[PPT] - On Implementing Multiplicatively Weighted Voronoi Diagrams Martin PowerPoint Presentation

SLIDE 1

On Implementing Multiplicatively Weighted Voronoi Diagrams

Martin Held1 Stefan de Lorenzo1

1University of Salzburg, Department of Computer Science

March 16, 2020

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

SLIDE 2

Problem Specification

Problem

Given: A set S of n input points in the plane, where every s ∈ S is associated with a weight w(s) > 0.

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Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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SLIDE 3

Problem Specification

Problem

Given: A set S of n input points in the plane, where every s ∈ S is associated with a weight w(s) > 0. Compute: The multiplicatively weighted Voronoi diagram (MWVD) VDw(S) of S.

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Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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SLIDE 4

Multiplicatively Weighted Voronoi Diagrams

The Voronoi edges are formed by straight-line segments, rays, and circular arcs.
The Voronoi regions are (possibly) disconnected.
The MWVD has a quadratic combinatorial complexity in the worst case.

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Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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SLIDE 5

Multiplicatively Weighted Voronoi Diagrams

The Voronoi edges are formed by straight-line segments, rays, and circular arcs.
The Voronoi regions are (possibly) disconnected.
The MWVD has a quadratic combinatorial complexity in the worst case.

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Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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SLIDE 6

Overview

We present a wavefront-based approach for computing MWVDs.
The wavefront covers an increasing portion of the plane over time.
It consists of wavefront arcs and wavefront vertices.
Whenever a wavefront arc vanishes or spawns, a new Voronoi node is discovered.

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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SLIDE 7

Offset Circles

Every site is associated with an offset circle.
Two moving intersection points trace out the bisector as time progresses.
Inactive arcs along the offset circles are eliminated.
The active arcs are stored in sorted angular order.

s2 s1

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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SLIDE 8

Offset Circles

Every site is associated with an offset circle.
Two moving intersection points trace out the bisector as time progresses.
Inactive arcs along the offset circles are eliminated.
The active arcs are stored in sorted angular order.

s2 s1

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

5

SLIDE 9

Offset Circles

Every site is associated with an offset circle.
Two moving intersection points trace out the bisector as time progresses.
Inactive arcs along the offset circles are eliminated.
The active arcs are stored in sorted angular order.

s2 s1

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

5

SLIDE 10

Offset Circles

Every site is associated with an offset circle.
Two moving intersection points trace out the bisector as time progresses.
Inactive arcs along the offset circles are eliminated.
The active arcs are stored in sorted angular order.

s2 s1

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

5

SLIDE 11

Offset Circles

Every site is associated with an offset circle.
Two moving intersection points trace out the bisector as time progresses.
Inactive arcs along the offset circles are eliminated.
The active arcs are stored in sorted angular order.

s2 s1

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

5

SLIDE 12

Offset Circles

Every site is associated with an offset circle.
Two moving intersection points trace out the bisector as time progresses.
Inactive arcs along the offset circles are eliminated.
The active arcs are stored in sorted angular order.

s2 s1

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

5

SLIDE 13

Event Handling

Collision and domination events mark the initial and last contact of a two offset circles.
Arc events happen whenever active arcs vanish or spawn.
These events are stored in a priority queue Q.
The angular order of active arcs only changes at events.

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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SLIDE 14

Event Handling

Collision and domination events mark the initial and last contact of a two offset circles.
Arc events happen whenever active arcs vanish or spawn.
These events are stored in a priority queue Q.
The angular order of active arcs only changes at events.

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

6

SLIDE 15

Event Handling

Collision and domination events mark the initial and last contact of a two offset circles.
Arc events happen whenever active arcs vanish or spawn.
These events are stored in a priority queue Q.
The angular order of active arcs only changes at events.

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

6

SLIDE 16

Event Handling

Collision and domination events mark the initial and last contact of a two offset circles.
Arc events happen whenever active arcs vanish or spawn.
These events are stored in a priority queue Q.
The angular order of active arcs only changes at events.

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

6

SLIDE 17

Event Handling

Collision and domination events mark the initial and last contact of a two offset circles.
Arc events happen whenever active arcs vanish or spawn.
These events are stored in a priority queue Q.
The angular order of active arcs only changes at events.

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

6

SLIDE 18

Event Handling

Collision and domination events mark the initial and last contact of a two offset circles.
Arc events happen whenever active arcs vanish or spawn.
These events are stored in a priority queue Q.
The angular order of active arcs only changes at events.

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

6

SLIDE 19

Event Handling

Collision and domination events mark the initial and last contact of a two offset circles.
Arc events happen whenever active arcs vanish or spawn.
These events are stored in a priority queue Q.
The angular order of active arcs only changes at events.

