Decomposing Coverings and the Planar Sensor Cover Problem

Transcription

1 Intro. previous work. Restricted Strip Cover Decomposing Multiple Coverings Decomposing Coverings and the Planar Sensor Cover Problem Matt Gibson Kasturi Varadarajan presented by Niv Gafni

2 Intro. previous work. Restricted Strip Cover Decomposing Multiple Coverings 1 Intro. Cover decomposable The Sensor cover problem Special cases of the Sensor cover problem 2 previous work. 3 Restricted Strip Cover RSC reminder Main algorithm Approximation Ratio 4 Decomposing Multiple Coverings Main theorem polygons to Wedges Level curves Algorithm for One Level curve Main Algorithm

3 Intro. previous work. Restricted Strip Cover Decomposing Multiple Cover Coverings decomposable The Sensor cover problem Special cases cover decomposable Definition we call a set of points P in the plane cover-decomposable if there exists a constant c > 0 (which may depend on P) such that any collection of translates of P, with the property that every point in the plane has c or more translates covering it, can be partitioned into two covers. Polygon Cover- each point with 3 polygons covering it is covered by polygon from both colors

4 Intro. previous work. Restricted Strip Cover Decomposing Multiple Cover Coverings decomposable The Sensor cover problem Special cases k-fold cover decomposability Definition given a collection of translates of P and any integer k, partition the collection into as many sub-collections as possible so that each sub-collection covers every point covered by k or more of the original translates Here, k=4 and each point with 4 polygons covering it is covered by polygon from all colors

5 Intro. previous work. Restricted Strip Cover Decomposing Multiple Cover Coverings decomposable The Sensor cover problem Special cases the Sensor cover problem Definitions given a set of sensors S in the universe U, for each s S, we let R(s) U denote the region that s covers (the range of s) and d(s) be the duration of s, which is a positive integer. for each x R(s) we say that s is live at x. X X s Here, R(s) is an hexagon. During d(s), s is live at x but not in x

6 Intro. previous work. Restricted Strip Cover Decomposing Multiple Cover Coverings decomposable The Sensor cover problem Special cases the Sensor cover problem Cont. schedule for a subset S S, we call an assignment of a positive integer t(s), s S a schedule of S, where t(s) is the start time of s. the sensors in S\S are said to be unassigned. A sensor s that is assigned a start time t(s) is said to be active at times {t(s), t(s) + 1,...t(s) + d(s) 1} cover given a schedule Sc over S, A point x U is said to be covered at time t > 0 if there is a sensor s such that x R(s) and s is active at time t. x U define the duration of x in the Sc to be M(Sc, x) = max {j : j j, s Sc, s covers x at time j}. If no sensor covers x at time 1, then define M(S, x) = 0. The duration of Sc is defined to be M(Sc) = min x M(Sc, x).

7 Intro. previous work. Restricted Strip Cover Decomposing Multiple Cover Coverings decomposable The Sensor cover problem Special cases schedule and cover Here, U is 1-dimentional, and r(s) spread from left to right. d(s) is vertical duration M(Sc) sensors U

8 Intro. previous work. Restricted Strip Cover Decomposing Multiple Cover Coverings decomposable The Sensor cover problem Special cases Sensor cover problem goal. the goal the goal of the Sensor cover problem is to compute a schedule Sc with maximum duration. Let OPT denote the duration of an optimal schedule The load at a point x U is L(x) = s S:x R(S) d(s). The load of the problem instance is L = min x L(x) Clearly, OPT L. so if we found an approximation with respect for L, it will also be with respect to OPT.

9 Intro. previous work. Restricted Strip Cover Decomposing Multiple Cover Coverings decomposable The Sensor cover problem Special cases special case 1: Restricted Strip Cover (RSC) Definitions smiler to the Sensor cover problem, only here U is a set of points on the real line,and the range R(s) of each sensor s is an interval on the real line. duration M(Sc) sensors U

10 Intro. previous work. Restricted Strip Cover Decomposing Multiple Cover Coverings decomposable The Sensor cover problem Special cases special case 2: Planar Sensor Cover Definitions here, U is a set of points in R 2,and the range R(s) of each sensor s is a translate of a fixed convex polygon.

11 Intro. previous work. Restricted Strip Cover Decomposing Multiple Coverings previous work- Cover decomposable In 1986, Mani and Pach showed that a unit disk is cover-decomposable (with the constant c being 33. In 1986, Pach showed that any centrally symmetric convex polygon is cover-decomposable. In 2007, Tardos and T oth showed that any triangle is cover-decomposable. In 2008, P alvolgyi and T oth shows that any convex polygon is cover-decomposable.

