A Mathematical Game and Its Applications to the Design of Interconnection Networks zy Chi-Hsiang Yeh Dept. of Electrical zyxwvutsr & Computer Engineering Queen's University Kingston, Ontario, K7L 3N6, Canada zyxwvu yeh@ee.queensu.ca Abstract zyxwvutsr In this papec we propose a mathematical game, called the ball-arrangement game (BAG). A game with a direrent set zyxwvut of rules (e.g., permissible moves) gives rise zyxwvutsr to a direrent network, and the algorithm that solves the game gives rise to a routing al- gorithm in that network. Based on the insights provided by BAG, we propose several new classes of symmetric and modular net- works, called super Cayley graphs, that have optimal (interclus- ter) diameters and average (intercluster)distances, small (inter- cluster) node degrees, high bisection bandwidth, strong embed- ding capabilio, and optimal communication algorithms given their (intercluster)node degrees. 1 Introduction In this paper, we introduce a mathematical game called the ball- arrangement game (BAG) and we use an interesting analogy to design several new classes of interconnectionnetworks and their algorithms. In the ball-arrangement game, we are given k balls, each stamped with a number. Different balls may be assigned the same or different numbers. The goal of the game is to re- arrange the balls so that the numbers on the balls appear in a desired order. At each step the player can take an arbitrary ac- tion from a set of d permissible moves, each being a particu- lar permutation of the balls. The set of permissible moves re- mains the same throughout the game, independent of the cur- rent configurationof the balls. If the k balls have different num- bers, then there are k! possible configurations of the balls (i.e., states) when playing the game. If we view each of the states as a network node and a permissible move leading from one state to another as a directed link connecting the nodes correspond- ing to those two states, then a network with k! nodes results, where each node has d outgoing links. In other words, the net- work can be obtained by drawing the state transition graph for the corresponding ball-arrangement game with specified move- ments. One can then relate playing a ball-arrangement game to routing in the corresponding network, where the initial and final states correspond to the source and destination nodes and the se- quence of movements performed to solve the game corresponds to the sequence of links along the routing path. Since the in-/out- degree of the derived network is upper bounded by the number d of permissible moves and the diameter is the maximum number of steps required to solve the game, we generally prefer to select a small number of permissible moves that allow us to solve the 'Work performed while with TU Delft, Netherlands Emmanouel A. Varvarigos* Dept. of Electrical & Computer Engineering University of California Santa Barbara, CA 93 106-9560, USA manos@ece.ucsb.edu game in a small (or optimal) number of steps for any initial and final states. A k-dimensional star graph, k-star [ 1, 21, is a well-known network that has a number of desirable properties, such as de- gree, diameter, and average distance smaller than those of a similar-size hypercube, symmetry properties, strong embedding capability, and fault tolerance properties. A variety of efficient algorithms have been proposed for star graphs and various prop- erties have been investigated in the literature [3,4,5, 10, 13, 14, 20, 21, 23, 25, 27, 28, 351. A k-star is derived from a special case of the ball-arrangementgame where each ball has a distinct number and at each step the player can interchange the leftmost ball with an arbitrary ball [l, 21. Akers, Harel, and Krishna- murthy presented a simple and efficient algorithm to solve the game in at most L3(k - 1)/2] steps [l, 21. Therefore, the de- gree of an N-node k-star is k - l = O(log/loglogN) and the di- ameter is [3(k- 1)/2] = O(log/loglogN), both of which are sublogarithmic. Moreover, it can be shown that the diameter of a star graph is optimal within a factor of 1.5 + o( zyx 1) ' from a universal lower bound given its node degree [32]. In [2], Ak- ers and Krishnamurthydevelop a group-theoretic model, called the Cayley graph model, for designing and analyzing symmetric interconnection networks. A Cayley graph can be defined by a corresponding permutation group, which corresponds to a ball- arrangement game where different balls have different numbers. In [2], Akers and Krishnamurthy showed that Cayley graphs are vertex-symmetric and that most vertex-symmetric graphs can be represented as Cayley graphs; it was also shown that every vertex-symmetric graph can be represented as a Cayley coset graph. In [31, 371, we derived an analogous result show- ing that every graph corresponds to a certain ball-arrangement game. Both the Cayley graph model and the Cayley coset graph model have been used to derived a wide variety of interesting networks for parallel processing and have since received con- siderable attention [2,9, 11, 12, 15, 18,26,32]. Many networks can be formulated by simple ball-arrangement games and that al- gorithms for networks derived from a similar set of moves can usually be developed in a unified manner. We have also used the underlying idea of the ball-arrangement game to derive a vari- ety of efficient networks that have certain desirable properties [31, 32, 33, 34, 36, 371. Although the star graph has many desirable properties, its node degree is still too large when the network size is large. The reason is that the corresponding ball-arrangement game per- mits k - 1 moves for rearranging the balls so that degree O(k) = 'The notation f(N) = o(g(N)) means that limhr,,f(N)/g(N) = 0. 0190-3918/01$10.00 zyxwvutsrqpo 0 2001 IEEE 21