Contents
Preface page xi
1 Background 1
1.1 Notion of Visibility 1
1.2 Polygon 2
1.3 Asymptotic Complexity 5
1.4 Triangulation 6
1.5 The Art Gallery Problem 8
1.6 Special Types of Visibility 11
2 Point Visibility 13
2.1 Problems and Results 13
2.2 Computing Visibility of a Point in Simple Polygons 16
2.2.1 Non-Winding Polygon: O(n) Algorithm 16
2.2.2 Removing Winding: O(n) Algorithm 23
2.3 Computing Visibility of a Point in Polygons with Holes 31
2.4 Recognizing Simple Polygons Visible from a Point 38
2.5 Notes and Comments 43
3 Weak Visibility and Shortest Paths 46
3.1 Problems and Results 46
3.2 Characterizing Weak Visibility 51
3.3 Computing Weak Visibility in Simple Polygons 58
3.3.1 Scanning the Boundary: O(n log n) Algorithm 58
3.3.2 Using Shortest Path Trees: O(n) Algorithm 65
viii Contents
3.4 Computing Weak Visibility in Polygons with Holes 66
3.5 Recognizing Weakly Internal Visible Polygons 68
3.5.1 Using Visibility Graph: O(E) Algorithm 68
3.5.2 Scanning the Boundary: O(n) Algorithm 73
3.6 Computing Shortest Path Trees 82
3.6.1 In Simple Polygons: O(n) Algorithm 82
3.6.2 In Weak Visibility Polygons: O(n) Algorithm 87
3.7 Recognizing Weakly External Visible Polygons 95
3.8 Notes and Comments 102
4 LR-Visibility and Shortest Paths 105
4.1 Problems and Results 105
4.2 Characterizing LR-Visibility 108
4.3 Computing LR-Visibility Polygons 110
4.4 Recognizing LR-Visibility Polygons 113
4.5 Walking in an LR-Visibility Polygon 115
4.6 Computing Shortest Path Trees using LR-Visibility 124
4.7 Notes and Comments 135
5 Visibility Graphs 136
5.1 Problems and Results 136
5.2 Computing Visibility Graphs of Simple Polygons 138
5.3 Computing Visibility Graphs of Polygons with Holes 143
5.3.1 Worst-Case: O(n2) Algorithm 143
5.3.2 Output-Sensitive: O(n log n + E) Algorithm 146
5.4 Computing Tangent Visibility Graphs 161
5.4.1 Convex Holes: O(n + h2 log h) Algorithm 161
5.4.2 Non-Convex Holes: O(n + h2 log h) Algorithm 165
5.5 Notes and Comments 169
6 Visibility Graph Theory 171
6.1 Problems and Results 171
6.2 Recognizing Visibility Graphs of Simple Polygons 174
6.2.1 Necessary Conditions 174
6.2.2 Testing Necessary Conditions: O(n2) Algorithm 180
6.3 Characterizing Visibility Graphs of Simple Polygons 183
6.4 Recognizing Special Classes of Visibility Graphs 187
6.4.1 Spiral Polygons: O(n) Algorithm 187
6.4.2 Tower Polygons: O(n) Algorithm 195
6.5 Characterizing a Sub-Class of Segment Visibility Graphs 201
6.6 A Few Properties of Vertex-Edge Visibility Graphs 205
6.7 Computing Maximum Clique in a Visibility Graph 208
6.8 Computing Maximum Hidden Vertex Set in a Visibility Graph 214
6.9 Notes and Comments 216
7 Visibility and Link Paths 218
7.1 Problems and Results 218
7.2 Computing Minimum Link Paths in Simple Polygons 221
7.2.1 Using Weak Visibility: O(n) Algorithm 221
7.2.2 Using Complete Visibility: O(n) Algorithm 224
7.3 Computing Minimum Link Paths in Polygons with Holes 231
7.4 Computing Link Center and Radius of Simple Polygons 238
7.5 Computing Minimum Nested Polygons 242
7.5.1 Between Convex Polygons: O(n log k) Algorithm 242
7.5.2 Between Non-Convex Polygons: O(n) Algorithm 248
7.6 Notes and Comments 253
8 Visibility and Path Queries 255
8.1 Problems and Results 255
8.2 Ray-Shooting Queries in Simple Polygons 259
8.3 Visibility Polygon Queries for Points in Polygons 267
8.3.1 Without Holes: O(log n + k) Query Algorithm 267
8.3.2 With Holes: O(n) Query Algorithm 272
8.4 Path Queries Between Points in Simple Polygons 278
8.4.1 Shortest Paths: O(log n + k) Query Algorithm 278
8.4.2 Link Paths: O(log n + k) Query Algorithm 289
8.5 Notes and Comments 292
Bibliography 295
Index 311
