the input files stored in memory but not currently used (because they are input files of inner nodes that are not yet ready). There are two cases, the candidate node c can be either a leaf node or an inner node: 1. c is a leaf node. The processing of a leaf node only starts if M used + f c ≤ M . Therefore, the processing of a leaf node never provokes the violation of the memory bound of M and, thus, a fortiori, of a memory limit of 2M . 2. c is an inner node. The processing of a candidate inner node always starts right away, regardless of the amount of available memory. When the processing of c starts, the amount of required memory becomes

M new = In in + Out in + Out lf + InIdle + f c .
T is by hypothesis a reduction tree. Therefore, the size of the output file f c does not exceed InIdle, that is, the size of all possible input files stored in memory right before the start of the processing of inner node c, but not used at that time, because this includes all the input files of inner node c. Also, the total size of the output files of the processed inner nodes, Out in , cannot exceed the total size of the input files of the processed inner nodes, In in . Therefore,

M new = In in +Out in +Out lf +InIdle +f c ≤ In in +Out in +Out lf +2InIdle ≤ 2In in +Out lf +2InIdle ≤ 2(In in + Out lf + InIdle).
So the new memory requirement M new is not greater than twice the memory occupied by all input files and all output files of leaf nodes. Because the tree is by hypothesis a reduction tree, executing an inner node never increases the total size of all input files and all output files of leaves. This can only happen by starting a leaf, but that is not done if it would exceed the required memory M . Therefore, In in +Out lf +InIdle never exceeds M and M new ≤ 2M . We now prove that when the algorithm ends the entire input tree has been processed. We reason by contradiction and assume that this is not the case. Ready inner nodes are processed without checking the amount of available memory. Therefore, when the algorithm terminates without having completed the processing of the tree, eventSet is empty but some leaves have not been processed. Then, let l be the first un-processed leaf, according to the order O. At the time Algorithm 4 terminates, it has processed exactly the same leaves as the sequential algorithm when it starts processing leaf l. Because eventSet is empty, there are no remaining ready inner nodes and no node is processed at the time of the algorithm termination. Because of the hypothesis that T is a reduction tree, the amount of available memory when Algorithm 4 terminates is not smaller than the amount of available memory under the sequential algorithm right before it starts processing leaf l. Because the sequential algorithm can process the whole tree with a peak memory usage of M seq ≤ M , the processing of leaf l can be started by Algorithm 4. This contradicts the assumption of early termination.

We define a variant ParDeepestFirstMemLimitOptim of ParDeepestFirstMemLimit, and a variant ParInnerFirstMemLimitOptim of ParInnerFirstMemLimit, by being more aggressive about starting leaves. Instead of checking for the condition M used + f c ≤ M before starting a leaf node c (line 12 of Algorithm 4), it is in fact sufficient to check that In in + 1 2 Out lf + InIdle + f c ≤ M (using the notation of the proof of Theorem 5). For Case (1) of the proof, one just needs to remark that after leaf c is started,

M new = In in + Out in + Out lf + InIdle + f c . Then, because the tree is a reduction tree, Out in ≤ In in . Therefore, M new ≤ 2In in + Out lf + f c + InIdle ≤ 2In in + Out lf + f c + 2InIdle,
which, in turn, is no greater than 2M because of the new condition. The modified condition has no impact on the study of Case (2), because the inequality

In in + 1 2 Out lf + InIdle ≤ M is sufficient to conclude that case.
Memory Theorem 5 establishes that the peak memory required by ParInnerFirstMemLimit and ParDeepestFirstMemLimit with p processors is at most twice that of their sequential execution (p = 1) with the same order. It should be noted that this peak requirement M seq does not correspond in general to the memory requirement of an optimal sequential algorithm. In particular, the sequential execution of ParInnerFirstMemLimit corresponds to a postorder traversal, which is not optimal for all instances, but generally provides good results [START_REF] Jacquelin | On optimal tree traversals for sparse matrix factorization[END_REF]. We propose to use ParInnerFirstMemLimit with a node order O that corresponds to an optimal sequential postorder, e.g., with [START_REF] Joseph | On the storage requirement in the out-of-core multifrontal method for sparse factorization[END_REF]. The memory requirement of the sequential ParDeepestFirstMem-Limit is unbounded compared to the optimal sequential memory requirement, because the same arguments apply as the ones discussed for ParDeepestFirst in Section 5.2.2.

