P. M. Vaidya, Geometry helps in matching, SIAM J. Comput., 18 (1989), pp. 1201--1225.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....large enough instances, it seems quite difficult to come up with small gaps within a very short (i.e. near linear in n) time. Moreover, the methods involved only use triangle inequality, and disregard the special properties of geometric instances. For the Minimum Weight Matching problem, Vaidya [19] showed that there is algorithm of complexity O(n 2:5 log 4 n) for planar geometric instances, which was improved by Varadarajan [20] to O(n 1:5 log 5 n) Cook and Rohe [6] also made heavy use of geometry to solve instances with up to 5,000,000 points in the plane within about 1.5 days of ....

P. M. Vaidya. Geometry helps in matching. SIAM J. Comput., 18 (1989), 1201--1225.

....this subject. In contrast to the TSP which is an NP problem the Euclidean matching can be formulated as linear programming problem and, therefore, has a polynomial running time exact solution of order n 3 . The fastest known exact solution for d = 2 is of the order O(n 2:5 (log n) 4 ) see Vaidya (1989)) But the improvement of the order of the running time to n(log n) d Gamma1 is considerable and of practical interest also for this combinatorial optimization problem. We analyze the partitioning algorithm in the random model where fX i g iin are independent random vectors uniformly ....

....can solve an Euclidean matching problem with k points in time f(k) Dk p with constants Birgit Anthes and Ludger Ruschendorf 5 p, D. This is fulfilled for p = 3 by Papadimitriou and Steiglitz (1982, Theorem 11.3, Problem 14) Improvements of this order to 0(n 2:5 (log n) 4 ) are given in Vaidya (1989) to some more general class of geometric algorithms in dimension two. By the assumption that X 1 ; X n are independent, uniformly distributed on [0; 1] d the number n i of points in C i are binomial distributed random variables P (n i = k) i n k j i 1 m d j k i 1 Gamma 1 m ....

Vaidya, P. M. (1989). Geometry helps in matching. SIAM J. Comp. 18, 1201--1225.

....is relaxation and establishment of a match between two sets of points based on the highest figure of merit computed for each possible translation. Some authors have considered the point pattern matching problem in the framework of graph matching. Proposed solutions include complete matching (e.g. [Vaidya 1989]) matching a sample graph with a subgraph of a larger model graph (e.g. Pearce et al. 1994] or matching graphs whose topologies are corrupted by errors in early stages of image processing (e.g. Wilson and Hakcock 1993] 1 The common characteristic shared by different methodologies ....

Vaidya, P.M. 1989. Geometry helps in matching. SIAM Journal on Computing, 18, 6, 1201-1225.

....isomorphisms. For general graphs, there are no polynomial time algorithms for both these problems. It is not known if the graph isomorphism problem is NP complete. However, the subgraph isomorphism problem is known to be NP complete. For special classes of graphs, more e#cient algorithms exist [4, 12, 21, 118]. Due to its discrete nature, graph matching is not very robust for errors: the success of the technique depends on the correct extraction of the graphs from the input. Another limitation of graph matching is a lack of discernment: large classes of patterns share the same graph. 10 CHAPTER 1. ....

....a lower bound for all other metric based pattern matching problems. Bottleneck matching The bottleneck distance between two finite point sets of equal cardinality is the minimum over all bijections between the sets over the maximum distance between each two points that are related in a bijection [54, 53, 118]. The formal definition is as follows. Let A and B be finite subsets of a space X with metric #. Assume that A and B have the same cardinality. Let F (A, B)betheset of all bijections from A onto B. Then, the # based bottleneck distance b # is defined as b # (A, B) min f#F (A,B) max a#A ....

P. M. Vaidya. Geometry helps in matching. SIAM J. Computing, 18(6):1201--1224, December 1989.

....Section 7) That is, we are given a set of n blue points and a set of n red points in the plane, and we wish to find a matching between the blue points and the red points, which minimizes the sum of the distances between the pairs of matched points. Our solution is based on the algorithm of Vaidya [67], which requires a data structure for answering nearest neighbor queries in a dynamic setting, where the distance to each of the maintained sites is the Euclidean distance plus some additive weight associated with the site. Using our dynamic nearestneighbor searching technique, we can improve the ....

....points, and the weight of a matching is the sum of the weights of its pairs. The standard Hungarian method [44, 45] yields an O(n 3 ) time algorithm for computing M . Exploiting the fact that the weights are Euclidean distances, Vaidya obtained an O(n 2:5 log n) time algorithm for computing M [67]. A number of efficient algorithms have been developed for computing a minimum weight bipartite Euclidean matching when P and Q have some special structure [10, 17, 48] but no progress has been made when P and Q are arbitrary sets of points in the plane. Recently, Alon and Itai [33] have proposed ....

