A Spectrum of Time–Space Trade-offs for Undirecteds-tConnectivity

Abstract

We present a family of randomized algorithms that enjoys a wide range of time–space trade-offs in deciding undirected S-T-connectivity. Our trade-offs cover the whole range between breadth first search and the random walk procedure of Aleliunaset al., and achieve a time-space product ofÕ(mn) (wherenis the number of vertices in the graph,mis the number of edges, andÕnotation is used in order to suppress logarithmic terms). Moreover, we obtain improved time–space trade-offs ofÕ(n2) for regular graphs. A convenient and informative way of expressing our trade-offs, that implies the trade-offs stated above, is asÕ((∑ni=1di)(∑ni=11/di)), wherediis the degree of vertexiin the input graph. In constructing our algorithms and analysing them, we build upon earlier work of Broderet al.(who achieved a time–space trade-off ofÕ(m2)), Barnes and Feige (who achieved a time–space trade-off ofÕ(m3/2n1/2)), and Aldous. In passing, we also improve previous results regarding the rate at which a random walk discovers new vertices in a graph.