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Alejandro Salinger University of Waterloo Joint work with Francisco Claude, Gautam K. Das, Reza Dorrigiv, Stephane Durocher, Robert Fraser, Alejandro Lpez-Ortiz, Bradford G. Nickerson Santiago, December 2010 Slide 2 Outline Discrete Unit Disc Cover Problem Line Separable Version Simple Greedy Algorithm Faster Dual Algorithm Approximating the General Problem Recent Advances 2 Slide 3 Discrete Unit Disk Cover (DUDC) Given m unit disks D (facilities) and n points Q (clients) in the plane, the discrete unit disk cover problem is to select a minimum subset of the disks to cover the points. 3 Slide 4 Applications 4 Slide 5 5 Slide 6 6 Slide 7 7 Slide 8 About Discrete Unit Disk Cover NP-Hard (Johnson, 1982) Geometric version of SET-COVER SET-COVER is not approximable within c log n DUDC admits constant factor approximation Related problems: Minimum Geometric Disk Cover: disk centres are not restricted. Discrete k-Centre: find set of k disks minimizing largest radius. 8 Slide 9 Approximation algorithms for DUDC: 108-approximate (C linescu et al., 2004) 72-approximate (Narayanappa & Voytechovsky, 2006) 38-approximate (Carmi et al., 2007) (1+)-approximate (Mustafa & Ray, 2009) Uses local improvement approach O(m 65 n) time in the worst case (3-approximation) This paper: 22-approximate, O(m 2 n 4 ) algorithm. 9 About Discrete Unit Disk Cover Slide 10 Line-separated DUDC d1d1 d2d2 d4d4 d3d3 d5d5 p1p1 p2p2 p4p4 p3p3 p5p5 q1q1 q2q2 q4q4 q3q3 q5q5 q6q6 q7q7 q9q9 q8q8 l 10 Slide 11 11 Slide 12 A Simple Greedy Algorithm Simplification rules: 1. If a disk d 1 covers no points from Q, we can remove it. 2. If a disk d 1 is dominated by a disk d 2, then we can remove d 1 from the problem instance. 3. If a point q 1 is only covered by a disk d 1, then d 1 must be part of the solution. d1d1 d2d2 d4d4 d3d3 d5d5 p1p1 p2p2 p4p4 p3p3 p5p5 q1q1 q2q2 q4q4 q3q3 q5q5 q6q6 q7q7 q9q9 q8q8 l Solution Set (D) d3d3 by Rule 3 Discarded d4d4 by Rule 2 d5d5 d2d2 d1d1 by Rule 1 12 by Rule 3 Slide 13 Simplification rules are not always sufficient. If no more simplification rules can be applied, then the leftmost disk is added to the solution set (disks are ordered by leftmost intersection with l). Greedy Step d1d1 d2d2 d3d3 p1p1 p2p2 p3p3 q3q3 q2q2 q1q1 l 13 Slide 14 The Greedy Algorithm 14 Slide 15 Faster Implementation 15 d1d1 d4d4 d3d3 d5d5 p1p1 p2p2 p4p4 p3p3 p5p5 q1q1 q2q2 q4q4 q3q3 q5q5 q6q6 q7q7 q9q9 q8q8 d2d2 1 d4d4 d1d1 d2d2 d3d3 d5d5 2 2 2 2 5 5 4 4 4 4 1 1 0 2 1 3 Slide 16 A faster algorithm p1p1 p2p2 p4p4 p3p3 p5p5 q1q1 q2q2 q4q4 q3q3 q5q5 q6q6 q7q7 q9q9 q8q8 l Solution Set (P) p 1 p 3 p 5 16 Slide 17 A faster algorithm p1p1 p2p2 p4p4 p3p3 p5p5 q1q1 q2q2 q4q4 q3q3 q5q5 q6q6 q7q7 q9q9 q8q8 l Solution Set (P) p1p1 p3p3 p5p5 17 Slide 18 Why is this optimal? 18 l d1d1 d k+1 dada Slide 19 Why is this optimal? (Case 1) 19 l d1d1 d k+1 dada s2s2 s1s1 Slide 20 Why is this optimal? (Case 2) 20 l d1d1 d k+1 dada s2s2 s1s1 Slide 21 Why is this optimal? (Case 3) 21 l d1d1 d k+1 dada s2s2 s1s1 Slide 22 Back to DUDC We have an exact algorithm for the line-separable case The goal is to obtain an approximation algorithm for the general problem We adapt the 38-approximation algorithm of Carmi et al. to obtain a 22-approximation to DUDC Their algorithm uses a variant of the line-separable discrete unit disk cover 22 Slide 23 Minimum Assisted Cover (MAC) u1u1 u2u2 u4u4 u3u3 u5u5 p1p1 p2p2 p4p4 p3p3 p5p5 q1q1 q2q2 q4q4 q3q3 q5q5 q6q6 q7q7 q9q9 q8q8 l l1l1 l2l2 Slide 24 Our LSDUDC algorithm plus greedy MAC gives a 2- approximation Use our algorithm to obtain a set U Use greedy MAC to obtain an improved solution A Minimum Assisted Cover (MAC) l Slide 25 How good is A? Separate A and OPT by l: OPT U, OPT L, A U, A L Let ac(U,OPT L ) be the smallest subset of U that forms a cover when assisted by OPT L A is the minimum size assisted cover based on U |A| = |A U |+|A L ||ac(U,OPT L )|+|OPT L | l U OPT L ac(U,OPT L ) Slide 26 Approximation Ratio d d Case 1Case 2 Slide 27 Approximation Ratio l Slide 28 l d vlvl vrvr Slide 29 29 Slide 30 Approximation of DUDC 3/2 Apply 2-approximation on each line in each direction. Each disk can participate in 8 applications of the algorithm. Carmi et al. give a 6- approximation for the single square problem. Approximation factor: 2 x 8 + 1 x 6 = 22 Worst case running time: O(m 2 n 4 ) Slide 31 Summary We presented an exact algorithm for the case when clients and facilities are separated by a line This allowed us to improve the approximation to the Minimum Assisted Cover problem We improved the approximation ratio from 38 to 22 for the general Discrete Unit Disk Cover problem O(m 2 n 4 ) running time 31 Slide 32 Recent Advances 32 Slide 33 Bonus Track: WALCOM paper Francisco Claude Gautam K. Das Reza Dorrigiv Stephane Durocher Bob Fraser Alex Lpez-Ortiz Bradford G. Nickerson Alejandro Salinger 33