Randomized Algorithms for Higher-Order Voronoi Diagrams

Maksym Zavershynskyi1joint work with

Cecilia Bohler2, Chih-Hung Liu2 and Evanthia Papadopoulou1

1University of Lugano, Faculty of Informatics, CH-6904 Lugano, Switzerland2University of Bonn, Institute of Computer Science I, D-53113 Bonn, Germany

This work was supported by the European Science Foundation (ESF) in the EUROCORES collaborative research project EuroGIGA/VORONOI, SNF 20GG21-134355. The work of the 1rd and 4rth author was supported in part by the Swiss National Science Foundation grant 200020-149658.

Randomized Algorithmsfor Higher-Order Voronoi Diagrams

Nearest Neighbor Voronoi Diagram

The nearest neighbor Voronoi diagram is the partitioning of the plane into regions, such that all points within a region have the same closest site.

Higher Order Voronoi Diagram

The order-k Voronoi diagram is the partitioning of the plane into regions, such that all points within an order-k region have the same k nearest sites.

VR2({p, q}, S)

p q

[Bohler et al.’13, Lee’82, Papadopoulou&Zavershynskyi’12]

Abstract Voronoi Diagrams

Defined by bisecting system rather than concrete geometric sites or distance measures.

J(p, q)D(p, q)

D(q, p)

[Bohler et al.’13, Klein et al.’09, Klein et al.’93]


First-order Voronoi region:

VR1({p}, S) =�

q∈S,q �=p

D(p, q)

For each

A1) is pathwise connectedA2) Each point in belongs to some A3) is not emptyA4) is unbounded Jordan-curveA5) and have finitely many intersection points that are transversal.

S� ⊆ S

VR1(p, S�)

VR1(p, S�)R2

VR1(p, S�)

J(p, q)

J(p, q) J(s, t)



Order-k Voronoi region:

VRk(H,S) =�

p∈H,q∈S\H

D(p, q)

• Points in convex distance metrics, Karlsruhe metric

• Disjoint line segments in Lp

• Disjoint convex polygons in Lp

Structural complexity is


≤ 2k(n− k)

Construction Algorithms

Construction Time Reference

Chazelle&Edelsbrunner’87

Clarkson’87

Aurenhammer’90

Mulmuley’91

Boissonnat et al.’93

Agarwal et al.’98

Chan’98

Ramos’99

O�n2 + b log2 n

�

O�n1+�

k�

O�nk

2 + n log n�

O�nk

3 + n log n�

O�n log3 n+ nk log n

�

O (n log n+ nk log n)

O

�n log n+ nk2c log

∗ k�

O�nk

2 log n�

Our Results

We Present 3 Algorithms for Abstract Diagrams

1.Randomized Divide-n-Conquer Algorithm

• Expected time complexityfor any constant

• Based on Clarkson-Shor technique and Clarkson’s algorithm for points.

• Uses two other algorithms for subroutines.

2.Traversal-based Algorithm

3. Iterative Algorithm

O(kn1+ε)ε > 0

Construction Algorithms

O(kn1+ε)

O(k2n log n)O(n22α(n) log n)

Randomized divide-n-conquer algorithm

Iterative algorithmTraversal-based algorithm

uses

uses

good for large k good for small k

Additional Axiom

Additional Axiom:

A6) Each bisector has constant number of points of vertical tangencies.

Divide-n-Conquer Algorithm

Vertical Decomposition

Vertical decomposition of :

1.Consider

2. Shoot vertical rays from each vertex and vertical tangent point.

Vk(S)

Vk(S)

Vk(S) ∪ Vk+1(S)

Vertical Decomposition

Vertical decomposition of :

1.Consider

2. Shoot vertical rays from each vertex and vertical tangent point.

Vk(S)

Vk(S) ∪ Vk+1(S)

Vk(S) ∪ Vk+1(S)

Random Sampling

Draw random sample such that it is a good estimator of .

R

S

Trapezoid is defined by at most elements of Rd

e.g.: d = 5s1

s2

s3

s4

p - dominatorp�

�

[Clarkson’87]

Random Sampling

Site strongly conflicts with if

d = 5s1

s2

s3

s4

p - dominatorp�

s � � ⊂ D(s, p)

D(s, p)

Random Sampling

Site weakly conflicts with if

d = 5s1

s2

s3

s4

p - dominatorp�

s �

D(s, p)

�∩D(s, p) �= ∅

Lemma 1

Lemma 1

For a random sample , with probability at least

1) number of strong conflicts with is

2) number of weak conflicts with is

for each trapezoid in vert. decomposition of�

R 1/2

S

≤ α|S|S

≥ |S|/(r − 5)

r = |R|,α = O(log r),β = O(log r/ log log r)

Allows to efficiently “bracket” the space!

