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Art Gallery Theorems Shih-Heng Chin Contents Introduction Theorems A Straightforward Method An Advanced Method Introduction

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art gallery theorems

Art Gallery Theorems

Shih-Heng Chin

contents
Contents
  • Introduction
  • Theorems
  • A Straightforward Method
  • An Advanced Method
introduction
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]
introduction cont
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
theorems
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]
a straightforward method
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}
a straightforward method cont
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}
an advanced method
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

monotone partitioning
Monotone Partitioning
  • sort vertices by decreasing height
  • remove upward-pointing interior splitting vertices
  • remove downward-pointing interior splitting vertices
monotone partitioning cont
Monotone Partitioning (cont.)
  • Remove upward-pointing interior splitting verticesS = [e0, v0]{S in the pattern …[ej-1, wj-1], [ej, wj], [ej+1, wj+1], [ej+2, wj+2]…} 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], [ej+2, wj+2] … 2: S = … [ej-1, vi], [e’, vi], [ej+1, wj+1] … 3: S = … [ej, vi], [e’, vi], [e’’, vi], [ej+1, wj+1] … if angle of vi > 180 then draw (vi, wj)

e

j

v

i

e

e

e’

j+1

j-1

w

v

i

e

e

e’’

e’

j+1

j-1

split monotone pieces into triangles
Split monotone pieces into triangles
  • Sort vertices by decreasing y-xoordinate
  • Scan vertices and draw diagonal

Case 1:

Case 2:

analyze
Analyze
  • Two ears throrem: O(n2)
  • Y-monotone: O(nlogn) + O(r)r: number of vertices in all monotone pieces
convex
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