Successive shortest paths in complete graphs with random edge weights

Consider a complete graphK(n)with edge weights drawn independently from a uniform distributionU(0, 1). The weight of the shortest (minimum-weight) pathP(1)between two given vertices is known to belnn/n, asymptotically. Define a second-shortest pathP(2)to be the shortest path edge-disjoint fromP(1), and consider more generally the shortest pathP(k)edge-disjoint from all earlier paths. We show that the costX(k)ofP(k)converges in probability to2k/n+lnn/nuniformly for allk <= n - 1. We show analogous results when the edge weights are drawn from an exponential distribution. The same results characterize the collectively cheapestkedge-disjoint paths, that is, a minimum-costk-flow. We also obtain the expectation ofX(k)conditioned on the existence ofP(k).