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P. Vaidya, Geometry helps in matching, SIAM J. Comput. 18 (1989), 1201--1225.

....Section 7) That is, we are given a set of n blue points and a set of n red points in the plane, and we wish to find a matching between the blue points and the red points, which minimizes the sum of the distances between the pairs of matched points. Our solution is based on the algorithm of Vaidya [67], which requires a data structure for answering nearest neighbor queries in a dynamic setting, where the distance to each of the maintained sites is the Euclidean distance plus some additive weight associated with the site. Using our dynamic nearestneighbor searching technique, we can improve the ....

....points, and the weight of a matching is the sum of the weights of its pairs. The standard Hungarian method [44, 45] yields an O(n 3 ) time algorithm for computing M . Exploiting the fact that the weights are Euclidean distances, Vaidya obtained an O(n 2:5 log n) time algorithm for computing M [67]. A number of efficient algorithms have been developed for computing a minimum weight bipartite Euclidean matching when P and Q have some special structure [10, 17, 48] but no progress has been made when P and Q are arbitrary sets of points in the plane. Recently, Alon and Itai [33] have proposed ....

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P. Vaidya, Geometry helps in matching, SIAM J. Comput. 18 (1989), 1201--1225.

.... matching for a general bipartite graph is known to have an O(N 3 ) time algorithm (see Lawler [18] for this and other background on matching) and for graphs with nodes in the plane with the Euclidean distance as cost function, there is a O(N 2:5 log N) time algorithm, due to Vaidya [22]. The minimum cost matching problem is substantially easier in the case where the nodes are in line like order or are circularly ordered. The simplest versions of line like circular orderings are where the points lie on a line or lie on a curve homeomorphic to a circle, and the cost c(x; y) of an ....

P. M. Vaidya, Geometry helps in matching, SIAM J. Comput., 18 (1989), pp. 1201--1225.

....so it is Eulerian, and an Euler tour of this graph has length at most 3 2 TSP(S) Eliminating repeats as above can only improve this tour. Figure 8.2 shows that the ratio of 3 2 is tight. The running time of Christofides algorithm depends on the time bound for Euclidean matching. Vaidya [Vai88] gave an O(n 2.5 log 4 n) matching algorithm. Optimal Christofides FIGURE 8.2 The tour found by Christofides algorithm can have length 3 2 times optimal. 8.2.2 HEURISTICS An insertion method [RSL77] adds the points of S one by one in some order. A newly inserted point s j is ....

P. Vaidya. Geometry helps in matching. In Proc. 20th ACM Symp. Theory of Computing, pages 422--425, 1988.

....17] we can achieve 3# TSP = 3(1 #) in a running time of O(N O( 1 # ) If we use the Christofides heuristic, we obtain an overall approximation factor of 3# TSP = 4.5, at a running time of f(N ) O(N 2. 5 log 4 N ) the bottleneck being the computation of the minimum weight matching [21]) Alternatively, we can apply the approximate matching result of Vaidya [22] within the Christofides heuristic, to achieve a factor of 4.5 # at a time complexity of f(N ) O(N 1.5 log 2.5 N ) Finally, by simply doubling the minimum spanning tree, we obtain a factor of 6, with a simple ....

P. M.Vaidya. Geometry helps in matching. SIAM Journal of Computing, 18, 1989 pp. 1201--1225.

....the solutions to several bichromatic closest pair problems, and to solving a dynamic graph MST problem in which edges are inserted or deleted according to the behavior of the closest pairs. If we were performing insertions only, maintaining bichromatic closest pairs would not be difficult. Vaidya [27] described a method for performing insertions in the red set, and both insertions and deletions in the blue set, in time O(n 1 2 log n) per operation. But no fully dynamic algorithm for this problem was previously known. A generalization of bichromatic closest pairs was described by Dobkin and ....

....that for the nearest neighbor data structure, O(n 1 # ) in R 2 , or worse bounds in higher dimensions. We note that for rectilinear (L 1 and L # ) metrics, orthogonal range query data structures can be used to answer dynamic bichromatic closest pair queries in O(log d n) time per update [27]. Theorem 3. The rectilinear MST of a set of n points in R d can be maintained in time O(n 1 2 log d n) per update. 11 7 Conclusions We have described algorithms for maintaining the minimum spanning tree of a changing point set, the bichromatic closest pair of a colored point set, and ....

P. M. Vaidya. Geometry helps in matching. SIAM J. Comput., 18:1201--1225, 1989.

....problem in lower dimensions (e.g. pointlocation [31, 30] or computing the area of rectangles [29] or for a di#erent problem (e.g. convex layers [6] or hidden surface removal [4] As a result, a lot of attention has been paid to studying dynamic geometric algorithms. For some examples see [3, 4, 16, 18, 20, 23, 28, 32, 33, 34], or see [11] for an excellent survey on this subject. Researchers have also studied the special cases when objects are only allowed to be inserted (deletions are not allowed) or when the sequence of insertions and deletions is known in advance. If the problem is decomposable (i.e. the solutions ....

