Improved Algorithm for Min-Cuts in Distributed Networks
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In this thesis, we present fast deterministic algorithm to find small cuts in distributed networks. Finding small min-cuts for a network is essential for ensuring the quality of service and reliability. Throughout this thesis, we use the CONGEST model which is a typical message passing model used to design and analyze algorithms in distributed networks. We survey various algorithmic techniques in the CONGEST model and give an overview of the recent results to find cuts. We also describe elegant graph theoretic ideas like cut spaces and cycle spaces that provide useful intuition upon which our work is built. Our contribution is a novel fast algorith to find small cuts. Our algorithm relies on a new characterization of trees and cuts introduced in this thesis. Our algorithm is built upon several new algorithmic ideas that, when coupled with our characterization of trees and cuts, help us to find the required min-cuts. Our novel techniques include a tree restricted semigroup function (TRSF), a novel sketching technique, and a layered algorithm. TRSF is defined with respect to a spanning tree and is based on a commutative semigroup. This simple yet powerful technique helps us to deterministically find min-cuts of size one (bridges) and min-cuts of size two optimally. Our sketching technique samples a small but relevant vertex set which is enough to find small min-cuts in certain cases. Our layered algorithm finds min-cuts in smaller sub-graphs pivoted by nodes at different levels in a spanning tree and uses them to make the decision about the min-cuts in the complete graph. This is interesting because it enables us to show that even for a global property like finding min-cuts, local information can be exploited in a coordinated manner.
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