Successive shortest paths in complete graphs with random edge weights

11/04/2019

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by   Stefanie Gerke, et al.

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Consider a complete graph K_n with edge weights drawn independently from a uniform distribution U(0,1). The weight of the shortest (minimum-weight) path P_1 between two given vertices is known to be ln n / n, asymptotically. We define a second-shortest path P_2 to be the shortest path edge-disjoint from P_1, and consider more generally the shortest path P_k edge-disjoint from all earlier paths. We show that the cost X_k of P_k is asymptotically (2k+ln n) / n. Specifically, X_k / (2k/n+ln n/n) p→ 1 uniformly for all k ≤ n-1. We also show an analogous result when the edge weights are drawn from an exponential distribution.