Geometric Optimization

Transcription

1 Geometric Optimization Mahdi Cheraghchi-Bashi-Astaneh Department of Computer Engineering Sharif Institute of Technology Monday May 31, home page:

2 Outline Introduction The Covering Problem The Shifting Strategy Problems Similar to Covering Euclidean Traveling Salesman Problem A Randomized Dynamic PTAS for ETSP Problems Similar to ETSP Discussion Mahdi Cheraghchi Talk: Geometric Optimization 2/25

3 Introduction (1/3) We are talking about maximization or minimization of something according to given constraints. So far, we have seen: Linear Programming Minimum Enclosing Disk Euclidean Shortest Path All these problems are solvable in polynomial time (2D case). Mahdi Cheraghchi Talk: Geometric Optimization 3/25

4 Introduction (2/3) Unfortunately, many interesting optimization problems are hard. Such hard problems include: Covering, Packing, and Piercing ETSP, k-tsp, MST, k-mst Exact algorithms are slow. Many optimization problems in combinatorics have geometric versions. We develop approximation schemes! Mahdi Cheraghchi Talk: Geometric Optimization 4/25

5 Introduction (3) Some interesting and similar optimization problems over data streams: [Ind04] Euclidean Minimum Spanning Tree (MSpT) Euclidean Minimum Weight Matching (MWM) Facility Location: For a parameter f > 0, find a facility set F P that minimizes f F +C(F,P), where C( F, P) k-median: Find a set Q of k points to minimize C(Q,P). pp min qf p q Mahdi Cheraghchi Talk: Geometric Optimization 5/25

6 The Covering Problem The goal is to cover n given points on the plane with a minimal number of disks of fixed diameter D. Some variants involve more dimensions and/or dierent shapes for covering objects. The problem is strongly NP-Complete. Solution: Develop a PTAS by the widelyused shifting technique. Mahdi Cheraghchi Talk: Geometric Optimization 6/25

7 The Shifting Strategy (1/5) A form of divide and conquer. Based on dividing the area rectangle into smallenough cells so that we can solve the problem for each cell by using a brute-force algorithm. Then we combine the results to obtain an approximation. We consider all possible shifts of the grid and maintain the best approximation. Let I be the bounding box, and positive integer be a fixed shifting parameter. Mahdi Cheraghchi Talk: Geometric Optimization 7/25

8 The Shifting Strategy (2/5) First, we subdivide the area of I into vertical (left closed and right open) strips of width D each. We consider groups of consecutive strips (a group can be wrapped around), thus larger strips of width D each. There are possible ways for such grouping. We order partitions so that each one can be obtained by shifting the previous one to the right over distance D. shifted partitions are denoted by S 1,S 2,,S. Mahdi Cheraghchi Talk: Geometric Optimization 8/25

9 The Shifting Strategy (3/5) Let A be any local algorithm, and A(S i ) be the algorithm that applies A on each strip of S i, and outputs the union of all disks. S A is defined as the algorithm that delivers the minimum answer among all possibilities of S i. r B, the Performance Ratio of an algorithm B is defined as supremum of Z B / OPT over all problem instances, where Z B is the value of solution delivered by B, and OPT is the optimal solution set. For our problem, r B 1. Mahdi Cheraghchi Talk: Geometric Optimization 9/25

10 The Shifting Strategy (4/5) r A The Shifting Lemma: Proof: By definition, for fixed i, S r A 1 1 and Z JS i A( S i ) OPT J r A JS i OPT OPT where OPT J is the optimal set of strip J, and OPT (i) is the set of disks in OPT covering points in any two adjacent D strips in S i. Clearly, the sets OPT (i) are disjoint for i = 1,2,,. J OPT (i) (1) (2) Mahdi Cheraghchi Talk: Geometric Optimization 10/25