Makespan

We have already stated that the above heuristics are p-approximation algorithms for makespan minimization. The following lemma refines this result:

Lemma 5. ParInnerFirstMemLimit and ParInnerFirstMemLimitOptim are both p-approximation algorithms for makespan minimization and this bound is tight.

Proof. We establish this result by studying the tree in Figure 9. This tree can be processed with a peak memory usage of M . We assume that ParInnerFirstMemLimit is called with this memory limit. The key observation is that in any schedule, among the three descendants of an a i node, the nodes c i and b i must be processed before the d i node: otherwise, keeping in memory the output file of size M/p of node d i makes it impossible to start processing the leaf node c i because of its output file of size M . And since under ParInnerFirstMemLimit, leaf nodes are processed according to a postorder traversal, the processing of the subtrees is sequentialized and the overall processing takes time p(2 + k). On the other hand, with respect to the makespan, it would be better to first sequentially process in that order c 1 , b 1 , c 2 , b 2 , ..., c p , and b p , which would take a time 2p, and then process in parallel the d i 's for an overall makespan of 2p + k. Hence, on this example, the approximation ratio of ParInnerFirstMemLimit is no smaller than p(2+k) 2p+k which tends to p when k tends to infinity.

We do not have a similar result for the memory limited deepest first algorithms as already the sequential traversal with these algorithms (which determines the given memory limit) can require significantly more memory than a postorder traversal. For the example in Figure 9, the minimum sequential memory for a deepest first traversal is equal to pM . The additional memory buys a lot of freedom for ParDeepestFirstMemLimit and makes the comparison harder.

Memory booking heuristic MemBookingInnerFirst

The two heuristics described in the previous section satisfy an achievable memory bound, M , in a relaxed way: the guarantee is that they never use more than twice the memory limit. Here, we root (0)

a 1 (0) b 1 (1) c 1 (1) M 0 d 1 (k) M/p 0 a 2 (0) b 2 (1) c 2 (1) M 0 d 2 (k) M/p 0 . . . a p (0) b p (1) c p (1) M 0 d p (k)
M/p 0 Figure 9: Tree used to establish the worst-case performance of most memory bounded heuristics. Node labels in parentheses are processing times, edge labels are memory weights. All nodes have null-size processing files. aim at designing a heuristic that satisfies an achievable memory bound M in the strong sense: the heuristic never uses more than a memory of M .

To achieve such a goal, we want to ensure that whenever an inner node i becomes ready there is enough memory to process it. Therefore, we book in advance some memory for its later processing. Our goal is to book as little memory as possible, and to do so as late as possible. The algorithm then relies on a sequential postorder schedule, denoted PO: for any node k in the task graph, PO(k) denotes the step at which node k is executed under PO. Let j be the last child of i to be processed. If the total size of the input files of j is larger than (or equal to) f i , then only that last child will book some memory for node i. In this case (part of) the memory that was used to store the input files of j will be used for f i . If the total size of the input files of j is smaller than f i , then the second to last child of i will also have to book some memory for f i , and so on. The following recursive formula states the amount of memory Contrib[j] a child j has to book for its parent i:

Contrib[j] = min      inputs(j), f i - j ′ ∈Children(i) PO(j ′ )>PO(j) Contrib[j ′ ]     
If j is a leaf, it may also have to book some memory for its parent. However, the behavior for leaves is quite different than for inner nodes. A leaf node cannot transfer some of the memory used for its input files (because it does not have any) to its parent for its parent output file. Therefore, the memory booked by a leaf node may not be available at the time of the booking. However, this memory will eventually become available (after some inner nodes are processed); booking the memory prevents the algorithm from starting the next leaf if it would use too much memory: this ensures that the algorithm completes the processing without violating the memory bound. The contribution of a leaf j for its parent i is:

