Multi - Terminal Shortest Paths

The present paper gives an algorithm that finds simultaneously the shortest paths between many pairs of nodes in a given network. In the book by Berge, the values of shortest paths between many pairs of nodes are found. Here, we use a special matrix multiplication technique to find the actual arcs that are used to form the shortest paths. In a network with n nodes, log [n-l] special matrix multiplications are needed to find all the shorte'-t paths. The present paper also gives an algorithm for constructing a network with prescribed shortest distances and with the total distances associated with arcs a minimum. The present paper gives an algorithm that finds simultaneously the shortest paths between many pairs of nodes in a given network, an algorithm for constructing a network with prescribed sho.-test path distances, and with the total distances associated with arcs a minimum. Many papers have been written about finding one shortest path from a given node to another node in a given network (see for example, [1J, [3], 1^1> [8], or the review [9] ). If we are interested in shortest paths between many pairs of nodes, then this problem can certainly be solved by using any of the existing algorlthnß for finding a shortest path and doing the algorithm over and over for every pair of nodes. Because in finding a shortest path many informations are obtained during the process, we would expect that some saving would be acheived if we want to find many shortest paths Also, for finding maximum flow values between many pairs of nodes, it is known that a great amount of computation can be saved (see [5] ), and the known dual relationship between maximum flow aud shortest path in an a-b planar network would also lead one to expect that many shortest paths can be found simultaneously„ In the book by Berger [2], the values of shortest paths between many pairs of noeds are found by a special matrix multiplication technique; in Gomory and Hu [6], a minimal cost network is found which can satisfy given flow requirements between many pairs of nodes simultaneously. We are now interested in finding the actual arcs that are used to form the shortest paths. Although the paper is self-contained, we shall use many of the notions of [2j and [6]. We consider a network consisting of nodes N and arcs A leading