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Efficient Algorithms for Minimum Unit Disk Covering in Euclidean Plane

Explore strategies to determine the minimum number of unit disks needed to cover a set of points in the Euclidean plane. Develop a partitioning approach to construct minimum unit disk covers in each cell. An approximation algorithm is proposed for faster computation. Investigate dominating set problems in unit disk graphs and connected dominating sets. Analyze the feasibility of PTAS and techniques for weighted dominating set and dominating set in intersection disk graphs.

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Efficient Algorithms for Minimum Unit Disk Covering in Euclidean Plane

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  1. Chapter 4 Partition (1) Shifting Ding-Zhu Du

  2. Disk Covering • Given a set of n points in the Euclidean plane, find the minimum number of unit disks to cover the n given points.

  3. a (x,x) Partition P(x)

  4. Construct Minimum Unit Disk Cover in Each Cell Each square with edge length 1/√2 can be covered by a unit disk. Hence, each cell can be covered By at most disks. 1/√2 Suppose a cell contains ni points. Then there are ni(ni-1) possible positions for each disk. Minimum cover can be computed In time ni 2 O(a )

  5. Solution S(x) associated with P(x) For each cell, construct minimum cover. S(x) is the union of those minimum covers. Suppose n points are distributed into k cells containing n1, …, nk points, respectively. Then computing S(x) takes time n1 + n2 + ··· + nk< n 2 2 2 2 O(a ) O(a ) O(a ) O(a )

  6. Approximation Algorithm For x=0, -2, …, -(a-2), compute S(x). Choose minimum one from S(0), S(-2), …, S(-a+2).

  7. Analysis • Consider a minimum cover. • Modify it to satisfy the restriction, i.e., a union of disk covers each for a cell. • To do such a modification, we need to add some disks and estimate how many added disks.

  8. Added Disks Count twice Count four times 2

  9. Shifting 2

  10. Estimate # of added disks Shifting

  11. Estimate # of added disks Vertical strips Each disk appears once.

  12. Estimate # of added disks Horizontal strips Each disk appears once.

  13. Estimate # of added disks # of added disks for P(0) + # of added disks for P(-2) + ··· + # of added disks for P(-a+2) < 3 opt where opt is # of disk in a minimum cover. There is a x such that # of added disks for P(x) < (6/a) opt.

  14. Performance Ratio P.R. < 1 + 6/a < 1 + ε when we choose a = 6 ⌠1/ε . 2 O(1/ε ) Running time is n.

  15. Unit disk graph < 1

  16. Dominating set in unit disk graph • Given a unit disk graph, find a dominating set with the minimum cardinality. • Theorem This problem has PTAS.

  17. Connected Dominating Set in Unit Disk Graph • Given a unit disk graph G, find a minimum connected dominating set in G. Theorem There is a PTAS for connected dominating set in unit disk graph.

  18. central area h+1 Boundary area h

  19. Why overlapping? cds for G cds for each connected component 1

  20. Construct PTAS For each partition P(a,a), construct C(a) as follows: 1. In each cell, construct MCDS for each connected component in the inner area. 2. Connect those minimum connected dominating sets with a part of 8-approximation lying in boundary area. Choose smallest C(a) for a = 0, h+1, 2(h+1), ….

  21. Existence of 8-approximation • There exists (1+ε)-approximation for minimum • dominating set in unit disk graph. 2. We can reduce one connected component with two nodes. Therefore, there exists 3(1+ε)-approximation for mcds.

  22. 8-approximation • A maximal independent set has size at most • 4 mcds +1. 2. There exists a maximal independent set, connecting it into cds need at most 4mcds nodes.

  23. MCDS (Time) • In a square of edge length , any node can • dominate every bode in the square. • Therefore, minimum dominating set has size • at most . a

  24. MCDS (Time) 2. The total size of MCDSs for connected components in an inner square area is at most . a

  25. MCDS (Size) • Modify a mcds for G into MCDSs in each cell. • mcds(G): mcds for G • mcdscell(inner): MCDS in a cell for connected components in inner area

  26. Connect & Charge charge

  27. Multiple Charge How many possible charges for each node? charge How many components can each node be adjacent to?

  28. 1. How many independent points can be packed by a disk with radius 1? 5! 1 >1

  29. Each node can be charged at most 10 times!!!

  30. Shifting h=2 a/(2(h+1)) = integer 2 O(a ) Time=n 3

  31. Weighted Dominating Set • Given a unit disk graph with vertex weight, find a dominating set with minimum total weight. • Can the partition technique be used for the weighted dominating set problem?

  32. Dominating Set in Intersection Disk Graph • An intersection disk graph is given by a set of points (vertices) in the Euclidean plane, each associated with a disk and an edge exists between two points iff two disks associated with them intersects. • Can the partition technique be used for dominating set in intersection disk graph?

  33. Thanks, End

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