Decentralized Search with Random Costs

A decentralized search algorithm is a method of routing on a random graph that uses only limited, local, information about the realization of the graph. In some random graph models it is possible to define such algorithms which produce short paths when routing from any vertex to any other, while for others it is not. We consider random graphs with random costs assigned to the edges. In this situation, we use the methods of stochastic dynamic programming to create a decentralized search method which attempts to minimize the total cost, rather than the number of steps, of each path. We show that it succeeds in doing so among all decentralized search algorithms which monotonically approach the destination. Our algorithm depends on knowing the expected cost of routing from every vertex to any other, but we show that this may be calculated iteratively, and in practice can be easily estimated from the cost of previous routes and compressed into a small routing table. The methods applied here can also be applied directly in other situations, such as ecient searching in graphs with varying vertex degrees.