In this paper, we show that any n point metric space can be embedded into a distribution over dominating tree metrics such that the expected stretch of any edge is O(log n). This improves upon the result of Bartal who gave a bound of O(log n log log n). Moreover, our result is existentially tight; there exist metric spaces where any tree embedding must have distortion Ω(log n)-distortion. This problem lies at the heart of numerous approximation and online algorithms including ones for group Steiner tree, metric labeling, buy-at-bulk network design and metrical task system. Our result improves the performance guarantees for all of these problems.

In this paper, we present an O(n log 3 n) time algorithm for finding shortest paths in an n-node planar graph with real weights. This can be compared to the best previous strongly polynomial time algorithm developed by Lipton, Rose, and Tarjan in 1978 which runs in O(n 3/2 ) time, and the best polynomial time algorithm developed by Henzinger, Klein, Subramanian, and Rao in 1994 which runs inÕ(n 4/3 ) time. We also present significantly improved data structures for reporting distances between pairs of nodes and algorithms for updating the data structures when edge weights change.

In this paper, we present an Ç´Ò ÐÓ ¿ Òµ time algorithm for finding shortest paths in a planar graph with real weights. This can be compared to the best previous strongly polynomial time algorithm developed by Lipton, Rose, and Tarjan in 1978 which ran in Ç´Ò ¿ ¾ µ time, and the best polynomial algorithm developed by Henzinger, Klein, Subramanian, and Rao in 1994 which ran in Ç´Ò ¿ µ time.We also present significantly improved algorithms for query and dynamic versions of the shortest path problems. Ô Òµ 1 Their algorithm, however, depended on the

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We consider the k -traveling repairmen problem, also known as the minimum latency problem, to multiple repairmen. We give a polynomial-time 8.497α-approximation algorithm for this generalization, where α denotes the best achievable approximation factor for the problem of finding the least-cost rooted tree spanning i vertices of a metric. For the latter problem, a (2 + ε)-approximation is known. Our results can be compared with the best-known approximation algorithm using similar techniques for the case k = 1, which is 3.59α. Moreover, recent work of Chaudry et al. [2003] shows how to remove the factor of α, thus improving all of these results by that factor. We are aware of no previous work on the approximability of the present problem. In addition, we give a simple proof of the 3.59α-approximation result that can be more easily extended to the case of multiple repairmen, and may be of independent interest.