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Finding the maximal empty disk containing a query point
 In Proc. 28th Annual ACM Symp. Comput. Geom. (SOCG) (2012
"... Let P be a set of n points in the plane. We present an efficient algorithm for preprocessing P, so that, for a given query point q, we can quickly report the largest disk that contains q but its interior is disjoint from P. The storage required by the data structure is O(nlogn), the preprocessing co ..."
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Let P be a set of n points in the plane. We present an efficient algorithm for preprocessing P, so that, for a given query point q, we can quickly report the largest disk that contains q but its interior is disjoint from P. The storage required by the data structure is O(nlogn), the preprocessing cost is O(nlog 2 n), and a query takes O(log 2 n) time. We also present an alternative solution with an improved query cost and with slightly worse storage and preprocessing requirements. 1
Localized geometric query problems
 CoRR
"... Abstract. A new class of geometric query problems are studied in this paper. We are required to preprocess a set of geometric objects P in the plane, so that for any arbitrary query point q, the largest circle that contains q but does not contain any member of P, can be reported efficiently. The geo ..."
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Abstract. A new class of geometric query problems are studied in this paper. We are required to preprocess a set of geometric objects P in the plane, so that for any arbitrary query point q, the largest circle that contains q but does not contain any member of P, can be reported efficiently. The geometric sets that we consider are point sets and boundaries of simple polygons.
Maximal empty boxes amidst random points
, 2013
"... We show that the expected number of maximal empty axisparallel boxes amidst n random points in the unit hypercube [0,1] d in R d is (1±o(1)) (2d−2)! (d−1)! nlnd−1 n, if d is fixed. This estimate is relevant for analyzing the performance of exact algorithms for computing the largest empty axisparal ..."
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We show that the expected number of maximal empty axisparallel boxes amidst n random points in the unit hypercube [0,1] d in R d is (1±o(1)) (2d−2)! (d−1)! nlnd−1 n, if d is fixed. This estimate is relevant for analyzing the performance of exact algorithms for computing the largest empty axisparallel box amidst n given points in an axisparallel box R, especially the algorithms that proceed by examining all maximal empty boxes. Our method for bounding the expected number of maximal empty boxes also shows that the expected number of maximal empty orthants determined by n random points in R d is (1 ± o(1))ln d−1 n, if d is fixed. This estimate is related to the expected number of maximal (or minimal) points amidst random points, and has application to algorithms for colored orthogonal range counting.
A Fast Implementation of FRDijkstra
"... The shortest path problem is the problem of finding the shortest distance from a specified source node to all other nodes in the graph. The best known algorithm for arbitrary graphs with realvalued weights is known as BellmanFord, and runs in O(nm) time, where m and n are the number of vertices an ..."
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The shortest path problem is the problem of finding the shortest distance from a specified source node to all other nodes in the graph. The best known algorithm for arbitrary graphs with realvalued weights is known as BellmanFord, and runs in O(nm) time, where m and n are the number of vertices and nodes in the graph, respectively. For an arbitrary graph with nonnegative weights, the
REPLACEMENT PATHS VIA ROW MINIMA OF CONCISE MATRICES∗
"... Abstract. Matrix M is kconcise if the finite entries of each column of M consist of k or fewer intervals of identical numbers. We give an O(n + m)time algorithm to compute the row minima of any O(1)concise n×m matrix. Our algorithm yields the first O(n+m)time reductions from the replacementpath ..."
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Abstract. Matrix M is kconcise if the finite entries of each column of M consist of k or fewer intervals of identical numbers. We give an O(n + m)time algorithm to compute the row minima of any O(1)concise n×m matrix. Our algorithm yields the first O(n+m)time reductions from the replacementpaths problem on an nnode medge undirected graph (respectively, directed acyclic graph) to the singlesource shortestpaths problem on an O(n)node O(m)edge undirected graph (respectively, directed acyclic graph). That is, we prove that the replacementpaths problem is no harder than the singlesource shortestpaths problem on undirected graphs and directed acyclic graphs. Moreover, our lineartime reductions lead to the first O(n + m)time algorithms for the replacementpaths problem on the following classes of nnode medge graphs: (1) undirected graphs in the wordRAM model of computation, (2) undirected planar graphs, (3) undirected minorclosed graphs, and (4) directed acyclic graphs.
Computational Geometry Column 60
, 2014
"... This column is devoted to maximal empty axisparallel rectangles amidst a point set. In particular, among these, maximumarea rectangles are of interest. ..."
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This column is devoted to maximal empty axisparallel rectangles amidst a point set. In particular, among these, maximumarea rectangles are of interest.
Faster Shortest Paths in Dense Distance Graphs, with Applications
, 2014
"... We show how to combine two techniques for efficiently computing shortest paths in directed planar graphs. The first is the lineartime shortestpath algorithm of Henzinger, Klein, Subramanian, and Rao [STOC’94]. The second is Fakcharoenphol and Rao’s algorithm [FOCS’01] for emulating Dijkstra’s alg ..."
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We show how to combine two techniques for efficiently computing shortest paths in directed planar graphs. The first is the lineartime shortestpath algorithm of Henzinger, Klein, Subramanian, and Rao [STOC’94]. The second is Fakcharoenphol and Rao’s algorithm [FOCS’01] for emulating Dijkstra’s algorithm on the dense distance graph (DDG). A DDG is defined for a decomposition of a planar graph G into regions of at most r vertices each, for some parameter r < n. The vertex set of the DDG is the set of Θ(n/ r) vertices of G that belong to more than one region (boundary vertices). The DDG has Θ(n) arcs, such that distances in the DDG are equal to the distances in G. Fakcharoenphol and Rao’s implementation of Dijkstra’s algorithm on the DDG (nicknamed FRDijkstra) runs in O(n log(n)r−1/2 log r) time, and is a key component in many stateoftheart planar graph algorithms for shortest paths, minimum cuts, and maximum flows. By combining these two techniques we remove the logn dependency in the running time of the shortestpath algorithm, making it O(nr−1/2 log2 r). This work is part of a research agenda that aims to develop new techniques that would lead to faster, possibly lineartime, algorithms for problems such as minimumcut, maximumflow, and shortest paths with negative arc lengths. As immediate applications, we show how to compute maximum flow in directed weighted planar graphs in O(n log p) time, where p is the minimum number of edges on any path from the source to the sink. We also show how to compute any part of the DDG that corresponds to a region with r vertices and k boundary vertices in O(r log k) time, which is faster than has been previously known for small values of k.