Average-case Performance Analysis Research Articles

Average-case performance analysis of scheduling random parallel tasks with precedence constraints on mesh connected multicomputer systems

We investigate the problem of scheduling parallel tasks with precedence constraints on mesh connected multicomputer systems. It is still an open problem on whether there exists an approximation algorithm with finite asymptotic worst-case and/or average-case performance bound for this scheduling problem. As an early attempt to solve our problem, we propose and analyze the performance of a level-by-level scheduling algorithm LL. In fact, we solve a special case of the problem when all tasks request for square submeshes and run on a square mesh system whose size is a power of 2. There are three basic techniques in algorithm LL, i.e., the level-by-level scheduling strategy for handling precedence constraints, the largest-task-first algorithm for scheduling tasks in the same level, and the two-dimensional buddy system for system partitioning and processor allocation. Algorithm LL does not have a finite worst-case performance bound; however, it has quite acceptable average-case performance. The main contribution of the paper is to show that under the assumptions that task sizes are independent and identically distributed (i.i.d.) random variables with a common probability distribution, and that task execution times are i.i.d. random variables with finite mean and variance, and that the probability distributions of task sizes and execution times are independent of each other, for wide task graphs and typical task size distributions, algorithm LL has an asymptotic average-case performance bound about two for all probability distributions of task execution times.

Average-Case Performance Analysis of a 2D Strip Packing Algorithm?NFDH

The two-dimensional strip packing problem is a generalization of the classic one-dimensional bin packing problem. It has many important applications such as costume clipping, material cutting, real-world planning, packing, scheduling etc. Average-case performance analysis of approximation algorithms attracts a lot of attention because it plays a crucial role in selecting an appropriate algorithm for a given application. While approximation algorithms for two-dimensional packing are frequently presented, the results of their average-case performance analyses have seldom been reported due to intractability. In this paper, we analyze the average-case performance of Next Fit Decreasing Height (NFDH) algorithm, one of the first strip packing algorithms proposed by Coffman, Jr. in 1980. We prove that the expected height of packing with NFDH algorithm, when the heights and widths of the rectangle items are independent and both obey (0, 1] uniform distribution, is about n/3, where n is the number of rectangle items. We also validate the theoretical result with experiments.

AVERAGE-CASE SCALABILITY ANALYSIS OF PARALLEL COMPUTATIONS ON k-ARY d-CUBES

We investigate the average-case scalability of parallel algorithms executing on multicomputer systems whose static networks are k-ary d-cubes. Our performance metrics are isoefficiency function and isospeed scalability. For the purpose of average-case performance analysis, we formally define the concepts of average-case isoefficiency function and average-case isospeed scalability. By modeling parallel algorithms on multicomputers using task interaction graphs, we are mainly interested in the effects of communication overhead and load imbalance on the performance of parallel computations. We focus on the topology of static networks whose limited connectivities are constraints to high performance. In our probabilistic model, task computation and communication times are treated as random variables, so that we can analyze the average-case performance of parallel computations. We derive the expected parallel execution time on symmetric static networks and apply the result to k-ary d-cubes. We characterize the maximum tolerable communication overhead such that constant average-case efficiency and average-case average-speed could be maintained and that the number of tasks has a growth rate Θ(P log P), where P is the number of processors. It is found that the scalability of a parallel computation is essentially determined by the topology of a static network, i.e., the architecture of a parallel computer system. We also argue that under our probabilistic model, the number of tasks should grow at least in the rate of Θ(P log P), so that constant average-case efficiency and average-speed can be maintained.

Average-case performance analysis of an approximation algorithm for maximum subset sum using recurrence relations

Given a positive integer M, and a set S = { x 1, x 2, …, x n } of positive integers, the maximum subset sum problem is to find a subset S′ of S such that ∑ x∈ S′ x is maximized under the constraint that the summation is no larger than M. In addition, the cardinality of S′ is also a maximum among all subsets of S which achieve the maximum subset sum. This problem is known to be NP-hard. We analyze the average-case performance of a simple on-line approximation algorithm assuming that all integers in S are independent and have the same probability distribution. We develop a general methodology, i.e., using recurrence relations, to evaluate the expected values of the content and the cardinality of S′ for any distribution. The maximum subset sum problem has important applications, especially in static job scheduling in multiprogrammed parallel systems. The algorithm studied can also be easily adapted for dynamic job scheduling in such systems.

A quantitative analysis of the simulated annealing algorithm: A case study for the traveling salesman problem

A quantitative study is presented of the typical behavior of the simulated annealing algorithm based on a cooling schedule presented previously by the authors. The study is based on the analysis of numerical results obtained by systematically applying the algorithm to a 100-city traveling salesman problem. The expectation and the variance of the cost are analyzed as a function of the control parameter of the cooling schedule. A semiempirical average-case performance analysis is presented from which estimates are obtained on the expectation of the average final result obtained by the simulated annealing algorithm as a function of the distance parameter, which determines the decrement of the control parameter.