Information Processing Letters 110 (2010) 854–860 Contents lists available at ScienceDirect Information Processing Letters www.elsevier.com/locate/ipl Routing automorphisms of the hypercube Paulin Melatagia Yonta a,∗ , Maurice Tchuente a,b , René Ndoundam a a Département d’Informatique, UMI 209, UMMISCO, Faculté des Sciences, Université de Yaoundé I, B.P. 812 Yaoundé, Cameroon b IRD, UMI 209, UMMISCO, 32 avenue Henri Varagnat, F-93143, Bondy, France article info abstract Article history: Received 23 March 2010 Received in revised form 22 June 2010 Accepted 1 July 2010 Available online 11 July 2010 Communicated by J. Torán Keywords: Graph algorithms Hypercube Queueless MIMD BPC permutation Arbitrarily routable We present an online algorithm for routing the automorphisms (BPC permutations) of the queueless MIMD hypercube. The routing algorithm has the virtue of being executed by each node of the hypercube without knowing the state of the others nodes. The algorithm is also vertex and link-contention free. We show, using the proposed algorithm, that BPC permutations are arbitrarily routable in the considered communication model.  2010 Elsevier B.V. All rights reserved. 1. Introduction Communication in computer networks is often per- formed using packet switching, that is, messages are put into packets together with a destination address and these packets are moved in store-and-forward fashion to their destination. Routing is a process of transmitting these packets among nodes to their destination and its efficiency is crucial for the performance of communication in a net- work. Graphs are mathematical objects which are generally used to model and study networks. Hereafter, we will use the terms network and graph indifferently. An undirected and connected graph G is a couple ( X , E ) with X the set of vertices or nodes and E the set of edges. Any problem of routing on graphs can be expressed as a four parameters problem (N, p, k 1 , k 2 ) where N is the number of messages (called packets) in the graph, each having a starting and a destination node. Initially, the N packets are on p nodes; each node has a maximum of k 1 packets. At the end of * Corresponding author. E-mail addresses: paulinyonta@gmail.com (P.M. Yonta), maurice.tchuente@ens-lyon.fr (M. Tchuente), ndoundam@yahoo.com (R. Ndoundam). the process, there is a maximum of k 2 packets on each node. The goal of the routing problem is to minimize the number of steps of communication required to route all the packets. Routing permutation is a special case of rout- ing problem in which the N packets are initially on the N nodes of the graph ( N = p), each node having only one packet at the beginning and at the end of the pro- cess (k 1 = k 2 = 1). The destination of the packet at node x (x ∈ X ) is π (x) where π ∈ X × X is a permutation of the nodes of the graph. When the permutation is known in advance, the routing strategy is said offline. Otherwise, the strategy is said online [1]. We consider, in this paper, the online problem of rout- ing a permutation of packets on the nodes of the n-cube. The n-dimensional hypercube (or n-cube) H n consists of 2 n nodes whose labels are the 2 n binary strings of length n [7]. Two nodes are connected through an edge if and only if their labels differ in exactly one bit. Each hyper- cube edge is considered to be bidirectional and therefore exchanges of packets between adjacent nodes are possible in one communication step. We consider a communication model in which each hypercube edge can only communi- cate in each of its direction one packet per communication step, and each node can only hold a single packet between 0020-0190/$ – see front matter  2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ipl.2010.07.002