There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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Convex Analysisの二,三の進展について

Efficient application scheduling is critical for achieving high performance in heterogeneous computing environments. The application scheduling problem has been shown to be NP-complete in general cases as well as in several restricted cases. Because of its key importance, this problem has been extensively studied and various algorithms have been proposed in the literature which are mainly for systems with homogeneous processors. Although there are a few algorithms in the literature for heterogeneous processors, they usually require significantly high scheduling costs and they may not deliver good quality schedules with lower costs. In this paper, we present two novel scheduling algorithms for a bounded number of heterogeneous processors with an objective to simultaneously meet high performance and fast scheduling time, which are called the Heterogeneous Earliest-Finish-Time (HEFT) algorithm and the Critical-Path-on-a-Processor (CPOP) algorithm. The HEFT algorithm selects the task with the highest upward rank value at each step and assigns the selected task to the processor, which minimizes its earliest finish time with an insertion-based approach. On the other hand, the CPOP algorithm uses the summation of upward and downward rank values for prioritizing tasks. Another difference is in the processor selection phase, which schedules the critical tasks onto the processor that minimizes the total execution time of the critical tasks. In order to provide a robust and unbiased comparison with the related work, a parametric graph generator was designed to generate weighted directed acyclic graphs with various characteristics. The comparison study, based on both randomly generated graphs and the graphs of some real applications, shows that our scheduling algorithms significantly surpass previous approaches in terms of both quality and cost of schedules, which are mainly presented with schedule length ratio, speedup, frequency of best results, and average scheduling time metrics.

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Performance-effective and low-complexity task scheduling for heterogeneous computing

This paper describes an extension to the set of Basic Linear Algebra Subprograms. The extensions are targeted at matrix-vector operations that should provide for efficient and portable implementations of algorithms for high-performance computers

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A set of level 3 basic linear algebra subprograms

The Grid 2: Blueprint for a New Computing Infrastructure

We consider the problem of allocating a large number of independent, equal-sized tasks to a heterogeneous computing platform. We use a nonoriented graph to model the platform, where resources can have different speeds of computation and communication. Because the number of tasks is large, we focus on the question of determining the optimal steady state scheduling strategy for each processor (the fraction of time spent computing and the fraction of time spent communicating with each neighbor). In contrast to minimizing the total execution time, which is NP-hard in most formulations, we show that finding the optimal steady state can be solved using a linear programming approach and, thus, in polynomial time. Our result holds for a quite general framework, allowing for cycles and multiple paths in the interconnection graph, and allowing for several masters. We also consider the simpler case where the platform is a tree. While this case can also be solved via linear programming, we show how to derive a closed-form formula to compute the optimal steady state, which gives rise to a bandwidth-centric scheduling strategy. The advantage of this approach is that it can directly support autonomous task scheduling based only on information local to each node; no global information is needed. Finally, we provide a theoretical comparison of the computing power of tree-based versus arbitrary platforms.

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Scheduling strategies for master-slave tasking on heterogeneous processor platforms

I Unidimensional Problems.- 1 Scheduling DAGs without Communications.- 2 Scheduling DAGs with Communications.- 3 Cyclic Scheduling.- II Multidimensional Problems.- 4 Systems of Uniform Recurrence Equations.- 5 Parallelism Detection in Nested Loops.

Scheduling and automatic parallelization

We address the issue of implementing matrix multiplication on heterogeneous platforms. We target two different classes of heterogeneous computing resources: heterogeneous networks of workstations and collections of heterogeneous clusters. Intuitively, the problem is to load balance the work with different speed resources while minimizing the communication volume. We formally state this problem in a geometric framework and prove its NP-completeness. Next, we introduce a (polynomial) column-based heuristic, which turns out to be very satisfactory: We derive a theoretical performance guarantee for the heuristic and we assess its practical usefulness through MPI experiments.

Matrix multiplication on heterogeneous platforms

Many applications in scientific and engineering domains are structured as large numbers of independent tasks with low granularity. These applications are thus amenable to straightforward parallelization, typically in master-worker fashion, provided that efficient scheduling strategies are available. Such applications have been called divisible-loads because a scheduler may divide the computation among worker processes arbitrarily, both in terms of number of tasks and of task sizes. Divisible load scheduling has been an active area of research for the last 15 years. A vast literature offers results and scheduling algorithms for various models of the underlying distributed computing platform. Broad surveys are available that report on, accomplishments in the field. By contrast, We propose a unified theoretical perspective that synthesizes previously published results, several novel results, and open questions, in a view to foster hover divisible load scheduling research. Specifically, we discuss both one-round and multiround algorithms, and we restrict our scope to the popular star and tree network topologies, which we study with both linear and affine cost models for communication and computation.

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Scheduling divisible loads on star and tree networks: results and open problems

We consider the problem of allocating a large number of independent, equal-sized tasks to a heterogeneous "grid" computing platform. We use a tree to model a grid, where resources can have different speeds of computation and communication, as well as different overlap capabilities. We define a base model, and show how to determine the maximum steady-state throughput of a node in the base model, assuming we already know the throughput of the subtrees rooted at the node's children. Thus, a bottom-up traversal of the tree determines the rate at which tasks can be processed in the full tree. The best allocation is bandwidth-centric: if enough bandwidth is available, then all nodes are kept busy; if bandwidth is limited, then tasks should be allocated only to the children which have sufficiently small communication times, regardless of their computation power.

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Yves Robert

Papers

Scheduling strategies for master-slave tasking on heterogeneous processor platforms

Scheduling and automatic parallelization

Matrix multiplication on heterogeneous platforms

Scheduling divisible loads on star and tree networks: results and open problems

Bandwidth-centric allocation of independent tasks on heterogeneous platforms