Successive shortest paths in complete graphs with random edge weights

Gerke, Stefanie, Mezei, Balazs F. and Sorkin, Gregory ORCID: 0000-0003-4935-7820 (2020) Successive shortest paths in complete graphs with random edge weights. Random Structures and Algorithms, 57 (4). 1205 - 1247. ISSN 1042-9832

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Abstract

Consider a complete graph Kn 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 ln n/n, asymptotically. Define a second-shortest path P2 to be the shortest path edge-disjoint from P1, and consider more generally the shortest path Pk edge-disjoint from all earlier paths. We show that the cost Xk of Pk converges in probability to 2k/n + ln n/n 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 Xk conditioned on the existence of Pk.

Item Type:	Article
Official URL:	https://onlinelibrary.wiley.com/journal/10982418
Additional Information:	© 2020 The Authors
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics
Date Deposited:	04 Sep 2020 10:06
Last Modified:	28 Jan 2025 01:57
URI:	http://eprints.lse.ac.uk/id/eprint/106487

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