Dynamic Graph Stream Algorithms in o(n) Space

In this paper we study graph problems in the dynamic streaming model, where the input is defined by a sequence of edge insertions and deletions. As many natural problems require [Formula: see text] space, where n is the number of vertices, existing works mainly focused on designing [Formula: see text] space algorithms. Although sublinear in the number of edges for dense graphs, it could still be too large for many applications (e.g., n is huge or the graph is sparse). In this work, we give single-pass algorithms beating this space barrier for two classes of problems. We present o(n) space algorithms for estimating the number of connected components with additive error [Formula: see text] of a general graph and [Formula: see text] -approximating the weight of the minimum spanning tree of a connected graph with bounded edge weights, for any small constant [Formula: see text] . The latter improves upon the previous [Formula: see text] space algorithm given by Ahn et al. (SODA 2012) for the same class of graphs. We initiate the study of approximate graph property testing in the dynamic streaming model, where we want to distinguish graphs satisfying the property from graphs that are [Formula: see text] -far from having the property. We consider the problem of testing k-edge connectivity, k-vertex connectivity, cycle-freeness and bipartiteness (of planar graphs), for which, we provide algorithms using roughly [Formula: see text] space, which is o(n) for any constant [Formula: see text] . To complement our algorithms, we present [Formula: see text] space lower bounds for these problems, which show that such a dependence on [Formula: see text] is necessary.