# 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 Annual ACM Symposium on Theory of Computing Annual ACM Symposium on Theory of Computing Association for Computing Machinery 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 22 Jul 2009 06:47 22 Jul 2009 06:47 http://eprints.iisc.ac.in/id/eprint/21483