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 α if its edges can be partitioned into at most α forests and that a planar graph has arboricity α = 3. Estimating the size of the maximum matching in such graphs has been a focus of recent data stream research [1–3,5, 7]. See also [6] for a survey of the general area of graph algorithms in the stream model. A surprising result on this problem was recently proved by Cormode et al. [3]. They designed an ingenious algorithm that returned a (22.5α + 6)(1 + ǫ) approximation using a single pass over the edges of the graph (ordered arbitrarily) and O(ǫα · log n · log1+ǫ n) space . We improve the approximation factor to (α+ 2)(1 + ǫ) via a tighter analysis and show that, with a modification of their algorithm, the space required can be reduced to O(ǫ log n).