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

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 $O(\epsilon^{-2} \log n)$.