Finding A Minimum Cost Acceptable Path in Parallel 1

We consider the problem of nding a minimum cost acceptable path on a toroidal grid graph, where each horizontal and each vertical edge have the same orientation. An acceptable path is closed path that makes a complete horizontal and vertical circuit. We exploit the structure of this graph to develop eecient parallel algorithms for a message passing computer. Given p processors and an m by n toroidal graph, our algorithm will nd the minimum cost acceptable path in O(mn log(m)=p) steps, if p = O(mn=((m + n) log(mn=(m + n)))), which is an optimal speedup. Also we show that the algorithm will send O(p 2 (m + n)) messages. The solution of this problem has applications in surface reconstruction 6, 5, 9]