12 Intro. previous work. Restricted Strip Cover Decomposing Multiple Coverings previous work- k-fold cover decomposable In 2007, Tardos and T oth implies that a k-fold cover with translates of a triangle can be partitioned into Ω(logk) covers. in 2009, Pach and T oth showed that a k-fold cover with a centrally symmetric convex polygon P can be decomposed into Ω( k) covers, where the constant as before depends on P. in 2009, Aloupis et al. improved this result and obtained an optimal bound, showing that one can obtain Ω(k) covers.

13 Intro. previous work. Restricted Strip Cover Decomposing Multiple Coverings previous work- RSC in 2007, Buchsbaum et al. introduce the RSC problem, showed that it is NP-hard, and give an algorithm with polynomial-time and O(log log log n)-approximation, where n is the number of sensors.

14 Intro. previous work. Restricted Strip Cover Decomposing Multiple RSCCoverings reminder Main algorithm Approximation Ratio RSC-reminder Definitions given a set of sensors S in the universe U= (1,2,..,m), for each s S, we let R(s) = l(s), l(s) + 1,..., r(s)wherel(s), r(s) U denote the region that s covers. for each x R(s) we say that s is live at x. duration M(Sc) sensors U

15 Intro. previous work. Restricted Strip Cover Decomposing Multiple RSCCoverings reminder Main algorithm Approximation Ratio RSC-defenitions Definition With respect to a schedule S, we say the sensor s dominates coordinate x to the right if s extends as far to the right as possible (maximizes r(s)) among all sensors that have not been assigned and are live at x. the definition for dominates to the left is symmetrical. S3 S4 S2 S1 S5 S6 S7 S8 U X Here, S3 dominates X to the left, and S6 dominates X to the right, assuming they are not scheduled yet.

16 Intro. previous work. Restricted Strip Cover Decomposing Multiple RSCCoverings reminder Main algorithm Approximation Ratio Main algorithm algorithm t 0 Sc φ while TRUE t M(Sc) + 1 i = the first uncovered coordinate at time t and j = max(x U all coordinates in [i, x] are uncovered at t). s = a sensor that dominates i to the right. If ne, Return Sc. If s is not live at j t(s ) t and Sc Sc s. Else let s be the sensor that dominates j to the left. If M(Sc, i 1) M(Sc, j + 1) t(s ) t and Sc Sc s. Else t(s ) t and Sc Sc s.

17 Intro. previous work. Restricted Strip Cover Decomposing Multiple RSCCoverings reminder Main algorithm Approximation Ratio Theorem of ratio For an instance S of RSC, let OPT denote the duration of an optimal solution for S. We have the following theorem which we will prove in the remainder of this section. Theorem Given any instance S of Restricted Strip Cover, the algorithm returns a schedule Sc such that M(Sc) OPT 5.

18 Intro. previous work. Restricted Strip Cover Decomposing Multiple RSCCoverings reminder Main algorithm Approximation Ratio Theorem of ratio Definition For x U and t 0, we define coverage(x, t) to be the number of sensors that cover x at time t in the schedule output by our algorithm. Lemma If coverage(x, t) c x U and t 0, then the duration t f of the schedule we output is at least L c. Lemma For any x U and t 0, coverage(x, t) 5.

19 Intro. previous work. Restricted Strip Cover Decomposing Multiple MainCoverings theorem polygons to Wedges Level curves Algorithm f Decomposing Multiple Coverings Main Theorem Theorem For any convex polygon P in the plane, there exists a constant α 1 so that for any k 1 and any finite collection of translates of P, we can partition the collection into k/α sub-collections, each of which covers any point in the plane that is covered by k or more translates in the original collection. Such a partition can be computed by an efficient algorithm Here, k=4 and each point with 4 polygons covering it is covered by polygon from all colors

20 Intro. previous work. Restricted Strip Cover Decomposing Multiple MainCoverings theorem polygons to Wedges Level curves Algorithm f adaptation For a planar set T and a point x in the plane, let T(x) denote the translate of T with centroid x. reflection Let P be the reflection of the polygon P. x, p U, p P(x) x P(p) P P p x

21 Intro. previous work. Restricted Strip Cover Decomposing Multiple MainCoverings theorem polygons to Wedges Level curves Algorithm f different version of Theorem 1. Theorem 1. it is sufficient for us to show that there exists a constant α 1 so that for any k 1 and any collection Q of points in the plane, it is possible to assign each point in Q a color from {1, 2,..., k α }, so that any translate of P with P Q k contains a point colored i, 1 i k α

22 Intro. previous work. Restricted Strip Cover Decomposing Multiple MainCoverings theorem polygons to Wedges Level curves Algorithm f possible example x We search for all the points of Q that are in P(x) instead of the points s of Q in which x is in P(s) the results is the same. x If P(x) contains more than k points (here it can be 5), We want a coloring in which P(x) will contain at all k α colors. Here it is 3 colors