Contrib[j] = f i - j ′ ∈Children(i) PO(j ′ )>PO(j) Contrib[j ′ ]
Note that the value of Contrib for each node can be computed before starting the algorithm, in a simple tree traversal. Using these formulas, we are able to guarantee that enough memory is booked for each inner node i:

j∈Children(i) Contrib[j] = f i .
Using these definitions, we design a new heuristic, MemBookingInnerFirst, which is described in Algorithm 5. In this algorithm, Booked [i] denotes the amount of memory currently booked for Input: tree T , number of processor p, postorder PO, memory limit M (not smaller than the peak memory of the sequential traversal defined by PO) 1 foreach node i do Booked [i] ← 0 2 M used ← 0 3 while the whole tree is not processed do the processing of an inner node i. We make use of a new notation: we denote by Ancestors(i) the set of nodes on the path from i to the root node (excluding i itself), that is, all ancestors of i. Note that, contrarily to ParInnerFirstMemLimit, MemBookingInnerFirst does not guarantee that there is always enough memory available to process an inner node i as soon as it becomes ready. This is why Lemma 6 only guarantees that an inner node i will eventually be processed if a leaf j with PO(j) > PO(i) is started by MemBookingInnerFirst. Lemma 6. Consider any inner node i. If some leaf j with PO(j) > PO(i) has been started by Algorithm 5, then at some point, there will be enough memory to process i.

Proof. By contradiction, assume that an available inner node i can never be processed because of memory constraints, that is, Algorithm 5 stops without processing i, and some leaf j with PO(j) > PO(i) has been started (in case of several such leaves, we consider the one with largest PO(j)). Note that i cannot be a parent of j (otherwise we would have PO(i) > PO(j)). We consider the amount A = M -M usedk / ∈Ancestors(j) Booked [k] and its evolution. Before starting j, we check that A ≥ f j . When starting j, the amount of available memory is decreased by f j and, thus, we have A ≥ 0. The following events may happen after the beginning of j:

• Some inner node u not in Ancestors(j) is terminated. Let us call v its parent. When u completes, M used decreases by inputs(u), while Booked [v] increases by Contrib[u] ≤ inputs(u). Thus, A does not decrease. • Some inner node k not in Ancestors(j) is started. In that case, the booked memory f k is traded for used memory, and the total memory amount A is preserved. • An inner node u in Ancestors(j) is started. In this case, the amount of available memory may temporarily decrease. However, because of the reduction property, the amount of memory freed when u completes is not smaller than the amount of additional memory temporarily used for the processing of u.

• A leaf node has completed: this modifies neither the amount of available or booked memory and, so, A is left unchanged. Therefore, when the algorithm stops with i available, A ≥ 0. Thus, M -M used ≥ Booked [i] = f i : there is enough memory to process i.

Using the previous lemma, we now prove Algorithm MemBookingInnerFirst. Theorem 6. MemBookingInnerFirst called with a postorder PO and a memory bound M processes the whole tree with memory M if M is not smaller than the peak memory of the sequential traversal defined by PO.

Proof. By assume that the algorithm stops while some nodes are unprocessed. We consider two cases:

• There is at least one available unprocessed inner node i (if there are several, we choose the one with the smallest PO value). Consider the step PO(i) when this node i is processed in the sequential postorder schedule. At this time, the set S of the leaves processed by the sequential postorder is exactly the set of the leaves j such that PO(j) < PO(i). Thanks to Lemma 6, we know that MemBookingInnerFirst has not processed any leaf j with PO(j) > PO(i). Therefore, the set of the leaves processed by MemBookingInnerFirst is a subset of S. Node i being available, MemBookingInnerFirst has processed all the leaves in the subtree ST rooted at i. MemBookingInnerFirst cannot start a leaf k if a leaf j with PO(j) < PO(k) has not been started. Therefore all leaves that precedes in the postorder the leaves of ST have also been processed by MemBookingInnerFirst. By definition of a postorder, there is no leaf that does not belong to ST that is scheduled after the first of the leaf of ST and before i. Therefore, MemBookingInnerFirst has processed the exact same set of leaves than the sequential postorder at step PO(i). We now prove that the same set of inner nodes have been processed by both algorithms:

-Assume that an inner node k has been processed by MemBookingInnerFirst but not by the sequential postorder at time PO(i). Since k has not yet been processed by the sequential postorder, PO(k) > PO(i). Since no leaf j with PO(j) > PO(i) has been processed by MemBookingInnerFirst, since PO(k) > PO(i), and since PO is a postorder, then k can only be a parent of i, which contradicts the fact that i is not processed.

