Are Static Schedules so Bad? A Case Study on Cholesky Factorization

Abstract

Our goal is to provide an analysis and comparison of static and dynamic strategies for task graph scheduling on platforms consisting of heterogeneous and unrelated resources, such as GPUs and CPUs. Static scheduling strategies, that have been used for years, suffer several weaknesses. First, it is well known that underlying optimization problems are NP-Complete, what limits the capability of finding optimal solutions to small cases. Second, parallelism inside processing nodes makes it difficult to precisely predict the performance of both communications and computations, due to shared resources and co-scheduling effects. Recently, to cope with this limitations, many dynamic task-graph based runtime schedulers (StarPU, StarSs, QUARK, PaRSEC) have been proposed. Dynamic schedulers base their allocation and scheduling decisions on the one side on dynamic information such as the set of available tasks, the location of data and the state of the resources and on the other hand on static information such as task priorities computed from the whole task graph. Our analysis is deep but we concentrate on a single kernel, namely Cholesky factorization of dense matrices on platforms consisting of GPUs and CPUs. This application encompasses many important characteristics in our context. Indeed, it involves 4 different kernels (POTRF, TRSM, SYRK and GEMM) whose acceleration ratios on GPUs are strongly different (from 2.3 for POTRF to 29 for GEMM) and it consists in a phase where the number of available tasks if large, where the careful use of resources is critical, and in a phase with few tasks available, where the choice of the task to be executed is crucial. In this paper, we analyze the performance of static and dynamic strategies and we propose a set of intermediate strategies, by adding more static (resp. dynamic) features into dynamic (resp. static) strategies. Our conclusions are somehow unexpected in the sense that we prove that static-based strategies are very efficient, even in a context where performance estimations are not very good.

Fast approximation algorithms for task‐based runtime systems

“…This area bound can be extended to the case of non‐independent tasks by adding a variable s i representing the start time of each task i , and the following set of constraints: …”

“…This H etero P rio allocation strategy has been studied in the context of StarPU for several linear algebra kernels and it has been proven experimentally that it enables to achieve a better utilization of slow resources than other standard HEFT‐based strategies.…”

Low-Cost Approximation Algorithms for Scheduling Independent Tasks on Hybrid Platforms

“…Task processing times are randomly generated and follow a gamma distribution with expected value 15 for the CPUs and 1 for the GPUs. These values are inspired from the measures in [2,7]. Moreover, the gamma distribution has been advocated for modeling job runtimes [19,29].…”

“…The problem has also been studied in practical situations using specific runtime environments as XKaapi [10] and StarPU [2]. The study with StarPU has been focused on a single application, the Cholesky factorization [1], and later extended to more than 2 types of processors [7].…”

Online Scheduling of Task Graphs on Hybrid Platforms

“…Tasks correspond to four linear algebra kernels: GEMM, SYRK, TRSM, and POTRF. Their respective processing times on a CPU are set to 170ms, 95ms, 88ms, and 33ms, and on a GPU to 5.95ms, 3.65ms, 8.11ms, and 15.6ms, which corresponds to measures [1,6] made using the Chameleon software [12].…”