Approximations for the general block distribution of a matrix

Bengt Aspvall
Department of Computer Science, University of Bergen, Bergen, Norway.

Magnús M. Halldórsson
Science Institute, University of Iceland, IS-107 Reykjavik, Iceland.
AND University of Bergen

Fredrik Manne
Department of Computer Science, University of Bergen, Bergen, Norway.

The general block distribution of a matrix is a rectilinear partition of the matrix into orthogonal blocks such that the maximum sum of the elements within a single block is minimized. This corresponds to partitioning the matrix onto parallel processors so as to minimize processor load while maintaining regular communication patterns. Applications of the problem include various parallel sparse matrix computations, compilers for high-performance languages, particle in cell computations, video and image compression, and simulations associated with a communication network.

We analyze the performance guarantee of a natural and practical heuristic based on iterative refinement, which has previously been shown to give good empirical results. When $p^2$ is the number of blocks, we show that the tight performance ratio is $\theta(\sqrt{p})$. When the matrix has rows of large cost, the details of the objective function of the algorithm are shown to be important, since a naive implementation can lead to a $\Omega(p)$ performance ratio. Extensions to more general cost functions, higher-dimensional arrays, and randomized initial configurations are also considered.

In comparison, Khanna et al.\ have shown the problem to be approximable within $O(1)$ factor \cite{KMS97}, while Grigni and Manne have shown it to be NP-hard to approximate within any constant factor less than two \cite{GM96}.

(Accepted revision, May 31, 1999)