Guard Placement For Wireless Localization

Transcription

1 Guard Placement For Wireless Localization David Eppstein Michael T. Goodrich Nodari Sitchinava Computer Science Dept., UC Irvine Guard Placement For Wireless Localization p. 1/1

2 Motivating example: Cyber-Café Supply wireless internet to paying customers inside Prevent access to non-paying customers outside But how to tell which customers are which? Guard Placement For Wireless Localization p. 2/1

3 Proposed Solution Use multiple directional transmitters Customers in range of both transmitters are inside Guard Placement For Wireless Localization p. 3/1

4 Sculpture Garden Problem Sculpture Garden Problem the problem of placing angle guards to define a given polygon d a b c Guard Placement For Wireless Localization p. 4/1

5 Sculpture Garden Problem Sculpture Garden Problem the problem of placing angle guards to define a given polygon d a a b d b c Guard Placement For Wireless Localization p. 4/1

6 Sculpture Garden Problem Sculpture Garden Problem the problem of placing angle guards to define a given polygon d a c d b c Guard Placement For Wireless Localization p. 4/1

7 Sculpture Garden Problem Sculpture Garden Problem the problem of placing angle guards to define a given polygon d a a b d c d b c F = abd + cd = (ab + c)d Guard Placement For Wireless Localization p. 4/1

8 Natural Angle Guards Natural angle guards are placed at vertices of the polygon with angle of vertex = angle of guard Non-natural guards: Guard Placement For Wireless Localization p. 5/1

9 Natural Guards Aren t Enough Theorem. There exists a polygon P such that it is impossible to solve SGP for P using a natural angle-guard vertex placement. Proof. Guard Placement For Wireless Localization p. 6/1

10 Natural Guards Aren t Enough Theorem. There exists a polygon P such that it is impossible to solve SGP for P using a natural angle-guard vertex placement. Proof. Guard Placement For Wireless Localization p. 6/1

11 Natural Guards Aren t Enough Theorem. There exists a polygon P such that it is impossible to solve SGP for P using a natural angle-guard vertex placement. Proof. Guard Placement For Wireless Localization p. 6/1

12 Natural Guards Aren t Enough Theorem. There exists a polygon P such that it is impossible to solve SGP for P using a natural angle-guard vertex placement. Proof. Guard Placement For Wireless Localization p. 6/1

13 Natural Guards Aren t Enough Theorem. There exists a polygon P such that it is impossible to solve SGP for P using a natural angle-guard vertex placement. Proof. Guard Placement For Wireless Localization p. 6/1

14 Natural Guards Aren t Enough Theorem. There exists a polygon P such that it is impossible to solve SGP for P using a natural angle-guard vertex placement. Proof. Guard Placement For Wireless Localization p. 6/1

15 Quadrilaterals Theorem. Any quadrilateral can be guarded with 2 natural angle guards. Guard Placement For Wireless Localization p. 7/1

16 Pentagons Theorem. Any pentagon can be guarded with 3 angle guards. C B B D C D A E F A E B C C D B D A E A E Guard Placement For Wireless Localization p. 8/1

17 Hexagons Theorem. Any hexagon can be guarded with 4 angle guards. C B A F D E Guard Placement For Wireless Localization p. 9/1

18 General Upper Bound Theorem. n + 2(h 1) guards are sufficient to define any n-vertex polygon with h holes. Proof. Triangulate into n + 2(h 1) triangles Partition triangulation into quadrilaterals, pentagons, and hexagons In each piece, # guards = # triangles Definition is concise: each region is defined by O(1) guards Guard Placement For Wireless Localization p. 10/1

19 Lower Bounds Theorem. At least n 2 guards are required to solve the SGP for any polygon with no two edges lying on the same line. Theorem. n 2 guards are always sufficient to solve SGP for any convex polygon. Guard Placement For Wireless Localization p. 11/1

20 Lower Bounds Theorem. Any n-sided polygon requires Ω( n) guards. Theorem. There exist n-sided simple polygons that can be guarded concisely by O( n) guards. Guard Placement For Wireless Localization p. 12/1

21 Orthogonal Polygons Definition. xy-monotone polygon is an orthogonal polygon which is monotone with respect to the x = y line. extreme vertex top edge left edge right edge bottom edge extreme vertex Guard Placement For Wireless Localization p. 13/1

22 Orthogonal Polygons n Theorem. 2 natural guards are sufficient to solve SGP for any orthogonal polygons, by placing natural angle guards in every other vertex starting with a left vertex of some top edge. Proof. By induction on unguarded reflex vertices. extreme vertex Base Case top edge left edge right edge bottom edge extreme vertex Guard Placement For Wireless Localization p. 14/1

23 Orthogonal Polygons n Theorem. 2 natural guards are sufficient to solve SGP for any orthogonal polygons, by placing natural angle guards in every other vertex starting with a left vertex of some top edge. Proof. By induction on unguarded reflex vertices. Inductive Hypothesis v P P v v p v p q q Guard Placement For Wireless Localization p. 14/1

24 Minimizing Number of Guards Open Problem: How hard is it to find the minimum number of guards for a particular polygon? Theorem (APPROXIMATION). For any polygon P, we can find a collection of guards for P, using a number of guards that is within a factor of two of optimal. Proof. Place an edge guard on the line bounding each edge of P Use a Peterson-style constructive-solid-geometry formula In optimal solution, each line must be guarded and each guard can cover at most two lines so optimal # guards is at least half the guards used Guard Placement For Wireless Localization p. 15/1

25 Summary Sometimes required Always Sufficient Arbitrary Ω( n) n + 2(h 1) General position n 2 n + 2(h 1) n n Orthogonal 2 2 general position Convex n 2 n 2 Obvious open question: close factor-of-two gap for non-orthogonal general position Guard Placement For Wireless Localization p. 16/1