[PPT] - A Wavefront-Like Strategy for Computing Multiplicatively Weighted PowerPoint Presentation

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A Wavefront-Like Strategy for Computing Multiplicatively Weighted Voronoi Diagrams

Martin Held1 Stefan de Lorenzo1

1University of Salzburg, Department of Computer Science

March 19, 2019

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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Problem Specification

Given:

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

Compute:

The multiplicatively weighted Voronoi diagram (MWVD) VDw(S) of S.

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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SLIDE 3

Problem Specification

Given:

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

Compute:

The multiplicatively weighted Voronoi diagram (MWVD) VDw(S) of S.

35 60 70 94 61 80 52 77 39 66 46 82 10 55 71 76 75 78 29 83 47 19 64 72 53 63 59 68 87 74

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

2

SLIDE 4

Problem Specification

Given:

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

Compute:

The multiplicatively weighted Voronoi diagram (MWVD) VDw(S) of S.

35 60 70 94 61 80 52 77 39 66 46 82 10 55 71 76 75 78 29 83 47 19 64 72 53 63 59 68 87 74

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

2

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Multiplicatively Weighted Voronoi Diagrams

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

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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Multiplicatively Weighted Voronoi Diagrams

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

p

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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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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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.

s1 s2

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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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.

s1 s2

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

5

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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.

s1 s2

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

5

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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.

s1 s2

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.

s1 s2

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

5

SLIDE 13

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.

s1 s2

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

5

SLIDE 14

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.

s1 s2

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

5

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Wavefront Structure

The wavefront is formed by parts of offset circles.
A wavefront vertex is a specific intersection point of two offset circles.
Active arcs do not necessarily coincide with wavefront arcs.

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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Event Handling

Collision and domination events mark the initial and last contact (of a pair
f offset circles), respectively.
Edge and break-through events happen whenever active arcs vanish or

spawn.

These events are stored in a priority queue Q.

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

7

SLIDE 17

Event Handling

Collision and domination events mark the initial and last contact (of a pair
f offset circles), respectively.
Edge and break-through events happen whenever active arcs vanish or

spawn.

These events are stored in a priority queue Q.

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

7

SLIDE 18

Event Handling

Collision and domination events mark the initial and last contact (of a pair
f offset circles), respectively.
Edge and break-through events happen whenever active arcs vanish or

spawn.

These events are stored in a priority queue Q.

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

7

SLIDE 19

Event Handling

Collision and domination events mark the initial and last contact (of a pair
f offset circles), respectively.
Edge and break-through events happen whenever active arcs vanish or

spawn.

These events are stored in a priority queue Q.

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

7

SLIDE 20

Event Handling

Collision and domination events mark the initial and last contact (of a pair
f offset circles), respectively.
Edge and break-through events happen whenever active arcs vanish or

spawn.

These events are stored in a priority queue Q.

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

7

SLIDE 21

Event Handling

Collision and domination events mark the initial and last contact (of a pair
f offset circles), respectively.
Edge and break-through events happen whenever active arcs vanish or

spawn.

These events are stored in a priority queue Q.

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

7

SLIDE 22

Event Handling

Collision and domination events mark the initial and last contact (of a pair
f offset circles), respectively.
Edge and break-through events happen whenever active arcs vanish or

spawn.

These events are stored in a priority queue Q.

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

7

SLIDE 23

Event Handling

Collision and domination events mark the initial and last contact (of a pair
f offset circles), respectively.
Edge and break-through events happen whenever active arcs vanish or

spawn.

These events are stored in a priority queue Q.

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

7

SLIDE 24

Event Handling

Collision and domination events mark the initial and last contact (of a pair
f offset circles), respectively.
Edge and break-through events happen whenever active arcs vanish or

spawn.

These events are stored in a priority queue Q.

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

7

SLIDE 25

Event Handling

Collision and domination events mark the initial and last contact (of a pair
f offset circles), respectively.
Edge and break-through events happen whenever active arcs vanish or

spawn.

These events are stored in a priority queue Q.

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

7

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Event Handling

Initially, the collisions of every pair of offset circles are computed.
Afterwards, the events are successively popped from Q.
A collision event is valid if the collision point is situated within active arcs.
On each edge and break-through event, three offset circles need to be

updated.

The angular order of active arcs only changes at events.

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

8

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Event Handling

Initially, the collisions of every pair of offset circles are computed.
Afterwards, the events are successively popped from Q.
A collision event is valid if the collision point is situated within active arcs.
On each edge and break-through event, three offset circles need to be

updated.

The angular order of active arcs only changes at events.

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

8

SLIDE 28

Wavefront Propagation

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Wavefront Propagation

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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Wavefront Propagation

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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Wavefront Propagation

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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Wavefront Propagation

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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Wavefront Propagation

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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Wavefront Propagation

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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Wavefront Propagation

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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Wavefront Propagation

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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Wavefront Propagation

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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Wavefront Propagation

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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Wavefront Propagation

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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Wavefront Propagation

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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Analysis

All topological changes of the wavefront are properly detected.
A quadratic number of collision events are computed in any case.
In the worst case O(n2) break-through events take place.
At most O(n2) active arcs are generated during the wavefront propagation.
Therefore, the algorithms runtime is O(n2 log n) in the worst case.

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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Discussion

Our algorithm is also able to handle additive weights.
Improve the average case behavior of our strategy.
Reduce the number of collisions that are computed.

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

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Discussion

Our algorithm is also able to handle additive weights.
Improve the average case behavior of our strategy.
Reduce the number of collisions that are computed.

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG

11

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The End Thank you for your attention!

Computational Geometry and Applications Lab UNIVERSIT¨ AT SALZBURG