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Steiner points in tree metrics don't (really) help

Content Provider	ACM Digital Library
Author	Gupta, Anupam
Abstract	Consider an edge-weighted tree T = (V, E, w : E ↣ $R^{+}),$ in which a subset R of the nodes (called the required nodes) are colored red and the remaining nodes in S = V\R are colored black (and called the Steiner nodes). The shortest-path distance according to the edge-weights defines a metric dT on the vertex set V.We now ask the following question: Is it possible to define another weighted tree T* = (R, E, w : E* ↣ $R^{+}),$ this time on just the red vertices so that the shortest-path metric dT* induced by T* on the vertices in R is “close” to the metric dT restricted to the red vertices? I.e., does there exist a weighted tree T* = (R, E, c) and a (small) constant α such that dT(u, v) ≤ dT* (u, v) ≤ α dT(u, v) for any two red vertices u, v ∈ R?We answer this question in the affirmative, and give a linear time algorithm to obtain a tree T* with α ≤ 8. We also give two applications of this result: an upper bound, in which we show that emulating multicasts using unicasts can be almost as good as general multicasts for certain performance measures; and a lower bound, in which we give a simple combinatorial proof of the fact that the metric generated by a graph of girthg must suffer a distortion of at least &OHgr;(g) when approximated by a tree.
Starting Page	220
Ending Page	227
Page Count	8
File Format	PDF
ISBN	0898714907
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2001-01-09
Access Restriction	Subscribed
Content Type	Text
Resource Type	Article

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