Art Gallery Theorems

1
Art Gallery Theorems

• Shih-Heng Chin

2
Contents

• Introduction
• Theorems
• A Straightforward Method
• An Advanced Method

3
Introduction

• Determine the minimum number of guards sufficient
  to cover the interior or an n-wall art gallery.
  Question posted by Victor Klee in 1973 and
  responded by Vasek Chvátal in 1975

4
Introduction (cont.)

• What shape can place just a guard/camera?
• Convex Polygon
• What kinds of polygon is always convex?
• Triangles
• So the problem is how to split a polygon in to
  triangles Triangulation, and how to set
  guards/cameras on these triangles

5
Theorems

• Every simple polygon (without intersections and
  holes) with n vertices consists of exactly n 2
  triangles. proof
• For a simple polygon with n vertices, need at
  least floor(3/n) cameras. 3-coloring approach

6
A Straightforward Method

• Two Ears Theorem Any simple polygon has two ears
  proof
• Triangulate(P) for v in P if
  is_a_ear_at(v) make_diagonal(pre(v),
  next(v)) P P v

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A Straightforward Method (cont.)

• is_a_ear_at(v) ear triangle(v, pre(v),
  next(v)) if intersect(ear, line(pre(v),
  pre(pre(v)))) or intersect(ear,
  line(next(v), next(next(v)))) return
  false for e in edges of (P v pre(v)
  next(v)) if intersect(e, line(pre(v),
  next(v))) return false return
  true

8
An Advanced Method

• Split a simple polygon into y-monotone pieces
• Split monotone pieces into triangles

A polygon P, for all lines l parallel to y 0,
if the number of intersected line segment of P
and l is at most 1, P is y-monotone
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Monotone Partitioning

• sort vertices by decreasing height
• remove upward-pointing interior splitting
  vertices
• remove downward-pointing interior splitting
  vertices

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Monotone Partitioning (cont.)

• Remove upward-pointing interior splitting
  verticesS e0, v0S in the pattern ej-1,
  wj-1, ej, wj, ej1, wj1, ej2, wj2
  for i 1 to n 1 do j index of edge
  which closest to vi case type_of_v(vi)
  1 S ej-1, vi, ej2, wj2
  2 S ej-1, vi, e, vi, ej1, wj1
  3 S ej, vi, e, vi, e, vi,
  ej1, wj1 if angle of vi gt 180
  then draw (vi, wj)

e
j
v
i
e
e
e
j1
j-1
w
v
i
e
e
e
e
j-1
j1
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Split monotone pieces into triangles

• Sort vertices by decreasing y-xoordinate
• Scan vertices and draw diagonal

Case 1
Case 2
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Analyze

• Two ears throrem O(n2)
• Y-monotone O(nlogn) O(r)r number of vertices
  in all monotone pieces

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Convex

• A subset S of the plane is called convex if and
  only if for any pair of points p,q in S the line
  segment pq is completely contained is S. back

Convex
Not convex
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3-Coloring Approach
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