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Parallel Dynamic Maximal Matching

Mohsen Ghaffari, Anton Trygub

TL;DR

This work presents the first (randomized) parallel dynamic algorithm for maximal matching, which can process an arbitrary number of updates simultaneously and readily generalizes to maximal matching in hypergraphs of rank r with a poly(r) increase in work, while retaining the poly(łog n) depth.

Abstract

We present the first (randomized) parallel dynamic algorithm for maximal matching, which can process an arbitrary number of updates simultaneously. Given a batch of edge deletion or insertion updates to the graph, our parallel algorithm adjusts the maximal matching to these updates in $poly(\log n)$ depth and using $poly(\log n)$ amortized work per update. That is, the amortized work for processing a batch of $k$ updates is $kpoly(\log n)$, while all this work is done in $poly(\log n)$ depth, with high probability. This can be seen as a parallel counterpart of the sequential dynamic algorithms for constant-approximate and maximal matching [Onak and Rubinfeld STOC'10; Baswana, Gupta, and Sen FOCS'11; and Solomon FOCS'16]. Our algorithm readily generalizes to maximal matching in hypergraphs of rank $r$ -- where each hyperedge has at most $r$ endpoints -- with a $poly(r)$ increase in work, while retaining the $poly(\log n)$ depth.

Parallel Dynamic Maximal Matching

TL;DR

This work presents the first (randomized) parallel dynamic algorithm for maximal matching, which can process an arbitrary number of updates simultaneously and readily generalizes to maximal matching in hypergraphs of rank r with a poly(r) increase in work, while retaining the poly(łog n) depth.

Abstract

We present the first (randomized) parallel dynamic algorithm for maximal matching, which can process an arbitrary number of updates simultaneously. Given a batch of edge deletion or insertion updates to the graph, our parallel algorithm adjusts the maximal matching to these updates in depth and using amortized work per update. That is, the amortized work for processing a batch of updates is , while all this work is done in depth, with high probability. This can be seen as a parallel counterpart of the sequential dynamic algorithms for constant-approximate and maximal matching [Onak and Rubinfeld STOC'10; Baswana, Gupta, and Sen FOCS'11; and Solomon FOCS'16]. Our algorithm readily generalizes to maximal matching in hypergraphs of rank -- where each hyperedge has at most endpoints -- with a increase in work, while retaining the depth.
Paper Structure (48 sections, 15 theorems, 16 equations)

This paper contains 48 sections, 15 theorems, 16 equations.

Table of Contents

  1. Introduction and related work
  2. Parallel computation
  3. Dynamic parallel algorithms
  4. Maximal matching, and sequential dynamic algorithms.
  5. Our result---dynamic parallel maximal matching.
  6. Preliminaries
  7. Hypergraphs and maximal matching.
  8. Update model.
  9. Parallel dictionary.
  10. Static Parallel Hypergraph Maximal Matching
  11. Luby's MIS algorithm.
  12. Algorithm
  13. Intuitive discussions and algorithm overview
  14. Difficulties in parallelism, when handling multiple updates simultaneously.
  15. General structure: levels, temporary deletions, and data structures
  16. ...and 33 more sections

Key Result

Theorem 1.1

There is a randomized parallel dynamic algorithm for hypergraph maximal matching, which adapts the maximal matching to any batch of simultaneous edge insertion/deletion updates. The batch update is processed, with high probability, in $\operatorname{\text{\rm poly}}(\log (nM))$ depth and using $poly

Theorems & Definitions (35)

  • Theorem 1.1: Imprecise statement; see \ref{['thm:main']} for precise bounds
  • Theorem 2.1: Luby luby
  • Theorem 2.2
  • proof
  • Claim 3.3
  • proof
  • Claim 3.4
  • proof
  • Theorem 4.1
  • Lemma 4.2
  • ...and 25 more