# An (O)over-tilde(mn) Gomory-Hu Tree Construction Algorithm for Unweighted Graphs

Bhalgat, Anand and Hariharan, Ramesh and Kavitha, Telikepalli and Panigrahi, Debmalya (2007) An (O)over-tilde(mn) Gomory-Hu Tree Construction Algorithm for Unweighted Graphs. In: ACM, Jun 11-13, San Diegeo, pp. 605-614.

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## Abstract

We present a fast algorithm for computing a Gomory-Hu tree or cut tree for an unweighted undirected graph G = (V, E). The expected running time of our algorithm is (O) over tilde (mc) where vertical bar E vertical bar = m and c is the maximum u-v edge connectivity, where u, v is an element of V. When the input graph is also simple (i.e., it has no parallel edges), then the u-v edge connectivity for each pair of vertices u and v is at most n - 1; so the expected run-ning time of our algorithm for simple unweighted graphs is (O) over tilde (mn). All the algorithms currently known for constructing a Gomory-Hu tree [8, 9] use n - 1 minimum s-t cut (i.e., max flow) subroutines. This in conjunction with the current fastest (O) over tilde (n(20/9)) max flow algorithm due to Karger and Levine[11] yields the current best running time of (O) over tilde (n(20/9)n) for Gomory-Hu tree construction on simple unweighted graphs with m edges and n vertices. Thus we present the first (O) over tilde (mn) algorithm for constructing a Gomory-Hu tree for simple unweighted graphs. We do not use a max flow subroutine here; we present an efficient tree packing algorithm for computing Steiner edge connectivity and use this algorithm as our main subroutine. The advantage in using a tree packing algorithm for constructing a Gomory-Hu tree is that the work done in computing a minimum Steiner cut for a Steiner set S subset of V can be reused for computing a minimum Steiner cut for certain Steiner sets S' subset of S.

Item Type: Conference Proceedings Publisher Copyright of this article belongs to Association for Computing Machinery. Steiner edge connectivity;cut trees;Gomory-Hu trees; min cuts Division of Electrical Sciences > Computer Science & Automation (Formerly, School of Automation) 22 Jul 2009 06:47 22 Jul 2009 06:47 http://eprints.iisc.ernet.in/id/eprint/21483