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- W4321320891 abstract "For a pair of positive parameters $D,chi$, a partition ${cal P}$ of the vertex set $V$ of an $n$-vertex graph $G = (V,E)$ into disjoint clusters of diameter at most $D$ each is called a $(D,chi)$ network decomposition, if the supergraph ${cal G}({cal P})$, obtained by contracting each of the clusters of ${cal P}$, can be properly $chi$-colored. The decomposition ${cal P}$ is said to be strong (resp., weak) if each of the clusters has strong (resp., weak) diameter at most $D$, i.e., if for every cluster $C in {cal P}$ and every two vertices $u,v in C$, the distance between them in the induced graph $G(C)$ of $C$ (resp., in $G$) is at most $D$. Network decomposition is a powerful construct, very useful in distributed computing and beyond. It was shown by Awerbuch etal cite{AGLP89} and Panconesi and Srinivasan cite{PS92}, that strong $(2^{O(sqrt{log n})},2^{O(sqrt{log n})})$ network decompositions can be computed in $2^{O(sqrt{log n})}$ distributed time. Linial and Saks cite{LS93} devised an ingenious randomized algorithm that constructs {em weak} $(O(log n),O(log n))$ network decompositions in $O(log^2 n)$ time. It was however open till now if {em strong} network decompositions with both parameters $2^{o(sqrt{log n})}$ can be constructed in distributed $2^{o(sqrt{log n})}$ time. In this paper we answer this long-standing open question in the affirmative, and show that strong $(O(log n),O(log n))$ network decompositions can be computed in $O(log^2 n)$ time. We also present a tradeoff between parameters of our network decomposition. Our work is inspired by and relies on the shifted shortest path approach, due to Blelloch etal cite{BGKMPT11}, and Miller etal cite{MPX13}. These authors developed this approach for PRAM algorithms for padded partitions. We adapt their approach to network decompositions in the distributed model of computation." @default.
- W4321320891 created "2023-02-19" @default.
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- W4321320891 date "2016-02-17" @default.
- W4321320891 modified "2023-09-29" @default.
- W4321320891 title "Distributed Strong Diameter Network Decomposition" @default.
- W4321320891 doi "https://doi.org/10.48550/arxiv.1602.05437" @default.
- W4321320891 hasPublicationYear "2016" @default.
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