Low-Congestion Shortcuts in Constant Diameter Graphs
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Low congestion shortcuts, introduced by Ghaffari and Haeupler (SODA 2016), provide a unified framework for global optimization problems in the congest model of distributed computing. Roughly speaking, for a given graph G and a collection of vertex-disjoint connected subsets S_1,…, S_ℓ⊆ V(G), (c,d) low-congestion shortcuts augment each subgraph G[S_i] with a subgraph H_i ⊆ G such that: (i) each edge appears on at most c subgraphs (congestion bound), and (ii) the diameter of each subgraph G[S_i] ∪ H_i is bounded by d (dilation bound). It is desirable to compute shortcuts of small congestion and dilation as these quantities capture the round complexity of many global optimization problems in the congest model. For n-vertex graphs with constant diameter D=O(1), Elkin (STOC 2004) presented an (implicit) shortcuts lower bound with c+d=Ω(n^(D-2)/(2D-2)). A nearly matching upper bound, however, was only recently obtained for D ∈{3,4} by Kitamura et al. (DISC 2019). In this work, we resolve the long-standing complexity gap of shortcuts in constant diameter graphs, originally posed by Lotker et al. (PODC 2001). We present new shortcut constructions which match, up to poly-logarithmic terms, the lower bounds of Das-Sarma et al. As a result, we provide improved and existentially optimal algorithms for several network optimization tasks in constant diameter graphs, including MST, (1+ϵ)-approximate minimum cuts and more.
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