A tight lower bound for the steiner point removal problem on trees

Authors:
T.-H. Chan;Donglin Xia;Goran Konjevod;Andrea Richa
Affiliations:
Computer Science Department, Carnegie Mellon University;Department of Computer Science and Engineering, Arizona State University;Department of Computer Science and Engineering, Arizona State University;Department of Computer Science and Engineering, Arizona State University
Venue:
APPROX'06/RANDOM'06 Proceedings of the 9th international conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th international conference on Randomization and Computation
Year:
2006

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Abstract

Gupta (SODA'01) considered the Steiner Point Removal (SPR) problem on trees. Given an edge-weighted tree T and a subset S of vertices called terminals in the tree, find an edge-weighted tree TS on the vertex set S such that the distortion of the distances between vertices in S is small. His algorithm guarantees that for any finite tree, the distortion incurred is at most 8. Moreover, a family of trees, where the leaves are the terminals, is presented such that the distortion incurred by any algorithm for SPR is at least 4(1 – o(1)). In this paper, we close the gap and show that the upper bound 8 is essentially tight. In particular, for complete binary trees in which all edges have unit weight, we show that the distortion incurred by any algorithm for the SPR problem must be at least 8 (1 – o(1)).