Sorting and selection on arrays with diagonal connections

Abstract

We examine the problems of sorting and selection on meshes and tori with diagonal connections. We are able to achieve dramatic reductions in time required for selection by making use of the diagonal connections. In particular, on an n × n mesh with diagonal connections, we show how to select the k-th largest of n ² elements in 0.65n steps, and on a torus in 0.59n steps. The best known results for the corresponding networks without diagonal connections are 1.15n and 1.13n respectively. We also give an algorithm for selection that works optimally on a random input on these networks. For the case of sorting, we show the surprising result that there can be no distance-optimal algorithm for sorting on a mesh with diagonal connections, on a model that does not use copies and allows only constant size queues. Combined with results due to Kunde et al. [7], our results imply that routing is easier than sorting on the mesh with diagonal connections when copies are disallowed and queues are restricted to size 10. We believe this is the first result of this type in the fixed connection network model. On a torus with diagonal connections, we show that any algorithm that does not use copies must take at least n-o(n) steps regardless of how large the queue size is. We also show nontrivial lower bounds for selection on meshes with diagonal connections.

Research was supported in part by the Natural Sciences and Engineering Research Council of Canada, research grant OGP0137991.