-Assume that an inner node k has been processed by the sequential postorder at time PO(i) but not by MemBookingInnerFirst. Since it has been processed before i in the sequential postorder, PO(k) < PO(i). This node, or one of its inner node predecessor, must be available in MemBookingInnerFirst (note that it cannot be a leaf, since all leaves with PO values smaller than PO(i) are already processed). This contradicts the fact that i is the available inner node with smallest PO value.

Thus, there is no difference in the state of the sequential postorder when it starts i and MemBookingInnerFirst when it stops, including in the amount of available memory. This contradicts the fact that i cannot be started because of memory issues.

• There is no unprocessed available inner node. Thus, some leaf is available and cannot be processed. Let j be the first of these leaves according to PO. None of the inner nodes for which some memory has been booked is available and, thus, they are all parents of j (because PO is a postorder and because the processing of all the leaves that precede j in PO has been completed). Thus, the memory condition which prevents j to be executed can be rewritten: M -M used < f j . However, since no inner node is available, this is the same situation as right before j is processed in the sequential postorder, which contradicts the fact that j can be processed in the sequential postorder.

Lemma 7. MemBookingInnerFirst is a p-approximation algorithm for makespan minimization, and this bound is tight.

This result is proved following the exact same arguments than for the bound on the performance of ParInnerFirstMemLimit, including the tree on Figure 9.

Complexity Algorithm 5 can be implemented with the same complexity as the other heuristics, namely O(n log(n)) (which comes from the management of the P Q queue). The only operations added to this algorithm which could increase this complexity is the test executed on line 17 to make sure that a new leaf can be started, that is, the computation of k / ∈Ancestors(j) Booked [k] for each leaf might take O(n 2 ) time if not done carefully. However, it is possible to avoid recomputing the values too many times. We first remark the following property: when leaf j has not been started, k / ∈Ancestors(j)

Booked [k] = PO(k)<PO(j) Booked [k].
Indeed, if leaf j has not been started, the postorder property ensures that any k / ∈ Ancestors(j) with PO(k) ≥ PO(j) has Booked [k] = 0, because none of its children have started their execution.

For an efficient implementation, we keep a record of R = PO(k)<PO(j) Booked [k] for the leaf j which was tested on the last execution of line 17. To keep this record, it is enough to • decrease R by Booked [i] each time an inner node i with PO(i) < PO(j) begins execution,

• increase R by Contrib[i] each time an inner node i with PO(i) < PO(j) finishes,

• and increase R by PO(j)≤PO(k)<PO(j ′ ) Booked [k] if a new leaf j ′ is being considered in the test of line 17.

In total, the number of updates to R over the course of the whole algorithm is bounded by 2n: each Contrib value is added at most once to R, and each Booked value is subtracted at most once. Furthermore, the cost of computing the sums PO(j)≤PO(k)<PO(j ′ ) Booked [k] is also bounded by n since each node is considered only once. Hence, these updates do not increase the total complexity of O(n log(n)) of the whole algorithm.

Experimental validation

In this section, we experimentally compare the heuristics proposed in the previous section, and we compare their performance to lower bounds.

Setup

All heuristics have been implemented in C. Special care has been devoted to the implementation to avoid complexity issues. Especially, priority queues have been implemented using binary heap to allow for O(log n) insertion and minimum extraction. We have also implemented Liu's algorithm [START_REF] Joseph | An application of generalized tree pebbling to sparse matrix factorization[END_REF] to obtain the minimum sequential memory peak, which is used as a lower bound on memory for comparing the heuristics.

