A Note on Logarithmic Space Stream Algorithms for Matchings in Low Arboricity Graphs

TL;DR

This paper improves the approximation factor and space efficiency of a streaming algorithm for estimating maximum matchings in low arboricity graphs, such as planar graphs, using a single pass.

Contribution

It provides a tighter analysis that reduces the approximation factor and modifies the existing algorithm to lower the space complexity significantly.

Findings

Improved approximation factor to (lpha+2)(1+)

Reduced space complexity to O(^{-2} \, ext{log} \, n)

Applicable to low arboricity graphs like planar graphs

Abstract

We present a data stream algorithm for estimating the size of the maximum matching of a low arboricity graph. Recall that a graph has arboricity $\alpha$ if its edges can be partitioned into at most $\alpha$ forests and that a planar graph has arboricity $\alpha=3$. Estimating the size of the maximum matching in such graphs has been a focus of recent data stream research. A surprising result on this problem was recently proved by Cormode et al. They designed an ingenious algorithm that returned a $(22.5\alpha+6)(1+\epsilon)$ approximation using a single pass over the edges of the graph (ordered arbitrarily) and $O(\epsilon^{-2}\alpha \cdot \log n \cdot \log_{1+\epsilon} n)$ space. In this note, we improve the approximation factor to $(\alpha+2)(1+\epsilon)$ via a tighter analysis and show that, with a modification of their algorithm, the space required can be reduced to $O(\epsilon^{-2} \log n)$.