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An approach to emulating separable graphs

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Abstract

A technique foremulating multicomputer interconnection networks that are based onseparable graphs (graphs having bounded degree and sublinear multicolor recursive bisectors) is presented. Efficient emulations among interconnection networks are necessary for porting programs designed for one network to another.

Emulations are formalized asgraph embeddings, where the nodes (processors) of theguest graph (emulated network) are assigned to nodes of thehost graph (emulator), while the edges (communication links) of the guest are routed via paths in the host. The communication slowdown in an emulation depens on thedilation (length of the longest routing path) and thecongestion (number of paths that contend for a host edge) of the embedding. Theexpansion of the embedding (the ratio of the sizes of the host to guest) determines the inefficiency of processor utilization.

Cell trees are introduced as interconnection networks whose special communication properties enable them to serve as intermediate devices in these emulations. Nodes in cell trees are organized into equinumerous parts calledcells; the cells are labeled by nodes of a complete binary tree. Communication in cell trees is restricted to two specific and distinct primitives:cell communication is confined within cells, whiletransfer communication occurs between adjacent cells. Rather than solved directly, the emulation problem for the original guest-host pair is decomposed into two independent parts: emulating the guest by the cell tree, and emulating the cell tree by the host.

In emulations of separable graphs by cell trees, the node assignment that ensures small dilation is derived from the separator-based decomposition of guest graphs. The congestion-free edge routing is achieved by coordinatingglobal andlocal phases, which are based on two characteristic cell-tree communication primitives.

The technique is instantiated by emulating cell trees on specific host graphs. Withshuffle-like hypercube-derivative networks as hosts new constant-expansion emulations are obtained that have both dilation and congestion logarithmic in the size of the multicolor bisector of guest graphs. These emulations are the first such to have optimal (up to constants)congestion; they provide the firstoptimal algorithm for emulating arbitrary separable graphs on shuffle-like networks. The application of the technique tohypercubes as hosts also produces optimal emulations that differ from those previously known by having smaller expansion constants.

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  1. Department of Computer Science, University of Massachusetts, 01003, Amherst, MA, USA

    Bojana Obrenić

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  1. Bojana Obrenić
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Obrenić, B. An approach to emulating separable graphs. Math. Systems Theory 27, 41–63 (1994). https://doi.org/10.1007/BF01187092

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  • DOI: https://doi.org/10.1007/BF01187092

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