23 Intro. previous work. Restricted Strip Cover Decomposing Multiple MainCoverings theorem polygons to Wedges Level curves Algorithm f adding a grid Let c be equal to half the minimum distance between two points on non-consecutive edges of P. We lay a square grid of side c on the plane. any translate of P intersects β O(1) grid cells,and each grid cell intersects at most two sides of a translate. moreover, if a grid cell does intersect two sides of a translate, then these sides must be adjacent in P. Minimum distance

24 Intro. previous work. Restricted Strip Cover Decomposing Multiple MainCoverings theorem polygons to Wedges Level curves Algorithm f adding a grid For a subset (region) R of the plane and for a subset X of points, denote load X (R) to be the number of points in X that lie in R. Since each grid cell intersects at most two edges of P(u), it must be that the intersection of a grid cell with P(u) is the same as the intersection of the grid cell with a wedge whose boundaries are parallel to two adjacent edges of P(u). If both boundaries are with vertex p i, then we call the wedge an i-wedge. definition For a point q in the plane, we denote W i (q) to be the i-wedge with apex q.

25 Intro. previous work. Restricted Strip Cover Decomposing Multiple MainCoverings theorem polygons to Wedges Level curves Algorithm f wedge example W 2(y) W 4(y) W 1(z) W 1(x) W 0(w) Wedges are named after the vertex indices Different wedges can be apex on different point

26 Intro. previous work. Restricted Strip Cover Decomposing Multiple MainCoverings theorem polygons to Wedges Level curves Algorithm f Theorem 2. Theorem 1 is established by applying the following theorem to the points Y within each grid cell G. Theorem 2. There exists a constant α 1 so that for any k 1 and any collection Y of points in the plane, it is possible to assign each point in {1, 2,..., k α }, so that any i-wedge that contains k or more points from Y contains a point colored j, 1 j k α.

27 Intro. previous work. Restricted Strip Cover Decomposing Multiple MainCoverings theorem polygons to Wedges Level curves Algorithm f definition definition Let W j i be the set of apices of all i-wedges W such that load Y (W ) = j i = 0, 1,...u 1. the Level curve C i (r) is defined as the boundary of the region j r W j i i = 0, 1,...u 1 C 1 (2) C 1 (5) C 4 (1) z

28 Intro. previous work. Restricted Strip Cover Decomposing Multiple MainCoverings theorem polygons to Wedges Level curves Algorithm f observations observation For any x C i (r), r load Y (W i (x)) r + 1. observation Any i-wedge W such that load Y (W ) r contains an i-wedge whose apex belongs to C i (r).

29 Intro. previous work. Restricted Strip Cover Decomposing Multiple MainCoverings theorem polygons to Wedges Level curves Algorithm f computecover(i, Q, t) it is sufficient to prove Theorem 2 for the i-wedges with apex on C i (k), for each 0 i u 1. In order to do this, we will use the procedure computecover(i, Q, t). this procedure takes as input one level curve C i (k), a positive integer t, and a subset Q Y. The input to the procedure has the guarantee that for any i-wedge W with apex on C i (k), we have W Q 2t. The output is a partial coloring of the points of Q with colors {1, 2,..., t} so that any i-wedge W with apex on C i (k): (a) contains a point colored j, for 1 j t (b) contains at most 2t colored points.

30 Intro. previous work. Restricted Strip Cover Decomposing Multiple MainCoverings theorem polygons to Wedges Level curves Algorithm f Main Algorithm Algorithm Y Y i {0, 1,..., u 1} L min{load Y (W z (x)) : W z (x) is a j-wedge on a level curve not yet calculated} X i a subset of Y which will be scheduled. L Run computecover(i, X i, 64u ) Y will be the remaining unscheduled points

31 Intro. previous work. Restricted Strip Cover Decomposing Multiple MainCoverings theorem polygons to Wedges Level curves Algorithm f Algorithm satisfy the theorem Lemma Suppose at the beginning of iteration i, all j-wedges with apex on C j (k) have load at least L from points in Y for j i, where L is larger than some absolute constant. After the i-th iteration of the algorithm, any j-wedge W j (x), for j > i, and with apex x on C j (k) has load at least L 5u from points in Y. L It is clear to see that after the call to computecover(i, Xi, 64u ), any i-wedge with apex on C i (k) contains points colored L 1, 2,..., 64u. the lemma implies that L, which equals k before the 0-th iteration, drops by a factor of at most 5u with each iteration. this means that L Ω(k), and that the algorithm produces a coloring as required in Theorem 2.