An Efficient Randomized Algorithm for Higher-Order Abstract Voronoi Diagrams

Given a set of n sites in the plane, the order- k Voronoi diagram is a planar subdivision such that all points in a region share the same k nearest sites. The order- k Voronoi diagram arises for the k -nearest-neighbor problem, and there has been a lot of work for point sites in the Euclidean metric...

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Veröffentlicht in:	Algorithmica Jg. 81; H. 6; S. 2317 - 2345
Hauptverfasser:	Bohler, Cecilia, Klein, Rolf, Liu, Chih-Hung
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York Springer US 01.06.2019 Springer Nature B.V
Schlagworte:	Algorithm Analysis and Problem Complexity Algorithms Axioms Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Mathematics of Computing Randomization Theory of Computation Voronoi graphs Order Computational geometry Voronoi diagrams Randomized geometric algorithms Abstract Voronoi diagrams
ISSN:	0178-4617, 1432-0541
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Abstract	Given a set of n sites in the plane, the order- k Voronoi diagram is a planar subdivision such that all points in a region share the same k nearest sites. The order- k Voronoi diagram arises for the k -nearest-neighbor problem, and there has been a lot of work for point sites in the Euclidean metric. In this paper, we study order- k Voronoi diagrams defined by an abstract bisecting curve system that satisfies several practical axioms, and thus our study covers many concrete order- k Voronoi diagrams. We propose a randomized incremental construction algorithm that runs in O ( k ( n - k ) log 2 n + n log 3 n ) steps, where O ( k ( n - k ) ) is the number of faces in the worst case. This result applies to disjoint line segments in the L p norm, convex polygons of constant size, points in the Karlsruhe metric, and so on. In fact, a running time with a polylog factor to the number of faces was only achieved for point sites in the L 1 or Euclidean metric before.
AbstractList	Given a set of n sites in the plane, the order-k Voronoi diagram is a planar subdivision such that all points in a region share the same k nearest sites. The order-k Voronoi diagram arises for the k-nearest-neighbor problem, and there has been a lot of work for point sites in the Euclidean metric. In this paper, we study order-k Voronoi diagrams defined by an abstract bisecting curve system that satisfies several practical axioms, and thus our study covers many concrete order-k Voronoi diagrams. We propose a randomized incremental construction algorithm that runs in O(k(n-k)log2n+nlog3n) steps, where O(k(n-k)) is the number of faces in the worst case. This result applies to disjoint line segments in the Lp norm, convex polygons of constant size, points in the Karlsruhe metric, and so on. In fact, a running time with a polylog factor to the number of faces was only achieved for point sites in the L1 or Euclidean metric before. Given a set of n sites in the plane, the order- k Voronoi diagram is a planar subdivision such that all points in a region share the same k nearest sites. The order- k Voronoi diagram arises for the k -nearest-neighbor problem, and there has been a lot of work for point sites in the Euclidean metric. In this paper, we study order- k Voronoi diagrams defined by an abstract bisecting curve system that satisfies several practical axioms, and thus our study covers many concrete order- k Voronoi diagrams. We propose a randomized incremental construction algorithm that runs in O ( k ( n - k ) log 2 n + n log 3 n ) steps, where O ( k ( n - k ) ) is the number of faces in the worst case. This result applies to disjoint line segments in the L p norm, convex polygons of constant size, points in the Karlsruhe metric, and so on. In fact, a running time with a polylog factor to the number of faces was only achieved for point sites in the L 1 or Euclidean metric before.
Author	Bohler, Cecilia Liu, Chih-Hung Klein, Rolf
Author_xml	– sequence: 1 givenname: Cecilia surname: Bohler fullname: Bohler, Cecilia organization: Department of Computer Science, University of Bonn – sequence: 2 givenname: Rolf surname: Klein fullname: Klein, Rolf organization: Department of Computer Science, University of Bonn – sequence: 3 givenname: Chih-Hung orcidid: 0000-0001-9683-5982 surname: Liu fullname: Liu, Chih-Hung email: chih-hung.liu@inf.ethz.ch organization: Department of Computer Science, ETH Zürich
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Snippet	Given a set of n sites in the plane, the order- k Voronoi diagram is a planar subdivision such that all points in a region share the same k nearest sites. The... Given a set of n sites in the plane, the order-k Voronoi diagram is a planar subdivision such that all points in a region share the same k nearest sites. The...
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithms Axioms Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Mathematics of Computing Randomization Theory of Computation Voronoi graphs
Title	An Efficient Randomized Algorithm for Higher-Order Abstract Voronoi Diagrams
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