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Practical Discrete Unit Disk Cover Using an Exact Line-Separable AlgorithmAlejandro 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 2010OutlineDiscrete Unit Disc Cover ProblemLine Separable VersionSimple Greedy AlgorithmFaster Dual AlgorithmApproximating the General ProblemRecent Advances22Discrete 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.3Applications

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Applications

7About Discrete Unit Disk CoverNP-Hard (Johnson, 1982)Geometric version of SET-COVERSET-COVER is not approximable within c log nDUDC admits constant factor approximationRelated problems:Minimum Geometric Disk Cover: disk centres are not restricted.Discrete k-Centre: find set of k disks minimizing largest radius.

8Approximation algorithms for DUDC:108-approximate (Clinescu et al., 2004)72-approximate (Narayanappa & Voytechovsky, 2006)38-approximate (Carmi et al., 2007)(1+)-approximate (Mustafa & Ray, 2009)Uses local improvement approachO(m65n) time in the worst case (3-approximation)This paper:22-approximate, O(m2n4) algorithm.9About Discrete Unit Disk CoverLine-separated DUDCd1d2d4d3d5p1p2p4p3p5q1q2q4q3q5q6q7q9q8l10

11A Simple Greedy AlgorithmSimplification rules:If a disk d1 covers no points from Q, we can remove it.If a disk d1 is dominated by a disk d2, then we can remove d1 from the problem instance. If a point q1 is only covered by a disk d1, then d1 must be part of the solution.d1d2d4d3d5p1p2p4p3p5q1q2q4q3q5q6q7q9q8lSolution Set (D)d3by Rule 3Discardedd4by Rule 2d5d2by Rule 2d1by Rule 112by Rule 3Simplification 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 Stepd1d2d3p1p2p3q3q2q1l13The Greedy Algorithm14Faster Implementation15d1d4d3d5p1p2p4p3p5q1q2q4q3q5q6q7q9q8d21d4d1d2d3d522544110213A faster algorithmp1p2p4p3p5q1q2q4q3q5q6q7q9q8lSolution Set (P)p1 p3 p5 16A faster algorithmp1p2p4p3p5q1q2q4q3q5q6q7q9q8lSolution Set (P)p1p3p517Why is this optimal?18ld1dk+1daWhy is this optimal? (Case 1)19ld1dk+1das2s1Why is this optimal? (Case 2)20ld1dk+1das2s1Why is this optimal? (Case 3)21ld1dk+1das2s1Back to DUDCWe have an exact algorithm for the line-separable caseThe goal is to obtain an approximation algorithm for the general problemWe adapt the 38-approximation algorithm of Carmi et al. to obtain a 22-approximation to DUDCTheir algorithm uses a variant of the line-separable discrete unit disk cover22Minimum Assisted Cover (MAC)u1u2u4u3u5p1p2p4p3p5q1q2q4q3q5q6q7q9q8ll1l2Our LSDUDC algorithm plus greedy MAC gives a 2-approximationUse our algorithm to obtain a set UUse greedy MAC to obtain an improved solution A

Minimum Assisted Cover (MAC)lHow good is A?Separate A and OPT by l: OPTU, OPTL, AU, ALLet ac(U,OPTL) be the smallest subset of U that forms a cover when assisted by OPTLA is the minimum size assisted cover based on U|A| = |AU|+|AL||ac(U,OPTL)|+|OPTL|lUOPTLac(U,OPTL) Approximation RatioddCase 1Case 2Approximation RatiolApproximation RatioldvlvrApproximation Ratio29Approximation of DUDC3/2Apply 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 = 22Worst case running time:O(m2n4)SummaryWe presented an exact algorithm for the case when clients and facilities are separated by a lineThis allowed us to improve the approximation to the Minimum Assisted Cover problemWe improved the approximation ratio from 38 to 22 for the general Discrete Unit Disk Cover problemO(m2n4) running time

31Recent Advances32Bonus Track: WALCOM paperFrancisco ClaudeGautam K. DasReza DorrigivStephane DurocherBob FraserAlex Lpez-OrtizBradford G. NickersonAlejandro Salinger33