11 The Shifting Strategy (5) Proof (continued): i 1 (OPT OPT ) ( 1) OPT Expressions (1), (2) and (3) imply ( i) (3) Z S A A( Si ) (1) min Z min( ra i1,, i1,, (1/ ) r OPT 1 A i J JS i (2,3) r 1 1/ OPT A JS i OPT J ) Q.E.D. Mahdi Cheraghchi Talk: Geometric Optimization 11/25

12 PTAS for Covering (1/2) PTAS Theorem: There is an algorithm H d For ball-covering in d-dimensions with a running time of O( d ( d ) (2n) d( d ) 1 and performance ratio of at most (1+1/) d. Construction: For d=1, the problem is easily solvable in linear time. For d >1, We apply the shifting strategy for d nested levels, until we reach cells of constant size. A brute-force exact algorithm in such cells take constant time with respect to n. Immediately from the shifting lemma, it follows that performance ratio is no more than (1+1/) d. d d ) Mahdi Cheraghchi Talk: Geometric Optimization 12/25

13 PTAS for Covering (2) Running time for 2D: Each cell is a square of side length D. A local algorithm can choose disks such that a disk covering at least two points has two on its boundary. 2 2 disks are enough to cover an entire cell. Just two ways to draw a circle of given diameter through two give points, thus 2 C(n, 2) possible positions. Thus a mapping from 2 2 choices to O(n 2 ) positions, O(n 4^2 ) arrangements. Validity of each arrangement is then verified in O( 2 n ) steps. The number of cells is 2, thus the total running time is O( 4 n 4^2+1 ). Proof for more dimensions is analogous. Mahdi Cheraghchi Talk: Geometric Optimization 13/25

14 Generalizations The strategy works for other shapes. For rectilinear blocks of given side lengths, H d delivers a cover in O( d n 2^d+1 ) time with performance ratio of at most (1+1/) d. (Application: image processing) The same strategy can also be applied to a number of other strongly NP-hard optimization problems, such as packing and piercing. In packing, we are given n fixed-diameter disks. We wish to find a maximal subset of disjoint disk. (Application: VLSI fibers) In piercing, again we are given n disks. We are asked to compute a minimal set of points that has a nonempty intersection with all disks. Mahdi Cheraghchi Talk: Geometric Optimization 14/25

15 Euclidean TSP (1/2) Given n nodes in the plane, find the smallest tour connecting them. We perform a recursive geometric partitioning of the instance, and then perform dynamic programming. The partitioning is a variant of quadtree. Again, a (dierent) shifting strategy is employed. Mahdi Cheraghchi Talk: Geometric Optimization 15/25

16 Euclidean TSP (2) Let I be the bounding box (smallest covering square), and L be the length of its side. Clearly, OPT L. The algorithm starts with a perturbation phase. We wish to compute a (1+1/c)-approximation. We place a grid of granularity L/8nc in the plane, and move each point to its nearest grid-point. The we scale distances by L/64nc. We get the following results: All nodes have integral coordinates. Nonzero internode distances are between 8 and (n). The size of the bounding box is O(nc). We need to compute a (1+3/4c)-approximation in this instance. A dissection is defined in the same manner as quadtree, but we stop partitioning a square only if it has size 1. Mahdi Cheraghchi Talk: Geometric Optimization 16/25

17 Dissection versus Quadtree (1/3) Mahdi Cheraghchi Talk: Geometric Optimization 17/25

18 Dissection versus Quadtree (2/3) Clearly, a dissection cell contains at most one rounded node. Both have depths of O(logn), but a dissection has O(n 2 ) squares while a quadtree has O(n logn). The (a, b)-shift of a dissection is defined as its translation modulo L so that its center is translated to coordinates (a, b). (Some squares wrap-around) The quadtree with shift (a, b) is obtained from corresponding shifted dissection by cutting-o the partitioning at squares that contain only 1 node. This can be done in O(n log 2 n) time. Mahdi Cheraghchi Talk: Geometric Optimization 18/25