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

6

SLIDE 20

Event Handling

Collision and domination events mark the initial and last contact of a two offset circles.
Arc events happen whenever active arcs vanish or spawn.
These events are stored in a priority queue Q.
The angular order of active arcs only changes at events.

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

6

SLIDE 21

Event Handling

Collision and domination events mark the initial and last contact of a two offset circles.
Arc events happen whenever active arcs vanish or spawn.
These events are stored in a priority queue Q.
The angular order of active arcs only changes at events.

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

6

SLIDE 22

Event Handling

Collision and domination events mark the initial and last contact of a two offset circles.
Arc events happen whenever active arcs vanish or spawn.
These events are stored in a priority queue Q.
The angular order of active arcs only changes at events.

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

6

SLIDE 23

Event Handling

All topological changes of the wavefront are properly detected.
A quadratic number of collision events are computed in any case.
A moving intersection can be charged with a constant number of arc events.
In the worst case O(n2) arc events take place.
All events can be handled in O(log n) time.
Therefore, the algorithms runtime is O(n2 log n) in the worst case.

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

7

SLIDE 24

Reducing the Number of Collisions

A vast number of collisions are invalid for general input.
The calculation of all possible collision requires a high computational effort.
Invalid collision are filtered in an additional preprocessing step.
Thus, the average case behavior of the algorithm is improved.

s1

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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SLIDE 25

Reducing the Number of Collisions

A vast number of collisions are invalid for general input.
The calculation of all possible collision requires a high computational effort.
Invalid collision are filtered in an additional preprocessing step.
Thus, the average case behavior of the algorithm is improved.

s1 s2

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

8

SLIDE 26

Reducing the Number of Collisions

A vast number of collisions are invalid for general input.
The calculation of all possible collision requires a high computational effort.
Invalid collision are filtered in an additional preprocessing step.
Thus, the average case behavior of the algorithm is improved.

s1 s2 s3

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

8

SLIDE 27

Reducing the Number of Collisions

A vast number of collisions are invalid for general input.
The calculation of all possible collision requires a high computational effort.
Invalid collision are filtered in an additional preprocessing step.
Thus, the average case behavior of the algorithm is improved.

s1 s2 s3 s4

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

8

SLIDE 28

Reducing the Number of Collisions

A vast number of collisions are invalid for general input.
The calculation of all possible collision requires a high computational effort.
Invalid collision are filtered in an additional preprocessing step.
Thus, the average case behavior of the algorithm is improved.

s1 s2 s3 s4 s5

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

8

SLIDE 29

Reducing the Number of Collisions

A vast number of collisions are invalid for general input.
The calculation of all possible collision requires a high computational effort.
Invalid collision are filtered in an additional preprocessing step.
Thus, the average case behavior of the algorithm is improved.

s1 s2 s3 s4 s5 s6 s7 s8 s9 s10

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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SLIDE 30

Overlay Arrangement

Only sites within the same candidate set may collide.
A candidate set contains O(log n) many sites in the expected case [HPR15].
The expected complexity of overlay arrangement is bound by O(n log n) [KRS11].
Thus, it is necessary to compute O(n log3 n) many collisions.

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Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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SLIDE 31

Experimental Evaluation

The implementation is based on the Computational Geometry Algorithms Library (CGAL).
We tested our strategy on 3800 randomly generated inputs.
All tests were carried out on an Intel Core i7-6700 clocked at 3.40GHz.

103 104 105 Input size 2.0 2.2 2.4 2.6 2.8 3.0 #Collision Events/n log n 103 104 105 Input size 10 11 12 13 14 #Arc Events/n

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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Experimental Evaluation

The implementation is based on the Computational Geometry Algorithms Library (CGAL).
We tested our strategy on 3800 randomly generated inputs.
All tests were carried out on an Intel Core i7-6700 clocked at 3.40GHz.

103 104 105 Input size 50 100 150 Runtime/n log2 n Overlay arrangement Event queue Overall runtime

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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SLIDE 33

Extensions

Our (basic) algorithm is also able to deal with additive weights.
The strategy can be easily extended to also handle weighted straight-line segments.

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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Discussion

We propose a fast, practical strategy to compute MWVDs.
The expected runtime is improved by using an overlay arrangement.
We provide a robust implementation using exact arithmetic.

Thank you for your attention!

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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Discussion

We propose a fast, practical strategy to compute MWVDs.
The expected runtime is improved by using an overlay arrangement.
We provide a robust implementation using exact arithmetic.

Thank you for your attention!

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

12

SLIDE 36

References I

Sariel Har-Peled and Benjamin Raichel. On the Complexity of Randomly Weighted Multiplicative Voronoi Diagrams. Discrete Comput. Geom., 53(3):547–568, 2015. Haim Kaplan, Edgar Ramos, and Micha Sharir. The Overlay of Minimization Diagrams in a Randomized Incremental Construction. Discrete Comput. Geom., 45(3):371–382, 2011.

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG