TY - GEN

T1 - Data structures for halfplane proximity queries and incremental Voronoi diagrams

AU - Aronov, Boris

AU - Bose, Prosenjit

AU - Demaine, Erik D.

AU - Gudmundsson, Joachim

AU - Iacono, John

AU - Langerman, Stefan

AU - Smid, Michiel

N1 - Copyright:
Copyright 2008 Elsevier B.V., All rights reserved.

PY - 2006

Y1 - 2006

N2 - We consider preprocessing a set S of n points in the plane that are in convex position into a data structure supporting queries of the following form: given a point q and a directed line I in the plane, report the point of S that is farthest from (or, alternatively, nearest to) the point q subject to being to the left of line ℓ We present two data structures for this problem. The first data structure uses O(n1+ε) space and preprocessing time, and answers queries in O(21/ε log n) time. The second data structure uses O(n log3 n) space and polynomial preprocessing time, and answers queries in O(log n) time. These are the first solutions to the problem with O(log n) query time and o(n2) space. In the process of developing the second data structure, we develop a new representation of nearest-point and farthest-point Voronoi diagrams of points in convex position, This representation supports insertion of new points in counterclockwise order using only O(log n) amortized pointer changes, subject to supporting O(log n)-time point-location queries, even though every such update may make ⊖(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) pointer changes per operation while keeping O(log n)-time point-location queries.

AB - We consider preprocessing a set S of n points in the plane that are in convex position into a data structure supporting queries of the following form: given a point q and a directed line I in the plane, report the point of S that is farthest from (or, alternatively, nearest to) the point q subject to being to the left of line ℓ We present two data structures for this problem. The first data structure uses O(n1+ε) space and preprocessing time, and answers queries in O(21/ε log n) time. The second data structure uses O(n log3 n) space and polynomial preprocessing time, and answers queries in O(log n) time. These are the first solutions to the problem with O(log n) query time and o(n2) space. In the process of developing the second data structure, we develop a new representation of nearest-point and farthest-point Voronoi diagrams of points in convex position, This representation supports insertion of new points in counterclockwise order using only O(log n) amortized pointer changes, subject to supporting O(log n)-time point-location queries, even though every such update may make ⊖(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) pointer changes per operation while keeping O(log n)-time point-location queries.

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U2 - 10.1007/11682462_12

DO - 10.1007/11682462_12

M3 - Conference contribution

AN - SCOPUS:33745587774

SN - 354032755X

SN - 9783540327554

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 80

EP - 92

BT - LATIN 2006

T2 - LATIN 2006: Theoretical Informatics - 7th Latin American Symposium

Y2 - 20 March 2006 through 24 March 2006

ER -