# On the definition of speedup

## Abstract

We propose an alternative definition for the *speedup* of parallel algorithms. Let *A* be a sequential algorithm and *B* a parallel algorithm for solving the same problem. If *A* and/or *B* are randomized or if we are interested in their performance on a probability distribution of problem instances, the running times are described by random variables *T*^{ A } and *T*^{ B }. The speedup is usually defined as *E[T*^{ A }]/*E*[*T*^{ B }] where *E* is the arithmetic mean. This notion of speedup delivers just a number, i.e. much information about the distribution is lost. For example, there is no variance of the speedup. To define a measure for possible fluctuations of the speedup, a new notion of speedup is required. The basic idea is to define speedup as *M*(*T*^{ A }/*T*^{ B }) where the functional form of *M* has to be determined. Also, we argue that in many cases *M*(*T*^{ A }/*T*^{ B }) is more informative than *E*[(*T*^{ A }]/*E*[*T*^{ B }] for a typical user of *A* and *B*. We present a set of intuitive axioms that any speedup function *M*(*T*^{ A }/*T*^{ B }) must fulfill and prove that the geometric mean is the only solution. As a result, we now have a uniquely defined speedup function that will allow the user of an improved system to talk about the average performance improvement as well as about its possible variations.

## Keywords

Speedup variation of running time geometric mean functional equations randomized algorithms## Preview

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## References

- [1]J. Aczél. The Notion of Mean Values.
*Norske Vid. Selsk. Forh. (Trondheim)*, 19:83–86, 1946.Google Scholar - [2]J. Aczél.
*Lectures on Functional Equations and Their Applications*, volume 19 of*Mathematics in Science and Engineering*. Academic Press, 1966.Google Scholar - [3]J. Aczél and C. Alsina. On Synthesis of Judgements.
*Socio-Econ. Plann. Sci.*, 20(6):333–339, 1986.CrossRefGoogle Scholar - [4]J. Aczél and T. L. Saaty. Procedures for Synthesizing Ratio Judgements.
*J. Math. Psych.*, 27:93–102, 1983.CrossRefGoogle Scholar - [5]W. Ertel. Random Competition: A Simple, but Efficient Method for Parallelizing Inference Systems. In
*Parallelization in Inference Systems*, pages 195–209. LNAI 590, Springer-Verlag, 1992.Google Scholar - [6]P. J. Fleming and J. J. Wallace. How not to Lie with Statistics: The Correct Way to Summarize Benchmark Results.
*Comm. of the ACM*, 29(3):218–221, 1986.CrossRefGoogle Scholar - [7]A. Goerdt and U. Kamps. On the Reasons for Average Superlinear Speedup in Parallel Backtrack Search. submitted for publication, 1993.Google Scholar
- [8]R. Mehrotra and E. F. Gehringer. Superlinear Speedup Through Randomized Algorithms. In
*International Conference on Parallel Processing*, pages 291–300, 1985.Google Scholar - [9]K. S. Natarajan. Expected Performance of Parallel Search. In
*International Conference on Parallel Processing*, pages 121–125, 1989.Google Scholar - [10]V. N. Rao and V. Kumar. Superlinear Speedup in Parallel Search. In
*Proc. of Foundations of Software Technology and Theor. Comp. Sci.*, New Dehli, 1988.Google Scholar - [11]F. S. Roberts. Merging Relative Scores.
*J. Math. Analysis and Applications*, 147:30–52, 1990.CrossRefGoogle Scholar - [12]E. Speckenmeyer, B. Monien, and O. Vornberger. Superlinear Speedup for Parallel Backtracking. In
*Supercomputing*, LNCS 297, pages 985–994. Springer Verlag, 1988.Google Scholar - [13]X.-H. Sun and J.L. Gustafson. Toward a Better Parallel Performance Metric.
*Parallel Computing*17, pages 1093–1109, 1991.Google Scholar