19 Dissection versus Quadtree (3) We allow a salesman tour to deviate from straight line at prespecified points, and have bent edges. Such a tour is called a salesman path. Let m, r be positive integers. An m-regular set of portals for a shifted dissection is a set of points on the edges of the squares in it, such that each square has a portal at each of its 4 corners and m other equally-spaced portals on each edge. A salesman path is (m, r)-light if it crosses each edge of each square in the dissection at most r times and always through m-portals. Mahdi Cheraghchi Talk: Geometric Optimization 19/25

20 A Randomized Dynamic Algorithm (1/3) Structure Theorem: If we pick shifts 0a,b<L randomly, with probability at least ½, the shifted dissection has an associated (m,r)- light salesman path of cost at most (1+1/c) OPT, where m=o(c logl) and r =O(c). First, we pick a random shift and compute the corresponding quadtree. Then we find the optimal (m,r)-light path. The tour crosses the boundary of each square at most 4r times. We try all possible arrangements. Mahdi Cheraghchi Talk: Geometric Optimization 20/25

21 A Randomized Dynamic Algorithm (2/3) We maintain a lookup table for subproblems specified by: a. One of the T squares in the shifted quadtree. b. A set of r portals on each of the four edges. c. A pairing between the portals in (b). Each choice in (b) and (c) determine an instance of multiple (team) TSP. Size of the table is O(T (m+3) 4r (4r)!). We build the table in a bottom-up fashion. Leaves contain at most one node and can be solved optimally in 2 O(r) time. For other squares, we enumerate all possible choices of the multiple-salesman tour. Mahdi Cheraghchi Talk: Geometric Optimization 21/25

22 A Randomized Dynamic Algorithm (3) For non-leave entries of the table, the algorithm enumerates all possible choices of portals and their arrangement. The number of choices is O((m+3) 4r (4r) 4r (4r)!). Thus the total running time is O(T (m+3) 8r (4r) 4r [(4r)!] 2 ), which is O(n(logn) O(c) ). We can make the algorithm deterministic by going through all possible shifts, which multiplies the running time by O(n 2 ). Mahdi Cheraghchi Talk: Geometric Optimization 22/25

23 Similar Problems By using the mentioned technique, we can develop approximation schemes for the following problems: a. Minimum Steiner Tree (MST). b. k-tsp (the smallest tour that visits at least k nodes) c. k-mst (Find k nodes with the shortest MST) d. Euclidean Mincost Perfect Matching (EMCPM) Mahdi Cheraghchi Talk: Geometric Optimization 23/25

24 Discussion Mahdi Cheraghchi Talk: Geometric Optimization 24/25

25 References [Ind04] Piotr Indyk, Algorithms for Dynamic Geometric Problems over Data Streams. STOC 2004, To Appear. [HM85] Dorit S. Hochbaum and Wolfgang Maass, Approximation Schemes for Covering and Packing Problems in Image Processing and VLSI. J. ACM 32.1, pp [Aro96] Sanjeev Arora, Polynomial Time Approximation Schemes for Euclidean TSP and other Geometric Problems. FOCS 1996, pp [Aro97] Sanjeev Arora, Nearly Linear Time Approximation Schemes for Euclidean TSP and other Geometric Problems. FOCS 1997, pp [Mit99] Joseph S. B. Mitchell, Guillotine subdivisions approximate polygonal subdivisions: A Simple Polynomial-Time Approximation Scheme for Geometric TSP, k-mst, and Related Problems. SIAM J. Comput. 28(4), pp (1999) [Mit97] Joseph S. B. Mitchell, Guillotine subdivisions approximate polygonal subdivisions: Part III Faster polynomial-time approximation schemes for geometric network optimization. Preprint. [Cha03] Timothy M. Chan, Polynomial-Time Approximation Schemes for Packing and Piercing Fat Objects. J. Algorithms 46(2), pp (2003) Mahdi Cheraghchi Talk: Geometric Optimization 25/25