Optimal Synthesis of Multiple Algorithms

TL;DR

This paper introduces a formal framework for defining and analyzing derived algorithms that can potentially outperform individual algorithms in mean execution time, supported by explicit formulas and empirical analysis methods.

Contribution

It provides a new definition of derived algorithms, explicit formulas for their mean execution times, and a maximum-likelihood estimation scheme for empirical data analysis.

Findings

Derived algorithms can have smaller mean execution times than their components.

Explicit formulas for mean execution times under various joint density forms.

Maximum-likelihood estimation method for empirical processing time data.

Abstract

In this paper we give a definition of "algorithm," "finite algorithm," "equivalent algorithms," and what it means for a single algorithm to dominate a set of algorithms. We define a derived algorithm which may have a smaller mean execution time than any of its component algorithms. We give an explicit expression for the mean execution time (when it exists) of the derived algorithm. We give several illustrative examples of derived algorithms with two component algorithms. We include mean execution time solutions for two-algorithm processors whose joint density of execution times are of several general forms. For the case in which the joint density for a two-algorithm processor is a step function, we give a maximum-likelihood estimation scheme with which to analyze empirical processing time data.