Decremental Dynamic Connectivity

Abstract

We consider Las Vegas randomized dynamic algorithms for on-line connectivity problems with deletions only. In particular, we show that starting from a graph with m edges and n nodes, we can maintain a spanning forest during m deletions in O( m log( n 2/ m) + n(log n) 3(log log n) 2) expected time, which is O( m) if m = Θ( n 2) and O( m log n) if m = Ω( n(log n log log n) 2). The deletions may be interspersed with connectivity queries, each of which is answered in constant time. The previous best bound was O( m log 2 n) by Henzinger and Thorup which covered both insertions and deletions. The result is based on a general randomized reduction for edge connectivity problems of many deletions-only queries to a few deletions and insertions queries. For 2-edge connectivity, the complexity is improved from O( m(log n) 5) to O( m log( n 2/ m) + n(log n) 6(log log n) 2). For the general decremental k-edge-connectivity problem, we get a total running time of O( k 2 n 2 polylog n). Here the previous best bound was O( kmn polylog n). Improved running times are also achieved for the static consensus tree problem, with applications to computational biology and relational data bases.