Data set

The data set contains assembly trees of a set of sparse matrices obtained from the University of Florida Sparse Matrix Collection (http://www.cise.ufl.edu/research/sparse/matrices/). The chosen matrices satisfy the following assertions: not binary, not corresponding to a graph, square, having a symmetric pattern, a number of rows between 20,000 and 2,000,000, a number of nonzeros per row at least equal to 2.5, and a number of nonzeros at most equal to 5,000,000; and each chosen matrix has the largest number of nonzeros among the matrices in its group satisfying the previous assertions. With these criteria we automatically select a set of medium to large matrices from different application domains with nontrivial number of nonzeros. At the time of testing there were 76 matrices satisfying these properties. We first order the matrices using MeTiS [START_REF] Karypis | MeTiS: A Software Package for Partitioning Unstructured Graphs, Partitioning Meshes, and Computing Fill-Reducing Orderings of Sparse Matrices Version 4[END_REF] (through the MeshPart toolbox [START_REF] Gilbert | Geometric mesh partitioning: Implementation and experiments[END_REF]) and amd (available in Matlab), and then build the corresponding elimination trees using the symbfact routine of Matlab. We also perform a relaxed node amalgamation [START_REF] Liu | The multifrontal method for sparse matrix solution: Theory and practice[END_REF] on these elimination trees to create assembly trees. We have created a large set of instances by allowing 1, 2, 4, and 16 (if more than 1.6 × 10 5 nodes) relaxed amalgamations per node.

At the end we compute memory weights and processing times to accurately simulate the matrix factorization: we compute the memory weight n i of a node as η 2 +2η(µ-1), where η is the number of nodes amalgamated, and µ is the number of nonzeros in the column of the Cholesky factor of the matrix which is associated with the highest node (in the starting elimination tree); the processing time w i of a node is defined as 2/3η 3 + η 2 (µ -1) + η(µ -1) 2 (these terms corresponds to one gaussian elimination, two multiplications of a triangular η × η matrix with a η × (µ -1) matrix, and one multiplication of a (µ -1) × η matrix with a η × (µ -1) matrix). The memory weights f i of edges are computed as (µ -1) 2 .

The resulting 608 trees contains from 2,000 to 1,000,000 nodes. Their depth ranges from 12 to 70,000 and their maximum degree ranges from 2 to 175,000. Each heuristic is tested on each tree using p = 2, 4, 8, 16, and 32 processors. Then the memory and makespan of the resulting schedules are evaluated by simulating a parallel execution. The comparison of the first set of heuristics (without memory bounds) is summarized in Table 1. It presents the fraction of the cases where each heuristic reaches the best memory (respectively makespan) among all heuristics, or when its memory (resp. makespan) is within 5% of the best one. It also shows the average normalized memory and makespan. For each scenario (consisting in a tree and a number of processors), the memory obtained by each heuristic is normalized by the optimal (sequential) memory, and the makespan is normalized using a classical lower bound, since makespan minimization is NP-hard even without memory constraint. The lower bound is the maximum between the total processing time of the tree divided by the number of processors, and the maximum weighted critical path.

Results for heuristics without memory bound

Table 1 shows that ParSubtrees and ParSubtreesOptim are the best heuristics for memory minimization. On average they use less than 2.5 times the amount of memory required by the optimal sequential traversal, when ParInnerFirst and ParDeepestFirst respectively need 3.79 and 4.13 times this amount of memory. ParInnerFirst and ParDeepestFirst perform best for makespan minimization, having makespans very close on average to the best achieved ones, which is consistent with their 2-approximation ratio for makespan minimization. Furthermore, given the critical-path-oriented node ordering, we can expect that ParDeepestFirst makespan is close to optimal. ParDeepestFirst outperforms ParInnerFirst for makespan minimization, at the cost of a noticeable increase in memory. ParSubtrees and ParSubtreesOptim may be better trade-offs, since they use (on average) almost only half the memory of ParDeepestFirst for at most a 35% increase in makespan. Figure 10 presents the evolution of the performance of these heuristics with the number of processors. On this figure, we plot the results for all 608 trees except 76 of them, for which the results are so different that it does not make sense to compute average values anymore. These outliers belong to four different classes of applications, and the specific results for these graphs are shown on Figure 11. Figure 10 shows that ParDeepestFirst and ParInnerFirst have a similar performance evolution, just like ParSubtrees and ParSubtreesOptim. The performance gap between these two groups, both for memory and makespan, increases with the number of processors. With a large number of processors, ParDeepestFirst and ParInnerFirst are able to decrease the normalized makespan (at the cost of an increase of memory), while ParSubtrees has a an almost constant normalized makespan with the number of processor.

