Efficient Algorithms for Steiner Edge Connectivity Computation and Gomory-Hu Tree Construction for Unweighted Graphs

We first consider the Steiner edge connectivity problem on an unweighted undirected or Eulerian directed graph with n vertices and m edges. This problem involves finding the edge connectivity of a specified subset S of vertices, i.e. the cardinality of the minimum cut in the graph that separates the vertices in S into two parts. We give a deterministic algorithm for this problem that runs in Õ(m + nc) time, where c is the Steiner edge connectivity of S. Our algorithm extends an algorithm due to Gabow[Gab95] that finds the minimum cut in a graph by constructing an edge-disjoint spanning tree packing. We apply this Steiner edge connectivity algorithm to solve our second problem, that of constructing Gomory-Hu trees for undirected unweighted graphs. A Gomory-Hu tree is a succinct data structure for storing pairwise edge connectivity for (or equivalently, maximum flow between) all pairs of vertices in an undirected graph. All previous algorithms for computing a Gomory-Hu tree [GH61, Gus90] use n− 1 maximum flow computations. The fastest Gomory-Hu tree algorithm on unweighted graphs with m edges and n vertices has an expected running time of Õ(mn + nF ), where F is the maximum pairwise connectivity between any pair of vertices in the graph. This algorithm uses an Õ(m + nf)-time Las Vegas algorithm for computing maximum flow due to Karger and Levine [KL02], where f is the maximum flow. We improve the time complexity of constructing a Gomory-Hu tree to Õ(mF ), which is Õ(mn) for simple graphs. The novelty of our approach is in replacing maximum flow computations by Steiner edge connectivity computations and showing that with high probability the entire time taken for the Gomory-Hu tree construction by this method is Õ(mF ). This paper combines results from [CH03], [HKP07] and [BHKP07]. This work was supported in part by NSF grants CCF 0515127 and IDM 0414763. University of Pennsylvania, Philadelphia, PA. [email protected]. Work partly done when at the Indian Institute of Science, Bangalore. New York University, New York, NY. [email protected]. Strand Life Sciences and House of Algorithms, Bangalore. [email protected]. Work partly done when at the Indian Institute of Science, Bangalore and while visiting New York University, New York, NY. Indian Institute of Science, Bangalore. [email protected]. Massachusetts Institute of Technology, Cambridge, MA. [email protected]. Work partly done when at Bell Labs Research, Bangalore and the Indian Institute of Science, Bangalore.