Matrix Chain Multiplication and Polygon Triangulation Revisited and Generalized

TL;DR

This paper revisits and simplifies the classic matrix-chain multiplication and polygon triangulation problems, providing new proofs, extending bounds, analyzing variants, and empirical testing of algorithms.

Contribution

It simplifies and clarifies the $O(n^2)$-time algorithm, extends its application to generalized problems, and empirically compares its performance with other methods.

Findings

The $O(n^2)$-time algorithm is correct and simple enough for teaching.

The algorithm's worst-case time complexity is $ heta(n^2)$.

Empirical tests show the variant runs in $ heta(n ext{log} n)$ time on random data.

Abstract

The {\it matrix-chain multiplication} problem is a classic problem that is widely taught to illustrate dynamic programming. The textbook solution runs in $\theta(n^3)$ time. However, there is a complex $O(n \log n)$-time method \cite{HU82}, based on triangulating convex polygons, and a description without proofs or implementation detail, of a much simpler $O(n^2)$-time method \cite{YAO82}. There is also a linear-time approximation algorithm with a small worst-case error bound \cite{HU-SHING1981}. In this paper, we make five contributions both to theory and pedagogy: 1) We simplify the approach in \cite{YAO82}, and provide complete, correct proofs and implementation details, to establish the $O(n^2)$-time bound. We believe that this exposition is simple enough for classroom use. 2) We extend the $O(n^2)$-time bound to a natural class of polygon-triangulation problems that generalizes the original polygon-triangulation problem in \cite{HU82}. 3) We show that the worst-case running time of the method in \cite{YAO82}, and of our version, is $\Theta(n^2)$. 4) We show that in a natural variant of the original polygon-triangulation problem, the approximation method of \cite{HU-SHING1981} does not achieve the same error bound, but does achieve an error bound about twice the original bound. 5) We detail empirical testing, showing that on random data our variant runs in $\Theta(n \log n)$ time, while the approach in \cite{YAO82} empirically takes $\Theta(n^2)$ time. Software for these tests is posted on the web.