Despite the very different values for makespan and memory utilization, and a much higher variability, the results for the outliers give the same conclusions about the relative performance of the heuristics. Furthermore, this graph also exhibits the absence of approximation ratios for ParDeepestFirst and ParInnerFirst for memory minmization. Indeed, even though the trees used in this set are taken from real-life applications, in contrast with the carefully crafted counterexamples of Section 5, the memory usage of ParDeepestFirst and ParInnerFirst on those trees can reach up to 100 times the optimal memory usage.

Results for memory-bounded heuristics

In addition to the previous heuristics, we also test the memory-bounded heuristics. Since they can be applied only to reduction trees with null processing sizes, we transform the trees used in the previous tests into reduction trees as explained in Section 5.3.1. For a given scenario (tree, number of processors, memory bound), the memory obtain by each heuristic is normalized by the optimal memory on the original tree (not the reduction one). Thus, the normalized memory represents the actual memory used by the heuristic compared to the one of a sequential processing. In particular, this allows a fair comparison between memory-bounded heuristics and the previous unbounded heuristics.

In order to compare the memory bounded heuristics, we have applied them on the previous data-set, using various memory bounds. For each tree, we first compute the minimum sequential memory M seq obtained by a postorder traversal of the original tree. Then, each heuristic is tested on the corresponding tree with a memory bound B = xM seq for various ratios x ≥ 1. Sometimes, the heuristic cannot run because the amount of available memory is too small. This is explained by the following factors:

• The memory-bounded heuristics use a reduction tree which may well need more memory than the original tree. In general, however, the transformation from original tree does not significantly increase the minimum amount of memory needed to process the tree. • ParInnerFirstMemLimit has a minimum memory guarantee which is twice the sequential memory of a postorder traversal, thus it cannot run with a memory smaller than 2M seq . • ParDeepestFirstMemLimit has a minimum memory guarantee which is twice the sequential memory of a deepest first sequential traversal of the tree. A deepest first traversal uses much more memory than a postorder traversal, and thus ParDeepestFirstMemLimit needs much more memory than ParInnerFirstMemLimit to process a tree. Figure 12 presents the results of these simulations. On this figure, points are shown only when a heuristic succeeds in more than 95% of the cases. The intuition is that with a success rate larger than 95%, the heuristic is presumably useful for this ratio. This figure shows that when the memory is very limited (B < 2M seq ), MemBookingInnerFirst is the only heuristic that can be run, and it achieves reasonable makespans. For a less strict memory bound (2M seq ≤ B < 5M seq or 2M seq ≤ B < 10M seq depending on the number of processors), ParInnerFirstMemLimit is able to process the tree, and achieves better makespans, especially when B is large. Finally, when memory is abundant, ParDeepestFirstMemLimit is the best among all heuristics. On this figure, we also tested the two variants ParInnerFirstMemLimitOptim and ParDeepest-FirstMemLimitOptim presented in Section 5.3.2 that are more aggressive when starting leaves, but with the same memory guarantee as ParInnerFirstMemLimit and ParDeepestFirst-

RR n°8606

MemLimit. We see that these strategies are able to better reduce the makespan in the case of a very limited memory (B close to 2M seq ). Finally, Figure 13 shows the ability of memory bounded heuristics to make use of the limited amount of available memory. On this figure, points corresponding to ParDeepestFirst-MemLimit (respectively ParDeepestFirstMemLimitOptim) are hardly distinguishable from ParInnerFirstMemLimit (resp. ParInnerFirstMemLimitOptim). We notice that Mem-BookingInnerFirst is able to fully use the very limited amount of memory when B is close to M seq . The good use of memory is directly correlated with good makespan performance: for a given amount of bounded memory, heuristics giving best makespans are the ones that uses the largest fraction of available memory. Especially, we can see that ParInnerFirstMemLimitOptim and ParDeepestFirstMemLimitOptim are able to use much more memory than their non-optimized counterpart, especially when memory is very limited.

