Otfried Cheong. Alon Efrat. Sariel Har-Peled. TU Eindhoven, Netherlands. University of Arizona UIUC

Transcription

1 On Finding a Guard that Sees Most and a Shop that Sells Most Otfried Cheong TU Eindhoven, Netherlands Alon Efrat University of Arizona Sariel Har-Peled UIUC On Finding a Guard that Sees Most and a Shop that Sells Most p.1/26

2 Problem You want to establish an art gallery: 1. Where to place it? 2. How to protect the gallery from thieves? 3. What to show in it? On Finding a Guard that Sees Most and a Shop that Sells Most p.2/26

3 Placing the Art Gallery Finding a shop that sells most. Assumption: Customer goes to closest shop. (Post office problem.) Given a city with shops, find where to place shop that controls the largest region. Assumptions: 1. City is a square. 2. Distance is Euclidean. On Finding a Guard that Sees Most and a Shop that Sells Most p.3/26

4 Finding best location On Finding a Guard that Sees Most and a Shop that Sells Most p.4/26

5 Finding best location On Finding a Guard that Sees Most and a Shop that Sells Most p.4/26

6 Previous Work [Cheong, Har-Peled, Matoušek, Linial, 02] The One-Round Voronoi Game One can always control > 1/2 + const of market, by inserting n sites. [Dehne, Klein and Seidel, 02] Find approx max area cell when sites are in convex position. On Finding a Guard that Sees Most and a Shop that Sells Most p.5/26

7 Result Input: T set of n points, δ > 0 Compute a point x app : µ(x app ) (1 δ)µ opt where µ(x) = area(v orcell T {x} (x)) and µ opt = max x µ(x). Running time: O ( n δ 4 + n log n ). On Finding a Guard that Sees Most and a Shop that Sells Most p.6/26

8 Second Problem A guard that seems most Input: A simple polygon P. Compute a point x P : µ(x) (1 δ)µ opt where µ(x) = area(v P (x)) and µ opt = max x µ(x). On Finding a Guard that Sees Most and a Shop that Sells Most p.7/26

9 Example On Finding a Guard that Sees Most and a Shop that Sells Most p.8/26

10 Example p On Finding a Guard that Sees Most and a Shop that Sells Most p.8/26

11 A Guard that Sees Most - Motivation 1. Guarding an Art Gallery Given P find min # of guards seeing all of P. 2. NP Hard 3. A continuous set cover problem. 4. Open problem: An approximation algorithm? 5. Heuristic: Pick guard that seems most, and repeat. 6. Bound uncovered area after m guards. On Finding a Guard that Sees Most and a Shop that Sells Most p.9/26

12 Result 1. P - Simple polygon (n vertices) 2. δ > 0 - approx param 3. Compute x P : µ(x) (1 δ)µ opt ( ) n 2 4. Running time: O δ 4 log3 (n/δ) 5. Constant approx is 3sum hard. On Finding a Guard that Sees Most and a Shop that Sells Most p.10/26

13 Further Motivation 1. For the guard that sees most, the function maximized is n P(x,y) f r (x,y) = R(x, y) i=1 inside region r. (a) There are O(n 3 ) regions. (b) Every region the function is different. (c) How to compute/approx max efficiently. 2. Max Voronoi cell Function max is similar. On Finding a Guard that Sees Most and a Shop that Sells Most p.11/26

14 First approach Direct sampling: R - Pick random sample inside P x P compute V P (x) Return largest visibility polygon computed. Problem: Does not work... On Finding a Guard that Sees Most and a Shop that Sells Most p.12/26

15 Second approach Range space X = (P, W) of all vis. polygons in P : W = { V P (x) x P } [Valtr,98] V C Dim(X) 23 Assuming µ(p) = 1... On Finding a Guard that Sees Most and a Shop that Sells Most p.13/26

16 Second approach - cont ε-approx theorem For any δ, a sample R of Õ(1/δ2 ) is a δ-approx: x P µ(v P(x)) R V P(x) R δ. µ opt = max x µ(v P (x)) 1/n. On Finding a Guard that Sees Most and a Shop that Sells Most p.14/26

