A Parallel Algorithm for Interpolation in Pancake Graph IOANA ZELINA, PETRICA POP, CORINA POP SITAR, IOANA TASCU Department of Mathematics and Computer Science North University of Baia Mare Victoriei 76, Baia Mare ROMANIA Abstract: In this paper a parallel algorithm for Lagrange interpolation is applied on a n-pancake graph. The n- pancake graph is a Cayley graph with N=n! vertices and with attractive properties regarding degree, diameter, symmetry, embeddings and self similarity. Using these properties the algorithm carries the calculation in O(N) steps of communication and arithmetic operations instead of O(N 2 ) steps for a single processor system. Key-Words: interconnection topology, pancake graph, Lagrange interpolation 1 Introduction An interconnection network consists of a set of processors, each with a local memory and a set of bidirectional links that serve for the exchange of data between processors. A convenient representation of an interconnection network is by an undirected (or sometimes directed) graph G=(V, E) where each processor P is a vertex in V and two vertices are connected by an edge in E if and only if there is a direct (bidirectional for undirected and unidirectional for directed graphs) communication link between processors. Such processors are called neighbors. The interconnection graph of a network is referred to as its topology. We will use node, vertex and processor with the same meaning and the terms edge and link are used as synonyms. Usually, all processors in a network are identical and each is assumed to have input and output abilities. Processors may execute the same or different programs. The time complexity of any algorithm has two components: computation time (covers local computation) and communication time (the time needed for the exchange of data between processors). We can say that a network topology is “good” if it has some properties as: small degree for the nodes, small diameter (that means small delay in communication), maximum connectivity (good fault tolerance), symmetry (minimum congestion, uniform loading), embedding properties (good simulation of other networks), modular structure (offering the possibility of recursive decomposition). The pancake topology used in this paper has a lot of such good properties that make this topology very attractive. 2 Problem Formulation Interpolation techniques are of great importance in numerical analysis since they are used in many science and engineering domains. The Lagrange interpolation for a given set of points (x 1 ,y 1 ), (x 2 , y 2 ),..., (x N , y N ) and a value x is defined as ∑ = × = N i i i x L y x f 1 ) ( ) ( , where N i L i , 1 , = are the Lagrange polynomials given by the formula ) )...( )( )...( ( ) )...( )( )...( ( ) ( 1 1 0 1 1 0 N i i i i i i N i i i x x x x x x x x x x x x x x x x x L − − − − − − − − = + − + − When the number of points N is very large, a long computation time and a large storage capacity may be required to carry out the calculation. To overcome this, a parallel implementation would be appropriate. This kind of parallel algorithms were introduced for Lagrange interpolation for different topologies: Goertzel [2] has introduced a parallel algorithm for a tree topology with N processors augmented with ring connections which requires N /2 + O(1og N) steps each composed of two substractions and four multiplications; a parallel algorithm has been discussed in [7] which uses a k- ary n-cube consisting of O(k n +kn) steps, each with 4 multiplications and substractions for N = k n node interpolation. In [6] is described a parallel algorithm for computing a N = n!-node Lagrange interpolation on a n-star graph. The algorithm in [6] consists of three phases and requires n!/2 steps, each consisting of 4 multiplications, 4 substractions and one communication operation. In [8], this Proceedings of the 6th WSEAS Int. Conf. on Software Engineering, Parallel and Distributed Systems, Corfu Island, Greece, February 16-19, 2007 98