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 P₁ between two given vertices is known to be , asymptotically. Define a second‐shortest path P₂ to be the shortest path edge‐disjoint from P₁, 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 converges in probability to uniformly for all k ≤ n − 1. We show analogous results when the edge weights are drawn from an exponential distribution. The same results characterize the collectively cheapest k edge‐disjoint paths, that is, a minimum‐cost k‐flow. We also obtain the expectation of X_k conditioned on the existence of P_k.

中文翻译：

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完整图形中具有连续边权重的连续最短路径

考虑具有独立于均匀分布U（0，1）绘制的边缘权重的完整图形K _n。已知两个给定顶点之间最短（最小权重）路径P ₁的权重是渐近的。将第二条最短路径P ₂定义为与P ₁的最短路径边缘不相交，并更普遍地考虑与所有较早路径的最短路径P _k边缘不相交。我们发现，成本X _ķ的P _ķ概率收敛于 统一所有ķ  ≤  ñ − 1.当边缘权重是从指数分布中得出时，我们显示出类似的结果。相同的结果表征了总体上最便宜的k条边缘不相交的路径，即最小成本的k流。我们还根据P _k的存在获得X _k的期望。