Sparsification—a technique for speeding up dynamic graph algorithms

Abstract

We provide data strutures that maintain a graph as edges are inserted and deleted, and keep track of the following properties with the following times: minimum spanning forests, graph connectivity, graph 2-edge connectivity, and bipartiteness in time O ( n 1/2 ) per change; 3-edge connectivity, in time O ( n 2/3 ) per change; 4-edge connectivity, in time O ( n α( n )) per change; k -edge connectivity for constant k , in time O ( n log n ) per change;2-vertex connectivity, and 3-vertex connectivity, in the O ( n ) per change; and 4-vertex connectivity, in time O ( n α( n )) per change. Further results speed up the insertion times to match the bounds of known partially dynamic algorithms. All our algorithms are based on a new technique that transforms an algorithm for sparse graphs into one that will work on any graph, which we call sparsification.