On the storage requirement in the out-of-core multifrontal method for sparse factorization

Liu, Joseph W. H.

doi:10.1145/7921.11325

Cited by 59 publications

(51 citation statements)

References 6 publications

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“…The trees arising in this context are in-trees (as said before, there is no difference between in-trees and out-trees). Liu [34] discusses how to find the best traversal for the MINMEMORY problem when the traversal is required to correspond to a postorder traversal of the tree. In the follow-up study [35], an exact algorithm is proposed to solve the MINMEMORY problem, without the postorder constraint on the traversal.…”

Section: Case Study: Memory-aware Schedulingmentioning

confidence: 99%

“…Liu [34] has characterized the best postorder traversal, leading to a fast but sub-optimal solution for MINMEMORY. In a nutshell, the best postorder is obtained by guaranteeing that in the resulting order, the children of a node are listed in the increasing order of the memory requirement of their respective subtrees.…”

Section: Postorder Traversalsmentioning

confidence: 99%

“…Section 0.5 is mainly based on [30]. Liu proposes two optimal algorithms for optimal memory traversal of trees, the first one restricted to postorder traversals [34], the second one for any traversal [35]. For more information on the pebble game model such as complexity results for general DAGs, see the seminal work of Sethi [48] and Gilbert, Lengauer and Tarjan [25]).…”

Section: Further Informationmentioning

confidence: 99%

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Scheduling for Large-Scale Systems

Vivien¹

2014

Computing Handbook, Third Edition

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In this chapter, scheduling is defined as the activity that consists in mapping a task graph onto a target platform. The task graph represents the application, nodes denote computational tasks, and edges model precedence constraints between tasks. For each task, an assignment (choose the processor that will execute the task) and a schedule (decide when to start the execution) are determined. The goal is to obtain an efficient execution of the application, which translates into optimizing some objective function.The traditional objective function in the scheduling literature is the minimization of the total execution time, or makespan. Section 0.2 provides a quick background on task graph scheduling, mostly to recall some definitions, notations, and well-known results.In contrast to pure makespan minimization, this chapter aims at discussing scheduling problems that arise at large scale, and involve one or several different optimization criteria, that depart from, or complement, the classical makespan objective. Section 0.3 assesses the importance of key ingredients of modern scheduling techniques: (i) multi-criteria optimization; (ii) memory management; and (iii) reliability. These concepts are illustrated in the following sections, which cover four case studies. Section 0.4 is devoted to illustrate multi-criteria optimization techniques, that mix throughput, energy and reliability. Sections 0.5 and 0.6 illustrate different memory management techniques to schedule large task graphs. Finally, Section 0.7 discusses the impact of checkpointing techniques to schedule parallel jobs at very large-scale. We conclude this chapter by stating some research directions in Section 0.8, by defining terms in Section 0.9, and by pointing to additional sources of information in Section 0.10. Background on Scheduling 0.2.1 Makespan MinimizationTraditional scheduling assumes that the target platform is a set of p identical processors, and that no communication cost is paid. In that context, a task graph is a directed acyclic vertex-weighted graph G = (V, E, w), where the set V of vertices represents the tasks, the set E of edges represents precedence constraints between tasks (e = (u, v) ∈ E if and only if u ≺ v). and the weight function w : V −→ N * gives the weight (or duration) of each task. Task weights are assumed to be positive integers. A schedule σ of a task graph is a function that assigns a start time to each task: σ : V −→ N * such that σ(u) + w(u) ≤ σ(v) whenever e = (u, v) ∈ E. In other words, a schedule preserves the dependence constraints induced by the precedence relation ≺ and embodied by the edges of the dependence graph; if u ≺ v, then the execution of u begins at time σ(u) and requires w(u) units of time, and the execution of v at time σ(v) must start after the end 1