Conclusion

In this study we have investigated the scheduling of tree-shaped task graphs onto multiple processors under a given memory limit and with the objective to minimize the makespan. We started by showing that the parallel version of the pebble game on trees is NP-complete, hence stressing the negative impact of the memory constraints on the complexity of the problem. More importantly, we have proved that there does not exist any algorithm that is simultaneously a constant-ratio approximation algorithm for both makespan minimization and peak memory usage minimization when scheduling tree-shaped task graphs. We have also established bounds on the achievable approximation ratios for makespan and memory when the number of processors is fixed. Based on these complexity results, we have then designed a series of practical heuristics; some of these heuristics are guaranteed to keep the memory under a given memory limit. Finally, we have assessed the performance of our heuristics using real task graphs arising from sparse matrices computation. These simulations demonstrated that the different heuristics achieve different tradeoffs between the minimization of peak memory usage and makespan; hence, the set of designed heuristics provide an efficient solution for each situation.

This work represents an important step towards a comprehensive theoretical analysis of memory/makespan minimization for applications organized as trees of tasks, as it provides both complexity results and memory bounded heuristics. Multifrontal sparse matrix factorization is an important application for this work, and a good incentive to refine the computation model. In a second step, we should consider trees of parallel tasks rather than of pure sequential tasks, as the computations corresponding to large tasks (at the top of the tree) are usually distributed across processors. Of course, one would need a proper computation model to derive relevant complexity results. To get even closer to reality, one would also need to consider distributed memory rather than shared memory, or a mix of both. Hence, many important but challenging findings remain to be done.

Figure 2 :

 2 Figure 2: Tree used for establishing Theorem 2.

Figure 3 :

 3 Figure 3: Tree used to establish Theorem 3 for p = 13 processors.

Figure 4 :

 4 Figure 4: Tree used for establishing Theorem 4.

Figure 5 :

 5 Figure 5: Tree used for the proof of Lemma 2.

Algorithm 2 : 8 seqSet ← seqSet ∪ node 9 10 C 11 s ← s + 1 12

 28910111 SplitSubtrees (T , p)1 foreach node i do compute W i (the total processing time of the tree rooted at i) 2 Initialize priority queue PQ with the tree root3 seqSet ← ∅ 4 Cost(0) = W root 5 s ← 1/* splitting rank */ 6 while W head(PQ) > w head(PQ) do 7 node ← popHead(PQ) Insert all children of node into priority queue PQ ParSubtrees max (s) = W head(PQ) + i∈seqSet w i + |PQ| i=PQ[p+1] W i Select splitting S with C ParSubtrees max (S) = min s-1 t=0 C ParSubtrees max (t) (break ties in favor of smaller t) Lemma 3. SplitSubtrees returns a splitting of T into subtrees that results in the makespanoptimal processing of T with ParSubtrees.

  W r . It follows that C ParSubtrees max (t) ≤ C * max , which is a contradiction to S opt 's shorter processing time. Complexity We first analyze the complexity of SplitSubtrees. Computing the weights W i costs O(n). Each insertion into PQ costs O(log n) and calculating C ParSubtrees max (s) in each step costs O(p). Given that there are O(n) steps, SplitSubtrees's complexity is O(n(log n + p)).

Algorithm 3 :1 4 popHead(eventSet) 5 Done ← set of the new ready nodes 6 7 P ← available processors 8 while P = ∅ and PQ = ∅ do 9 proc ← popHead (P) 10 node

 345678910 List scheduling(T , p, O) Insert leaves in priority queue PQ according to order O 2 eventSet ← {0} /* ascending order */ 3 while eventSet = ∅ do /* event:node finishes */ Insert nodes from Done in PQ according to order O /* available parents of nodes completed at event */ ← popHead (PQ) 11

Figure 7 :

 7 Figure 7: No memory bound for ParInnerFirst.

Figure 8 :

 8 Figure 8: Tree with long chains.

Algorithm 4 :1 5 popHead(eventSet) 6 Done ← set of the new ready nodes 7 8 P ← available processors 9 M 13 M 16 Assign node to proc 17 eventSet

