Log Diameter Rounds MST Verification and Sensitivity in MPC
Pages 269 - 280
Abstract
We consider two natural variants of the problem of minimum spanning tree (MST) of a graph in the parallel setting: MST verification (verifying if a given tree is an MST) and the sensitivity analysis of an MST (finding the lowest cost replacement edge for each edge of the MST). These two problems have been studied extensively for sequential algorithms and for parallel algorithms in the PRAM model of computation. In this paper, we extend the study to the standard model of Massive Parallel Computation (MPC).
It is known that for graphs of diameter D, the connectivity problem can be solved in O(log D + log log n) rounds on an MPC with low local memory (each machine can store only O(nδ ) words for an arbitrary constant δ > 0) and with linear global memory, that is, with optimal utilization. However, for the related task of finding an MST, we need Ω(log DMST ) rounds, where DMST denotes the diameter of the minimum spanning tree. The state of the art upper bound for MST is O(log n) rounds; the result follows by simulating existing PRAM algorithms. While this bound may be optimal for general graphs, the benchmark of connectivity and lower bound for MST suggest the target bound of O(log DMST ) rounds, or possibly O(log DMST + log log n) rounds. As for now, we do not know if this bound is achievable for the MST problem on an MPC with low local memory and linear global memory. In this paper, we show that MST verification and sensitivity analysis of an MST can be completed in O(log DT) rounds on an MPC with low local memory and with linear global memory, that is, with optimal utilization; here DT is the diameter of the input "candidate MST" T. The algorithms asymptotically match our lower bound, conditioned on the 1-vs-2-cycle conjecture.
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Index Terms
- Log Diameter Rounds MST Verification and Sensitivity in MPC
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Published: 17 June 2024
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