Optimal Emulation of Meshes on Meshes of Trees

Many problems can be solved more efficiently on a mesh of trees network than on a mesh. Until now it has been an open problem whether the mesh of trees is always at least as fast as the mesh. In this paper, we present an emulation of N-node meshes on O(N)-node meshes of trees with constant slowdown, even though any embedding of a mesh into a mesh of trees requires dilation Ω(log N). This demonstrates that the mesh of trees is strictly more powerful than the mesh. As an application, we show how to construct an optimal O(√N) sorting algorithm for the mesh of trees that improves on the best previously known algorithm by a logarithmic factor.

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