Node-to-Set Disjoint Paths Problem in Cross-Cubes

Abstract

The increasing demand for high-performance computing in recent years has led to active research on massively parallel systems. The interconnection network in a massively parallel system interconnects hundreds of thousands of processing elements so that they can process large tasks while communicating among others. By regarding the processing elements as nodes and the links between processing elements as edges, respectively, we can discuss various problems of interconnection networks in the framework of the graph theory. Many topologies have been proposed for interconnection networks of massively parallel systems. The hypercube is a very popular topology and it has many variants. The cross-cube is such a topology, which can be obtained by adding one extra edge to each node of the hypercube. The cross-cube reduces the diameter of the hypercube, and allows cycles of odd lengths. Therefore, we focus on the cross-cube and propose an algorithm that constructs disjoint paths from a node to a set of nodes. We give a proof of correctness of the algorithm. Also, we show that the time complexity and the maximum path length of the algorithm are O(n³ log n) and 2n - 3, respectively. Moreover, we estimate that the average execution time of the algorithm is O(n²) based on a computer experiment.