The bridge-connectivity augmentation problem with a partition constraint

Authors:
Yen-Chiu Chen;Hsin-Wen Wei;Pei-Chi Huang;Wei-Kuan Shih;Tsan-sheng Hsu
Affiliations:
Department of Computer Science, National Tsing-Hua University, No 101, Section 2, Kuang-Fu Road, Hsinchu, Taiwan;Institute of Information Science, Academia Sinica, No 128, Section 2, Academia Road, Nankang, Taipei, Taiwan;Department of Computer Science, National Tsing-Hua University, No 101, Section 2, Kuang-Fu Road, Hsinchu, Taiwan;Department of Computer Science, National Tsing-Hua University, No 101, Section 2, Kuang-Fu Road, Hsinchu, Taiwan;Institute of Information Science, Academia Sinica, No 128, Section 2, Academia Road, Nankang, Taipei, Taiwan
Venue:
Theoretical Computer Science
Year:
2010

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Abstract

In this paper, we consider the augmentation problem of an undirected graph with k partitions of its vertices. The main issue is how to add a set of edges with the smallest possible cardinality so that the resulting graph is 2-edge-connected, i.e., bridge-connected, while maintaining the original partition constraint. To solve the problem, we propose a simple linear-time algorithm. To the best of our knowledge, the most efficient sequential algorithm runs in O(n(m+nlogn)logn) time. However, we show that it can also run in O(logn) parallel time on an EREW PRAM using a linear number of processors, where n is the number of vertices in the input graph. If a simple graph exists, our main algorithm ensures that it is as simple as possible.