# Rearrangeable graphs

## Abstract

Interconnection networks play an important role in parallel computing systems. A lot of effort has been made to construct rearrangeable networks. This paper generalizes the concept of rearrangeability to general, directed graphs. Given a directed graph *G =(V,E )*, where *V* = 1, 2, ..., *n*, an *n*-permutation *π* is said to be *realizable* on *G* if there exist *n* edge-disjoint paths in *G* that connect vertex *i* to vertex *π*(*i*) (1 ≤ *i* ≤ *n*). Graph *G* is said to be *rearrangeable* if all the *n!**n*-permutations are realizable on *G*. This paper presents tight lower bounds on the number of edges in a rearrangeable graph. One of the lower bounds is *m* ≥ 2(*n*-1), where *m* and *n* are the number of edges and the number of vertices, respectively. Moreover, this paper introduces a new graph parameter, *rearrangeable number*, denoted by *ψ*, which is the minimal multiplicity every edge in *G* needs to be duplicated so that the resultant graph becomes rearrangeable. A lower bound on the value of *ψ* is derived in the paper.

## Key words

Permutation capability Permutation realization Rearrangeable graph## Preview

Unable to display preview. Download preview PDF.

## Reference

- [1]T. Szymanski, “On the Permutation Capability of a Circuit-switched Hypercube”, 1989 Int, Conf on Parallel Processing, pp. I:103–110.Google Scholar
- [2]D. Nassimi and S. Sahni, “Optimal BPC permutations on a Cube Connected SIMD Computer”,
*IEEE Trans. Comput.*Vol. C-31(4), Apr. 1982, pp. 338–341.Google Scholar - [3]K. Hwang and F.A. Briggs,
*Computer Architecture and Parallel Processing*, McGraw-Hill, 1984.Google Scholar - [4]A. Lubiw, A Counter-Example to a Conjecture by Szymanski on Hypercube Routing,
*Inform. Process. Lett.*35 (1990) pp57, 61Google Scholar - [5]Xiaojun Shen, Qing Hu, Hao Dai, Xiangzu Wang, Optimal Routing of Permutations on Rings, Lecture Notes in Computer Science 834, pp360–368, Springer-Verlag, 1994Google Scholar
- [6]H.J. Siegel,
*Interconnection Networks for Large-Scale Parallel Processing*, McGraw-Hill, 1990.Google Scholar - [7]F.T. Leighton,
*Introduction to Parallel Algorithms and Architectures: Arrays. Trees. Hypercubes*, Morgan Kanfman Publisher. 1992.Google Scholar - [8]M. R. Garey and D. S. Johnson,
*Computers and Intractability. A guide to the Theory NP-Completeness*, W.H. Freeman, 1979.Google Scholar - [9]X. Shen and Q. Hu, “Realizability of an Arbitrary Permutation on a CircuitSwitched HYpercube”,
*Information Processing Letters*51 (1994) pp. 237–243.Google Scholar - [10]S.E. Orcutt, “Implementation of Permutation in ILLIAC IV type Computer”,
*IEEE Trans. Comput.*Vol. C-31(12), Dec. 1982, pp. 1202–1214.Google Scholar - [11]V.E. Benes,
*The Mathematical Theory of Connecting Networks and Telephone Traffic*. Academic Press, New York, 1965.Google Scholar