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Section: Case Study: Memory-aware Schedulingmentioning

confidence: 99%

Section: Postorder Traversalsmentioning

confidence: 99%

Section: Further Informationmentioning

confidence: 99%

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Scheduling for Large-Scale Systems

Vivien¹

2014

Computing Handbook, Third Edition

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“…The methods used to achieve this goal are described in [65,85,86]. The survey by George and Liu [88] lists, inter alia, the following improvements and algorithmic follow-ups: mass eliminations [90], where it is shown that, in case of finite-element problems, after a minimum degree vertex is eliminated a subset of adjacent vertices can be eliminated next, together at the same time; indistinguishable nodes [87], where it is shown that two adjacent nodes having the same adjacency can be merged and treated as one; incomplete degree update [75], where it is shown that if the adjacency set of a vertex becomes a subset of the adjacency set of another one, then the degree of the first vertex does not need to be updated before the second one has been eliminated; element absorption [66], where based on a compact representation of elimination graphs, redundant structures (cliques being subsets of other cliques) are detected and removed; multiple elimination [134], where it was shown that once a vertex v is eliminated, if there is a vertex with the same degree that is not adjacent to the eliminated vertex, then that vertex can be eliminated before updating the degree of the vertices in adj (v), that is the degree updates can be postponed; external degree [134], where instead of the true degree of a vertex, the number of adjacent and indistinguishable nodes is used as a selection criteria. Some further improvements include the use of compressed graphs [11], where the indistinguishable nodes are detected even before the elimination process and the graph is reduced, and the extensions of the concept of the external degree [44,94].…”

Section: Labelling or Orderingmentioning

confidence: 99%

Combinatorial Problems in Solving Linear Systems

Duff¹,

Uçar²

2012

Combinatorial Scientific Computing

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Abstract. Numerical linear algebra and combinatorial optimization are vast subjects; as is their interaction. In virtually all cases there should be a notion of sparsity for a combinatorial problem to arise. Sparse matrices therefore form the basis of the interaction of these two seemingly disparate subjects. As the core of many of today's numerical linear algebra computations consists of the solution of sparse linear system by direct or iterative methods, we survey some combinatorial problems, ideas, and algorithms relating to these computations. On the direct methods side, we discuss issues such as matrix ordering; bipartite matching and matrix scaling for better pivoting; task assignment and scheduling for parallel multifrontal solvers. On the iterative method side, we discuss preconditioning techniques including incomplete factorization preconditioners, support graph preconditioners, and algebraic multigrid. In a separate part, we discuss the block triangular form of sparse matrices.

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“…The frontal matrices that are allocated at any given time reside on one path from a root of the etree to some of its descendants. By delaying the allocation of the frontal matrix until after we factor the first child of j and by cleverly restructuring the etree, one can reduce the maximum size of the frontal-matrices stack [Liu 1986]. …”

Section: Left-looking-factor( a S)mentioning

confidence: 99%

The design and implementation of a new out-of-core sparse cholesky factorization method

Rotkin

Toledo

2004

ACM Trans. Math. Softw.

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We describe a new out-of-core sparse Cholesky factorization method. The new method uses the elimination tree to partition the matrix, an advanced subtree-scheduling algorithm, and both rightlooking and left-looking updates. The implementation of the new method is efficient and robust. On a 2 GHz personal computer with 768 MB of main memory, the code can easily factor matrices with factors of up to 48 GB, usually at rates above 1 Gflop/s. For example, the code can factor AUDIKW, currenly the largest matrix in any matrix collection (factor size over 10 GB), in a little over an hour, and can factor a matrix whose graph is a 140-by-140-by-140 mesh in about 12 hours (factor size around 27 GB).

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On the storage requirement in the out-of-core multifrontal method for sparse factorization

Cited by 59 publications

References 6 publications

Scheduling for Large-Scale Systems

Scheduling for Large-Scale Systems

Combinatorial Problems in Solving Linear Systems

The design and implementation of a new out-of-core sparse cholesky factorization method

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