Computational Geometry Exercise session #4: Polygon decompositions - - PDF document

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Computational Geometry Exercise session #4: Polygon decompositions - - PDF document

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Computational Geometry Exercise session #4: Polygon decompositions Art Gallery Theorem Trapezoidal decomposition Convex decomposition Solution to homework 1 1 Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005 The

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Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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Computational Geometry

Exercise session #4: Polygon decompositions

  • Art Gallery Theorem
  • Trapezoidal decomposition
  • Convex decomposition
  • Solution to homework 1

Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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The art gallery problem

  • Given a simple polygon, place point guards

whose visibility regions cover the interior.

n=36, guards=9

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Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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Art Gallery Theorem

  • Theorem: For any simple polygon, ⎣n/3⎦ point

guards are sufficient to guard the entire

  • polygon. A placement of ⎣n/3⎦ guards can be

computed in O(nlogn) time.

  • Proof idea: triangulate polygon, then position

guards to cover each triangle.

Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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Placing the guards

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Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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3-Coloring of Polygon Vertices

Green: 8 Black: 8 White: 9

Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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Correctness & Complexity

  • Dual graph of triangulation is a tree.
  • BFS traversal of tree visits vertices once.
  • Neighboring triangle vertices get different

colors.

  • One vertex guard sees the vertices of different

colors.

  • Running time: triangulation + linear traversal
  • f dual graph. Can be done in O(n) time.
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Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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Guarding with Edge guards

How many edge guards are needed to guard a gallery?

Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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Lower bound for edge guards

⎣n/4⎦ edge guards needed What about star-shaped polygons?

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Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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Trapezoidal decomposition

  • Trapezoids are quadrilaterals with two parallel

edges, or triangles (degenerate case).

  • Trapezoids are easily triangulated.
  • Decompose polygon into slabs which are

easily searched.

  • Computation using sweep line is easily

achieved.

  • Drawback: new vertices are added to the

decomposition.

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Example

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Sweep events I

e1 e2 e3 e4 v Sweepline status: (e1,e2,e3,e4) (e1,e4)

Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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Sweep events II

e1 e2 e3 e4 v Sweepline status: (e1,e4) (e1,e2,e3,e4)

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Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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Sweep events III

e1 e2 e3 e4 v Sweepline status: (e1,e2,e4) (e1,e3,e4)

Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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Convex decompositions

  • Partition polygon into minimal number of

convex pieces with disjoint interiors such they cover the polygon.

  • Two versions:

Partition using only diagonals Partition with additional vertices (‘Steiner points’)

  • For a given polygon, which version results in

less convex pieces?

  • Which is harder to compute?
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Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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Decomposition example

Decomposition with diagonals Decomposition with Steiner points

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Bounds on optimal decomposition

  • Let Φ be the fewest number of convex pieces

into which a polygon may be partitioned.

  • Theorem: for a polygon of r reflex vertices:

⎡r/2⎤+1 ≤ Φ ≤ r+1

  • Proof:

Drawing a segment that bisects a reflex vertex removes all reflex angles upper bound (Steiner). At most two reflex angles can be resolved by a single segment lower bound (both versions).

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Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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4-Approximation algorithm

  • Hertel & Melhorn 1983.
  • Quickly partition polygon such that number of

pieces is at most four times the optimal.

  • Definition: a diagonal d is essential for vertex

v if removal of d creates a non-convex piece. Otherwise it is inessential.

  • Algorithm: triangulate, then remove

inessential diagonals one after one until none remain.

Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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Quality of approximation

  • Algorithm outputs convex partitions, since it

starts with one and maintains this invariant throughout all steps.

  • Lemma 1: There can be at most two diagonals

essential for any reflex vertex.

  • Lemma 2: The Hertel-Melhorn algorithm is

never worse than four times the optimal number of pieces.

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Proof of lemma 1

v+ v- v H+ H- a b c

Yaron Ostrovsky-Berman, Computational Geometry, Spring 2005

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Proof of lemma 2

  • When algorithm terminates, all diagonals are

essential.

  • According to lemma 1, every reflex vertex is

responsible for at most 2 diagonals.

  • Number of diagonals cannot be more than 2r.
  • Number of convex pieces produced: M ≤ 2r+1.
  • Lower bound for Φ implies M ≤ 4Φ.
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Optimal Convex partitioning

  • Best algorithm for partitioning with diagonals:

[Keil, 1985], runs in O(r2nlogn) time.

  • Best algorithm for partitioning with Steiner

points: [Chazelle, 1980], runs in O(n+r3) time.