The Gromov-Hausdorff (GH) distance is a natural way to measure distance between two metric spaces. We prove that it is NP-hard to approximate the Gromov-Hausdorff distance better than a factor of 3 for geodesic metrics on a pair of trees. We complement this result by providing a polynomial time O(min{n, √ rn})approximation algorithm for computing the GH distance between a pair of metric trees, where r is the ratio of the longest edge length in both trees to the shortest edge length. For metric trees with unit length edges, this yields an O( √ n)-approximation algorithm.The Gromov-Hausdorff distance (or GH distance for brevity) [10] is one of the most natural distance measures between metric spaces, and has been used, for example, for matching deformable shapes [3,15], and for analyzing hierarchical clustering trees [5]. Informally, the Gromov-Hausdorff distance measures the additive distortion suffered when mapping one metric space to another using a correspondence between their points. Multiple approaches have been proposed to estimate the Gromov-Hausdorff distance [3,14,15]. Despite much effort, the problem of computing, either exactly or approximately, GH distance has remained elusive. The problem is not known to be NP-hard, and computing the GH distance, even approximately, for graphic metrics 1 is at least as hard as the graph isomorphism problem. Indeed, the metrics for two graphs have GH distance 0 if and only if the two graphs are isomorphic. Motivated by this trivial hardness result, it is natural to ask whether GH distance becomes easier in more restrictive settings such as geodesic metrics over trees, where efficient algorithms are known for checking isomorphism [1].Related work. Most work on associating points between two metric spaces involves embedding a given high dimensional metric space into an infinite host space of lower dimensional metric spaces. However, there is some work on finding a bijection between points in two given finite metric spaces that minimizes typically multiplicative distortion of distances between points and their images, with some limited results on additive distortion. Kenyon et al. [13] give an optimal algorithm for minimizing the multiplicative distortion of a bijection between two equal-sized finite metric spaces, and a parameterized polynomial time algorithm that finds the optimal bijection between an arbitrary unweighted graph metric and a bounded-degree tree metric.Papadimitriou and Safra [17] show that it is NP-hard to approximate the multiplicative distortion of any bijection between two finite 3-dimensional point sets to within any additive constant or to a factor better than 3.Hall and Papadimitriou [11] discuss the additive distortion problem -given two equal-sized point sets S, T ⊂ R d , find the smallest ∆ such that there exists a bijection f :They show that it is NP-hard to approximate by a factor better than 3 in R 3 , and also give a 2-approximation for R 1 and a 5-approximation for the more general problem of embedding an arbitrary metric space onto R 1 . ...

We give a deterministic algorithm to find the minimum cut in a surface-embedded graph in near-linear time. Given an undirected graph embedded on an orientable surface of genus g, our algorithm computes the minimum cut in g O(g) n log log n time, matching the running time of the fastest algorithm known for planar graphs, due to Łącki and Sankowski, for any constant g. Indeed, our algorithm calls Łącki and Sankowski's recent O(n log log n) time planar algorithm as a subroutine.Previously, the best time bounds known for this problem followed from two algorithms for general sparse graphs: a randomized algorithm of Karger that runs in O(n log 3 n) time and succeeds with high probability, and a deterministic algorithm of Nagamochi and Ibaraki that runs in O(n 2 log n) time. We can also achieve a deterministic g O(g) n 2 log log n time bound by repeatedly applying the best known algorithm for minimum (s, t)-cuts in surface graphs. The bulk of our work focuses on the case where the dual of the minimum cut splits the underlying surface into multiple components with positive genus.

At SODA 2009, Demaine et al. presented a novel connection between binary search trees (BSTs) and subsets of points on the plane. This connection was independently discovered by Derryberry et al. As part of their results, Demaine et al. considered GreedyFuture,an offline BST algorithm that greedily rearranges the search path to minimize the cost of future searches. They showed that GreedyFuture is actually an online algorithm in their geometric view, and that there is a way to turn GreedyFuture into an online BST algorithm with only a constant factor increase in total search cost. Demaine et al. conjectured this algorithm was dynamically optimal, but no upper bounds were given in their paper. We prove the first non-trivial upper bounds for the cost of search operations using GreedyFuture including giving an access lemma similar to that found in Sleator and Tarjan's classic paper on splay trees.1 In fact, the exact optimization problem becomes NP-hard if we must access an arbitrary number of specified nodes during each search [4].

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Due to its optimality on a single machine for the problem of minimizing average flow time, Shortest-RemainingProcessing-Time (SRPT) appears to be the most natural algorithm to consider for the problem of minimizing average flow time on multiple identical machines. It is known that SRPT achieves the best possible competitive ratio on multiple machines up to a constant factor. Using resource augmentation, SRPT is known to achieve total flow time at most that of the optimal solution when given machines of speed 2 − 1 m . Further, it is known that SRPT's competitive ratio improves as the speed increases; SRPT is s-speed ii To my parents, Jan and Lonnie Fox.iii