9 research outputs found

    Data Structures for Halfplane Proximity Queries and Incremental Voronoi Diagrams

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    We consider preprocessing a set SS of nn points in convex position in the plane into a data structure supporting queries of the following form: given a point qq and a directed line ℓ\ell in the plane, report the point of SS that is farthest from (or, alternatively, nearest to) the point qq among all points to the left of line ℓ\ell. We present two data structures for this problem. The first data structure uses O(n1+Δ)O(n^{1+\varepsilon}) space and preprocessing time, and answers queries in O(21/Δlog⁥n)O(2^{1/\varepsilon} \log n) time, for any 0<Δ<10 < \varepsilon < 1. The second data structure uses O(nlog⁥3n)O(n \log^3 n) space and polynomial preprocessing time, and answers queries in O(log⁥n)O(\log n) time. These are the first solutions to the problem with O(log⁥n)O(\log n) query time and o(n2)o(n^2) space. The second data structure uses a new representation of nearest- and farthest-point Voronoi diagrams of points in convex position. This representation supports the insertion of new points in clockwise order using only O(log⁥n)O(\log n) amortized pointer changes, in addition to O(log⁥n)O(\log n)-time point-location queries, even though every such update may make Θ(n)\Theta(n) combinatorial changes to the Voronoi diagram. This data structure is the first demonstration that deterministically and incrementally constructed Voronoi diagrams can be maintained in o(n)o(n) amortized pointer changes per operation while keeping O(log⁥n)O(\log n)-time point-location queries.Comment: 17 pages, 6 figures. Various small improvements. To appear in Algorithmic

    Proximity problems on line segments spanned by points

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    AbstractFinding the closest or farthest line segment (line) from a point are fundamental proximity problems. Given a set S of n points in the plane and another point q, we present optimal O(nlogn) time, O(n) space algorithms for finding the closest and farthest line segments (lines) from q among those spanned by the points in S. We further show how to apply our techniques to find the minimum (maximum) area triangle with a vertex at q and the other two vertices in S∖{q} in optimal O(nlogn) time and O(n) space. Finally, we give an O(nlogn) time, O(n) space algorithm to find the kth closest line from q and show how to find the k closest lines from q in O(nlogn+k) time and O(n+k) space

    Veröffentlichungen und VortrĂ€ge 2004 der Mitglieder der FakultĂ€t fĂŒr Informatik

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    Farthest-point queries with geometric and combinatorial constraints

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    In this paper we discuss farthest-point problems in which a set or sequence S of n points in the plane is given in advance and can be preprocessed to answer various queries efficiently. First, we give a data structure that can be used to compute the point farthest from a query line segment in O(log2n) time. Our data structure needs O(nlogn) space and preprocessing time. To the best of our knowledge no solution to this problem has been suggested yet. Second, we show how to use this data structure to obtain an output-sensitive query-based algorithm for polygonal path simplification. Both results are based on a series of data structures for fundamental farthest-point queries that can be reduced to each other

    Farthest-point queries with geometric and combinatorial constraints

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    In this paper we discuss farthest-point problems in which a set or sequence S of n points in the plane is given in advance and can be preprocessed to answer various queries efficiently. First, we give a data structure that can be used to compute the point farthest from a query line segment in O(log2n) time. Our data structure needs O(nlogn) space and preprocessing time. To the best of our knowledge no solution to this problem has been suggested yet. Second, we show how to use this data structure to obtain an output-sensitive query-based algorithm for polygonal path simplification. Both results are based on a series of data structures for fundamental farthest-point queries that can be reduced to each other

    Farthest-Point Queries with Geometric and Combinatorial Constraints

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    In this paper we discuss farthest-point problems, in which a sequence S = (p_1, p_2, ..., p_n) of n points in the plane is given in advance and can be preprocessed to answer various queries efficiently. We first consider the general setting where query points can be arbitrary, then we investigate a special setting where each point in S is queried exactly once. To describe our problem..

    Farthest-point queries with geometric and combinatorial constraints

    No full text
    In this paper we discuss farthest-point problems in which a set or sequence S of n points in the plane is given in advance and can be preprocessed to answer various queries efficiently. First, we give a data structure that can be used to compute the point farthest from a query line segment in O(log2 n) time. Our data structure needs O(n log n) space and preprocessing time. To the best of our knowledge no solution to this problem has been suggested yet. Second, we show how to use this data structure to obtain an output-sensitive query-based algorithm for polygonal path simplification. Both results are based on a series of data structures for fundamental farthest-point queries that can be reduced to each other
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