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Nearly Optimal ExpectedCase Planar Point Location
"... We consider the planar point location problem from the perspective of expected search time. We are given a planar polygonal subdivision S and for each polygon of the subdivision the probability that a query point lies within this polygon. The goal is to compute a search structure to determine which ..."
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We consider the planar point location problem from the perspective of expected search time. We are given a planar polygonal subdivision S and for each polygon of the subdivision the probability that a query point lies within this polygon. The goal is to compute a search structure to determine which cell of the subdivision contains a given query point, so as to minimize the expected search time. This is a generalization of the classical problem of computing an optimal binary search tree for onedimensional keys. In the onedimensional case it has long been known that the entropy H of the distribution is the dominant term in the lower bound on the expectedcase search time, and further there exist search trees achieving expected search times of at most H + 2. Prior to this work, there has been no known structure for planar point location with an expected search time better than 2H, and this result required strong assumptions on the nature of the query point distribution. Here we present a data structure whose expected search time is nearly equal to the entropy lower bound, namely H + o(H). The result holds for any polygonal subdivision in which the number of sides of each of the polygonal cells is bounded, and there are no assumptions on the query distribution within each cell. We extend these results to subdivisions with convex cells, assuming a uniform query distribution within each cell.
An Algorithmic Approach to Social Networks
 PhD thesis at MIT References 118 Science and Artificial Intelligence Laboratory
, 2005
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Navigating LowDimensional and Hierarchical Population Networks
 In 14th European Symposium on Algorithm (ESA), LNCS 4168
, 2006
"... Social networks are navigable small worlds, in which two arbitrary people are likely connected by a short path of intermediate friends that can be found by a “decentralized ” routing algorithm using only local information. We develop a model of social networks based on an arbitrary metric space of p ..."
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Cited by 9 (4 self)
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Social networks are navigable small worlds, in which two arbitrary people are likely connected by a short path of intermediate friends that can be found by a “decentralized ” routing algorithm using only local information. We develop a model of social networks based on an arbitrary metric space of points, with population density varying across the points. We consider rankbased friendships, where the probability that person u befriends person v is inversely proportional to the number of people who are closer to u than v is. Our main result is that greedy routing can find a short path (of expected polylogarithmic length) from an arbitrary source to a randomly chosen target, independent of the population densities, as long as the doubling dimension of the metric space of locations is low. We also show that greedy routing finds short paths with good probability in treebased metrics with varying population distributions. 1
Achieving Spatial Adaptivity while Finding Approximate Nearest Neighbors
"... We present the first spatially adaptive data structure that answers approximate nearest neighbor (ANN) queries to points that reside in a geometric space of any constant dimension d. The Ltnorm approximation ratio is O(d 1+1/t), and the running time for a query q is O(d 2 lg δ(p, q)), where p is th ..."
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Cited by 4 (2 self)
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We present the first spatially adaptive data structure that answers approximate nearest neighbor (ANN) queries to points that reside in a geometric space of any constant dimension d. The Ltnorm approximation ratio is O(d 1+1/t), and the running time for a query q is O(d 2 lg δ(p, q)), where p is the result of the preceding query and δ(p, q) is the number of input points in a suitablysized box containing p and q. Our data structure has O(dn) size and requires O(d 2 n lg n) preprocessing time, where n is the number of points in the data structure. The size of the bounding box for δ depends on d, and our results rely on the Random Access Machine (RAM) model with word size Θ(lg n). 1
DISTRIBUTIONSENSITIVE POINT LOCATION IN CONVEX SUBDIVISIONS
"... A data structure is presented for point location in convex planar subdivisions when the distribution of queries is known in advance. The data structure has an expected query time that is within a constant factor of optimal. ..."
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Cited by 4 (3 self)
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A data structure is presented for point location in convex planar subdivisions when the distribution of queries is known in advance. The data structure has an expected query time that is within a constant factor of optimal.
Adaptive Binary Search Trees
, 2009
"... A ubiquitous problem in the field of algorithms and data structures is that of searching for an element from an ordered universe. The simple yet powerful binary search tree (BST) model provides a rich family of solutions to this problem. Although BSTs require Ω(lg n) time per operation in the wors ..."
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Cited by 1 (0 self)
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A ubiquitous problem in the field of algorithms and data structures is that of searching for an element from an ordered universe. The simple yet powerful binary search tree (BST) model provides a rich family of solutions to this problem. Although BSTs require Ω(lg n) time per operation in the worst case, various adaptive BST algorithms are capable of exploiting patterns in the sequence of queries to achieve tighter, inputsensitive, bounds that can be o(lg n) in many cases. This thesis furthers our understanding of what is achievable in the BST model along two directions. First, we make progress in improving instancespecific lower bounds in the BST model. In particular, we introduce a framework for generating lower bounds on the cost that any BST algorithm must pay to execute a query sequence,
ENTROPY, TRIANGULATION, AND POINT LOCATION IN PLANAR SUBDIVISIONS
, 2009
"... A data structure is presented for point location in connected planar subdivisions when the distribution of queries is known in advance. The data structure has an expected query time that is within a constant factor of optimal. More specifically, an algorithm is presented that preprocesses a connecte ..."
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Cited by 1 (1 self)
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A data structure is presented for point location in connected planar subdivisions when the distribution of queries is known in advance. The data structure has an expected query time that is within a constant factor of optimal. More specifically, an algorithm is presented that preprocesses a connected planar subdivision G of size n and a query distribution D to produce a point location data structure for G. The expected number of pointline comparisons performed by this data structure, when the queries are distributed according to D, is ˜ H + O ( ˜ H2/3 + 1) where ˜ H = ˜ H(G, D) is a lower bound on the expected number of pointline comparisons performed by any linear decision tree for point location in G under the query distribution D. The preprocessing algorithm runs in O(n log n) time and produces a data structure of size O(n). These results are obtained by creating a Steiner triangulation of G that has nearminimum entropy.