Parallel Approximate Maximum Flows in Near-Linear Work and Polylogarithmic Depth
CoRR(2024)
摘要
We present a parallel algorithm for the (1-ϵ)-approximate maximum
flow problem in capacitated, undirected graphs with n vertices and m edges,
achieving O(ϵ^-3polylog n) depth and O(m ϵ^-3polylog n) work in the PRAM model. Although near-linear time sequential
algorithms for this problem have been known for almost a decade, no parallel
algorithms that simultaneously achieved polylogarithmic depth and near-linear
work were known.
At the heart of our result is a polylogarithmic depth, near-linear work
recursive algorithm for computing congestion approximators. Our algorithm
involves a recursive step to obtain a low-quality congestion approximator
followed by a "boosting" step to improve its quality which prevents a
multiplicative blow-up in error. Similar to Peng [SODA'16], our boosting step
builds upon the hierarchical decomposition scheme of Räcke, Shah, and
Täubig [SODA'14].
A direct implementation of this approach, however, leads only to an algorithm
with n^o(1) depth and m^1+o(1) work. To get around this, we introduce a
new hierarchical decomposition scheme, in which we only need to solve maximum
flows on subgraphs obtained by contracting vertices, as opposed to
vertex-induced subgraphs used in Räcke, Shah, and Täubig [SODA'14]. In
particular, we are able to directly extract congestion approximators for the
subgraphs from a congestion approximator for the entire graph, thereby avoiding
additional recursion on those subgraphs. Along the way, we also develop a
parallel flow-decomposition algorithm that is crucial to achieving
polylogarithmic depth and may be of independent interest.
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