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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| Vydáno v: | Algorithmica Ročník 81; číslo 6; s. 2317 - 2345 |
|---|---|
| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
New York
Springer US
01.06.2019
Springer Nature B.V |
| Témata: | |
| ISSN: | 0178-4617, 1432-0541 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | 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. |
|---|---|
| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0178-4617 1432-0541 |
| DOI: | 10.1007/s00453-018-00536-7 |