( k− k) Routing on multidimensional mesh-connected arrays

Abstract

In this paper we study the problem of routing packets on an r-dimensional mesh-connected array of processors. The focus of this paper is on routing with each processor containing exactly k packets, k ⩾ 2, initially and finally (so-called k−k routing). For two-dimensional n × n grids the number of transport steps is at most 5 4 kn + O(kn/f(n)) with a buffer size of O(kf(n)). In the special case of a sequence of k permutation routing problems this step count can be reduced to kn + O( kn/ f( n)). For an r-dimensional grid, r ⩾ 3, with side length n the same technique yields an algorithm with step count (r − 1)(1 + 1/r 2kn + O(n/f(n) 1 (r − 1) ) and buffer rk · f( n). For sequences of permutation routing problems this drops to [k/r] (2r − 2)n + O(kn/f(n) 1 (r−1) ) and a buffer size of O(kf(n)). Furthermore it is shown that splitting large packets into smaller ones has benefits for permutation routing problems. For grids with wrap-around connections these step counts and times generally can be reduced by one-half.