17 Second approach - algorithm Algorithm: 1. Set δ = ε/n 2. R: compute δ-approx of P 3. Find x = arg max x P R V P (x). 4. Return V P (x) Q: How to compute x? 1. Compute W = { V P (y) y R } 2. Find most covered point in A(W). On Finding a Guard that Sees Most and a Shop that Sells Most p.15/26

18 Second approach - Alg. Pick a random sample. On Finding a Guard that Sees Most and a Shop that Sells Most p.16/26

19 Second approach - Alg. Compute the visibility polygon of each random sample. On Finding a Guard that Sees Most and a Shop that Sells Most p.16/26

20 Second approach - Alg. x 1. Compute the arrangement of the visibility polygons. 2. Find the point x with largest # of vis. polygons covering it. On Finding a Guard that Sees Most and a Shop that Sells Most p.16/26

21 Second approach - Alg. V P (x) x Compute visibility polygons V P (x), and return it. On Finding a Guard that Sees Most and a Shop that Sells Most p.16/26

22 Second approach - Result Lemma: ( ) n 5 Running time O δ 4 log3 (n/δ) Return x P, s.t. µ(x) (1 δ)µ opt. Similar to [Natafos and Tsoukalas, 94] Algorithm is slow... On Finding a Guard that Sees Most and a Shop that Sells Most p.17/26

23 Third approach Estimating area directly Lemma: x P, δ > 0 µ(x) = area(v P (x)) ν S: uniform sample from P : S estimates the area V P (x) correctly...with high probability ε-approx sample size: Õ 1 δ 2 ν 2 ( 1 S = Õ δ 2 ν ) On Finding a Guard that Sees Most and a Shop that Sells Most p.18/26

24 Third approach Estimating area directly Lemma: x P, δ > 0 µ(x) = area(v P (x)) ν S: uniform sample from P : S = Õ ( 1 δ 2 ν ) S estimates the area V P (x) correctly...with high probability ε-approx sample size: Õ 1 δ 2 ν 2 On Finding a Guard that Sees Most and a Shop that Sells Most p.18/26

25 Third approach - cont Estimate area visible from a single point Problem: Estimate area for all points of P. Idea: Prove that there is a small witness set. Same sample works for all points of P On Finding a Guard that Sees Most and a Shop that Sells Most p.19/26

26 Third approach - cont Algorithm: 1. R: A random sample of P 2. ν = 1/n 3. R = Õ( 1/(νδ 2 ) ) = Õ( n/δ 2) 4. Find x = arg max x P R V P (x). 5. Return V P (x) Running time: Õ( n 3 /δ 4). On Finding a Guard that Sees Most and a Shop that Sells Most p.20/26

27 Third approach - analysis Why the alg. slow? A: Lower bound ν on µ opt too small. Idea: Exponential search on ν On Finding a Guard that Sees Most and a Shop that Sells Most p.21/26

28 Fourth approach ν i = 1/2 i At ith iter: 1. Check if µ opt ν i. (using prev. algorithm) 2. If so, got approx 3. Otherwise, µ opt ν i On Finding a Guard that Sees Most and a Shop that Sells Most p.22/26

29 Fourth approach - cont At ith iter: 1. Sample R of size Õ(1/ν i) points. 2. Compute Visibility arrangement of R 3. Find most covered point in AP(R) Expected depth of AP(R) is ν i 1 Õ(1/ν i) = Õ(1). AP(R) is shallow, Complexity of AP(R) is Õ( R 2 ). On Finding a Guard that Sees Most and a Shop that Sells Most p.23/26

30 Fourth approach - cont Running time dominated by last iteration. Overall running time Õ( n 2 /δ 4) On Finding a Guard that Sees Most and a Shop that Sells Most p.24/26

31 Conclusions Presented fast algorithms for approx. nasty area related functions. Basic idea: indirect sampling. Technique seems to be generic. Open problems: Approximating the minimum # of guards. On Finding a Guard that Sees Most and a Shop that Sells Most p.25/26

32 The end... On Finding a Guard that Sees Most and a Shop that Sells Most p.26/26