[Clarkson’87]

Vβ(R)

Lemma 2

r = |R|,α = O(log r),β = O(log r/ log log r)[Clarkson’87]

- set of sitesS

Vk(S) - order-k diagramv - vertex

v

Lemma 2


- set of sitesS

- random sampleR ⊂ S

where each in vert. decomp. of has strong conflicts

Vβ(R)

Vk(S) - order-k diagram

�

≥ k

v - vertex

v

Lemma 2


- set of sitesS

- random sampleR ⊂ S

where each in vert. decomp. of has strong conflicts

Vβ(R)

Vk(S) - order-k diagram

�

v - vertex

S� ⊂ S- sites in weak conflictVk(S

�)then is vertex ofv

v

Allows to perform divide-n-conquer!

≥ k


Construction of can be reduced to finding all the vertices of .

Vk(S)

Vk(S)

[Clarkson’87]


If then use traversal-based algorithm.

Else choose “good” random sample .

• Construct and the decomposition, using iterative algorithm.

• For each trapezoid in the decomposition recursively compute vertices in .

R

|S| ≤ k(r − 5)

Vβ(R)

�Vk(S

�)

where - the sites in weak conflict with .S� ⊂ S �[Clarkson’87]

Traversal-Based Algorithm

Influence Region

Influence region of site - union of all order-k Voronoi regions induced by the site .

s

s

Vk(S)

influence region of site s [Papadopoulou&Zavershynskyi, Bohler et al.’13]


We can construct influence region of sitein time , using Har-Peled’s random walk method.

s

O(n2α(n) log n)

Vk(S)

influence region of site s [Har-Peled’00]


Union of all influence regions is the order-k Voronoi diagram.

Total running time: O(n22α(n) log n)

[Har-Peled’00, Chazelle&Edelsbrunner’87]

Iterative Algorithm

Iterative Algorithm

Order-k Voronoi diagram can be computed from the order-(k-1) Voronoi diagram in expected time.O(k(n− k) log n)

Total running time: O(k2n log n)

[Bohler et al.’13, Lee’82, Papadopoulou&Zavershynskyi, Klein et al.’93]

SUMMARY

We have developed 3 randomized algorithms for abstract higher-order Voronoi diagrams.

1.Divide-n-conquer algorithm with time complexity.

2.Traversal-based algorithm with time complexity. Good for large orders.

3.Iterative algorithm with time complexity. Good for small orders.

O(kn1+ε)

O(n22α(n) log n)

O(k2n log n)

Thank you!

References

1) C. Bohler, P. Cheilaris, R. Klein, C.-H. Liu, E. Papadopoulou, and M. Zavershynskyi. On the complexity of higher order abstract Voronoi diagrams. 40th International Colloquium on Automata, Languages and Programming (ICALP’13), pp. 208–219, 2013.

2) K. L. Clarkson. New applications of random sampling in computational geometry. Discrete and Computational Geometry 2(1), pp. 195–222, 1987.

3) S. Har-Peled. Taking a walk in a planar arrangment. SIAM Journal on Computing 30(4), pp. 1341-1367, 2000.

4) D. T. Lee. On k Nearest Neighbor Voronoi Diagrams in the Plane. IEEE Trans. Computers 31(6), pp. 478-487, 1982.

5) E. Papadopoulou and M. Zavershynskyi. On Higher Order Voronoi Diagrams of Line Segments. 23rd International Symposium on Algorithms and Computation (ISAAC ’12), LNCS 7676, pp. 177–186, 2012.

6) R. Klein, E. Langetepe, and Z. Nilforoushan. Abstract Voronoi Diagrams Revisited. Computational Geometry: Theory and Applications 42(9), pp. 885-902, 2009.

7) R. Klein, K. Mehlhorn, and St. Meiser. Randomized Incremental Construction of Abstract Voronoi Diagrams. Computational Geometry: Theory and Applications 3(1), pp. 157–184 1993.

8) B. Chazelle and H. Edelsbrunner. An improved algorithm for constructing kth-order Voronoi Diagram. IEEE Transactions on Computers 36(11), pp. 1349–1454, 1987.

Misc

projection

Misc

planes that correspond to extremal positions of β-sets

Misc


normals

Misc


normals

triangulation of the cone

Misc

normals

points that belong to the union of the halfplanes - analog of weak conflictpoints that belong to the intersection of the halfplanes - analog of strong conflict

Randomized Algorithms for Higher-Order Voronoi Diagrams

Technology

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