 456789131617 Pseudo list scheduling with memory limit (T , p, O, M ) Insert leaves in priority queue PQ according to order O 2 eventSet ← {0} /* ascending order */ 3 M used ← 0 /* amount of memory used */ 4 while eventSet = ∅ do /* event:node finishes */ Insert nodes from Done in PQ according to order O /* available parents of nodes completed at event */ used ← M used -j∈Done inputs(j) 10 while P = ∅ and PQ = ∅ do 11 c ← head(PQ) 12 if |Children(c)| > 0 or M used + f c ≤ M then used ← M used + f c 14 proc ← popHead(P) 15 node ← popHead(PQ) ← eventSet ∪ f inishT ime(node)

Algorithm 5 :

 5 MemBookingInnerFirst (T , p, PO, M )

4 6 M 7 Booked 8 Done ← set of the new ready nodes 9 15 Booked [j] ← 0 16 Make P u process j 17 else 18 M 20 Make P u process j 21 else 22 push

 678915161718202122 Wait for an event (task finish time or starting point of the algorithm)5 foreach finished non-leaf node j with parent i do used ← M usedinputs(j) [i] ← Booked [i] + Contrib[j]Insert nodes from Done in PQ according to order PO 10 WaitForNextTermination ← false 11 while WaitForNextTermination = false and there is an available processor P u and PQ is not empty do 12 j ← pop(PQ)13 if j is an inner node and M used + f j ≤ M then 14 M used ← M used + f j if j is a leaf and M used + f j + k / ∈Ancestors(j) Booked [k] ≤ M then used ← M used + f j19Booked [parent of j] ← Booked [parent of j] + Contrib[j]

Figure 10 :Figure 11 :

 1011 Figure 10: Performance (makespan and memory) to the respective lower bounds for the first set of heuristics. Vertical bars represents confidence intervals when we exclude trees with extreme performance.

Figure 12 :

 12 Figure 12: Memory and makespan performance of memory-bounded heuristics (logarithmic scale).

Figure 13 :

 13 Figure 13: Real use of limited memory for memory-bounded heuristics.

  ParSubtrees is at best a p-approximation for the makespan. the memory used during the sequential part is bounded byM S ≤ M seq +f C1 +f C2 + i∈Q\{C1,C2} f i .Since a sequential execution must process node X, we obtain f C1 + f C2 ≤ M seq . And since ∀i, f i ≤ M seq , we can bound the memory used during the sequential part by M S ≤ 2M seq + (p -2)M seq ≤ pM seq .

			root
	1	2	...	p • k -1	p • k
	Figure 6:

both cases we have C max (S) ≥ C max (S -1), which contradicts the definition of X (the select phase, Step 12 of SplitSubtrees, would have selected step S -1 rather than step S). Let us now denote by C 1 and C 2 two children of X which are processed in the parallel phase. Remember that

Table 1 :

 1 Proportions of scenarii when heuristics reach best (or close to best) performance, and average deviations from optimal memory and best achieved makespan.

	Heuristic	Best memory	Within 5% of Normalized Best makespan best memory memory	Within 5% of Normalized best makespan makespan
	ParSubtrees	81.1 %	85.2 %	2.34	0.2 %	14.2 %	1.40
	ParSubtreesOptim	49.9 %	65.6 %	2.46	1.1 %	19.1 %	1.33
	ParInnerFirst	19.1 %	26.2 %	3.79	37.2 %	82.4 %	1.07
	ParDeepestFirst	3.0 %	9.6 %	4.13	95.7 %	99.9 %	1.04

The concept of file is used here in a very general meaning and does not necessarily correspond to a classical file on a disk. Essentially, a file is a set of data.RR n°8606

RR n°8606

Acknowledgement

We thank Bora Uçar for his help in creating and managing the data set used in the experiments. We gratefully acknowledge that this work is partially supported by the Marsden Fund Council from Government funding, Grant 9073-3624767, administered by the Royal Society of New Zealand, and by the ANR RESCUE and SOLHAR projects, funded by the French National Research Agency.

RESEARCH CENTRE GRENOBLE -RHÔNE-ALPES