Optimal Systolic Design for the Transitive Closure and the Shortest Path Problems

Due to VLSI technological progress, algorithm-oriented array architectures, such as systolic arrays, appear to be very effective, feasible, and economic. This paper discusses how to design systolic arrays for the transitive closure and the shortest path problems. We shall focus on the Warshall algorithm for the transitive closure problem and the Floyd algorithm for the shortest path problem. These two algorithms share exactly the same structural formulation; therefore, they lead to the same systolic array design. In this paper, we first present a general method for mapping algorithms to systolic arrays. Using this methodology, two new systolic designs for the Warshall-Floyd algorithm will be derived. The first one is a spiral array, which is easy to derive and can be further simplified to a hexagonal array. The other is an orthogonal systolic array which is optimal in terms of pipelining rate, block pipelining rate, and the number of input/output connections.