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 Ω(n) space, where n is the number of vertices, existing works mainly focused on designing O(n · poly log n) 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 εn of a general graph and (1 + ε)-approximating the weight of the minimum spanning tree of a connected graph with bounded edge weights, for any small constant ε > 0. The latter improves upon the previous O(n · poly log n) 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 ε-far from having the property. We consider the problem of testing k-edge connectivity, k-vertex connectivity, cyclefreeness and bipartiteness (of planar graphs), for which, we provide algorithms using roughly O(n 1−ε · poly log n) space, which is o(n) for any constant ε. To complement our algorithms, we present Ω(n 1−O(ε) ) space lower bounds for these problems, which show that such a dependence on ε is necessary.
There has also been work on property testing in the streaming model REF .
Dynamic Graph Stream Algorithms in o(n) Space
Algorithmica
Algorithmica
Mathematics Computer Science