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

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 such that dT (u, v) dT (u, v) 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 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 g...

• Type: Conference
• Year: 2001
• Where: SODA
• Authors: Anupam Gupta
• Keywords: Algorithms, Metric DT, Red Vertices, SODA 2001, Weighted Tree