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Journal of Information Science and Engineering, Vol. 22 No. 6, pp. 1409-1425 (November 2006)

FasterDSP: A Faster Approximation Algorithm for Directed Steiner Tree Problem*

Ming-I Hsieh, Eric Hsiao-Kuang Wu and Meng-Feng Tsai
Department of Computer Science and Information Engineering
National Central University
Chungli, 320 Taiwan
E-mail: mihs@wmlab.csie.ncu.edu.tw
E-mail: {hsiao; mftsai}@csie.ncu.edu.tw

Given a weighted directed graph G = (V, E, c), where c : E ¡÷ R+ is an edge cost function, a subset X of vertices (terminals), and a root vertex vr, the directed Steiner tree problem (DSP) asks for a minimum-cost tree which spans the paths from root vertex vr to each terminal. Charikar et al.'s algorithm is well-known for this problem. It achieves an approximation guarantee of l(l - 1)kl in O(nlk2l) time for any fixed level l > 1, where l is the level of the tree produced by the algorithm, n is the number of vertices, |V|, and k is the number of terminals, |X|. However, it requires a great amount of computing power, and there are some problems in the proof of the approximation guarantee of the algorithm. This paper provides a faster approximation algorithm improving Charikar et al.'s DSP algorithm with a better time complexity, O(nlkl + n2k + nm), where m is the number of edges, and an amended 8k -£_lnk factor for the 2-level Steiner tree, where £_ = 6 - 2 = 0.4494.

Keywords: directed Steiner tree, approximation algorithm, multicast distribution tree, content delivery network, multicast routing

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Received September 21, 2004; revised December 10, 2004; accepted January 17, 2005.
Communicated by Gen-Huey Chen.
* This work was supported by Chungshan Institute of Science and Technology under the ¡§Wide Band Mobile Communication System Integration Technology¡¨ project, No. 95-EC-17-A-03-R7-02C5 and NSC 95-2524- S-008-001 project.