Brief Announcement: Near Optimal Bounds for Replacement Paths and Related Problems in the CONGEST Model

Vignesh Manoharan, Vijaya Ramachandran
2022 Proceedings of the 2022 ACM Symposium on Principles of Distributed Computing  
We present round complexity results in the CONGEST model for Replacement Paths (RPaths), Minimum Weight Cycle (MWC), and All Nodes Shortest Cycles (ANSC). We study these fundamental problems in both directed and undirected graphs, weighted and unweighted. Many of our results are optimal to within a polylog factor: For an 𝑛-node graph 𝐺 we establish near linear lower and upper bounds for RPaths if 𝐺 is directed and weighted, and for MWC and ANSC if 𝐺 is weighted, directed or undirected; near √ 𝑛
more » ... lower and upper bounds for undirected weighted RPaths; and Θ(𝐷) bound for undirected unweighted RPaths. We also present lower and upper bounds for approximation: a (2 − (1/𝑔))-approximation algorithm for undirected unweighted MWC that runs in Õ ( √ 𝑛 + 𝐷) rounds, improving on the previous best bound of Õ ( √ 𝑛𝑔 +𝐷) rounds, where 𝑔 is the MWC length, and a (1 + 𝜖)-approximation algorithm for directed weighted RPaths and (2 + 𝜖)-approximation for weighted undirected MWC, for any constant 𝜖 > 0, that beat the round complexity lower bound for an exact solution.
doi:10.1145/3519270.3538473 fatcat:s3b2sphdp5ehvocrsuykmaoqyi