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Bounded dilation maps of hypercubes into Cayley graphs on the symmetric group

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Abstract

LetG andH be graphs with ¦V(G)≤ ¦V(H)¦. Iff:V(G) →V(H) is a one-to-one map, we letdilation(f) be the maximum of dist H (f x),f(y)) over all edgesxy inG where dist H denotes distance inH. The construction of maps fromG toH of small dilation is motivated by the problem of designing small slowdown simulations onH of algorithms that were originally designed for the networkG.

LetS(n), thestar network of dimension n, be the graph whose vertices are the elements of the symmetric group of degreen, two verticesx andy being adjacent ifx o (1,i) =y for somei. That is,xy is an edge ifx andy are related by a transposition involving some fixed symbol (which we take to be 1). Also letP(n), thepancake network of dimension n, be the graph whose vertices are the elements of the symmetric group of degreen, two verticesx andy being adjacent if one can be obtained from the other by reversing some prefix. That is,xy is an edge ifx andy are related byx o (1,i(2,i-1) ⋯ ([i/2], [i/2]) =y. The star network (introduced in [AHK]) has nice symmetry properties, and its degree and diameter are sublogarithmic as functions of the number of vertices, making it compare favorably with the hypercube network. These advantages ofS(n) motivate the study of how well it can simulate other parallel computation networks, in particular, the hypercube.

The concern of this paper is to construct low dilation maps of hypercube networks into star or pancake networks. Typically in such problems, there is a tradeoff between keeping the dilationsmall and simulating alarge hypercube. Our main result shows that at the cost ofO (1) dilation asn→ ∞, one can embed a hypercube of near optimum dimension into the star or pancake networksS(n) orP(n).

More precisely, lettingQ (d) be the hypercube of dimensiond, our main theorem is stated below. For simplicity, we state it only in the special case when the star network dimension is a power of 2. A more general result (applying to star networks of arbitrary dimension) is obtained by a simple interpolation.

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Authors and Affiliations

  1. Department of Mathematics, Miami University, 45056, Oxford, OH, USA

    Z. Miller

  2. Department of Statistics, Miami University, 45056, Oxford, OH, USA

    D. Pritikin

  3. Computer Science Program, University of Texas-Dallas, 75083, Richardson, TX, USA

    I. H. Sudborough

Authors
  1. Z. Miller
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  2. D. Pritikin
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  3. I. H. Sudborough
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This author's research was done during the Spring Semester 1991, as a visiting professor in the Department of Mathematics and Statistics at Miami University.

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Miller, Z., Pritikin, D. & Sudborough, I.H. Bounded dilation maps of hypercubes into Cayley graphs on the symmetric group. Math. Systems Theory 29, 551–572 (1996). https://doi.org/10.1007/BF01301964

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  • DOI: https://doi.org/10.1007/BF01301964

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