....) Recently Chazelle et al. 7] showed that a bi chromatic farthest pair can be computed in time O(n 2(1 1 (#d 2# 1) # ) The diameter can be computed using this algorithm by setting R = B = S. Dobkin and Suri [16] gave e#cient solutions to these problems in the semi online model. Vaidya [34] described an algorithm for closest pairs which allows insertions and deletions to R, but only allows insertions in B. Supowit [33] showed that if we allow only deletions, then the diameter of a planar point set can be maintained in O( # n log n) amortized time. But even in two dimensions, no ....

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P.M. Vaidya. Geometry helps in matching. SIAM J. Comput. 18 (1989) 1201--1225. 10

....of size two. The weight wt(M) of a perfect matching M is de ned as the sum of the Euclidean lengths of all edges in M . The minimum weight matching MWM (S) of S is the perfect matching of S that has minimum weight. The best known algorithm that computes a minimum weight matching is due to Vaidya [9]; its running time is bounded by O(n 5=2 (log n) 4 ) if d = 2, and O(n 3 1=c d ) if d 2, for some constant c 1. Exercise 7 Prove that the minimum weight matching of a set of 2n real numbers i.e. one dimensional points can be computed in O(n log n) time. Let us consider the (hopefully) ....

P. M. Vaidya. Geometry helps in matching. SIAM J. Comput., 18:1201{ 1225, 1989. 26

.... to maintain closest and furthest bichromatic pairs among a changing set of red and blue points, and for maintaining the Euclidean minimum spanning tree of a changing set of points [32] This technique generalizes and improves previous results of Bentley and Saxe [15] for insertions only) Vaidya [63] (where only one set permits deletions) Eppstein [34] for o ine insertions and deletions) and Dobkin and Suri [30] where each object s deletion time is given when it is inserted) The original statement of the theorem [32] required T(n) to be a worst case bound on the query and update times ....

P. Vaidya. Geometry helps in matching. SIAM J. Comput. 18:1201-1225, 1989.

....the analogous nonbipartite version of the problem, which involves just one set S of 2n points, and the complete graph on S. The goal is to explore the underlying geometric structure of these graphs, to obtain faster algorithms than those available for general abstract graphs. It was shown in [68] that both the bipartite and the nonbipartite versions of the problem can be solved in time close to n 2:5 . Recently, a fairly sophisticated application of vertical decomposition in 3 dimensional arrangements, given in [3] has improved the running time for the bipartite case to O(n 2 ) ....

P.M. Vaidya, Geometry helps in matching, SIAM J. Comput. 18 (1989), 1201--1225.

....Figure 6: Hausdor distance. setting, where the problem is to nd a matching in a graph (V; E) with vertices V = A [ B, and given edges E with weights. Exploiting the geometric nature if the vertices are points, and the weights are distances between points, results in more e cient algorithms, see [Vai89] for example. 4.1 Bottleneck matching Let A and B be two point sets of size n, and d(a; b) a distance between two points. The bottleneck distance is the minimum over all 1 1 correspondences f between A and B of the maximum distance d(a; f(a) The results on bottleneck distance mentioned in ....

....of the distances d(a; f(a) It can be computed in O(n 2 ) time [AES95] Here, the constant stands for a positive constant which can be chosen arbitrarily small with an appropriate choice of other constants of the algorithm. For the L1 distance, it can be computed in time O(n 2 log 3 n) Vai89] 4.3 Uniform matching The most uniform distance is the minimum over all 1 1 correspondences f between A and B of the di erence between the maximum and the minimum d(a; f(a) The most uniform matching is also called balanced or fair matching. The distance can be computed in time O(n 10=3 ....

Pravin M. Vaidya. Geometry helps in matching. SIAM Journal of Computing, 18(6):1201-1224, 1989.

.... matching for a general bipartite graph is known to have an O(N 3 ) time algorithm (see Lawler [18] for this and other background on matching) and for graphs with nodes in the plane with the Euclidean distance as cost function, there is a O(N 2:5 log N) time algorithm, due to Vaidya [22]. The minimum cost matching problem is substantially easier in the case where the nodes are in line like order or are circularly ordered. The simplest versions of linelike circular orderings are where the points lie on a line or lie on a curve homeomorphic to a circle, and the cost c(x; y) of an ....

P. M. Vaidya, Geometry helps in matching, SIAM J. Comput., 18 (1989), pp. 1201--1225.

....bipartite graphs (with m n nodes and mn edges) takes O(n(mn m log m) time. This time complexity can be achieved by using the Hungarian method [Kuh55, AMO89, GTT89] Since this problem has a relatively high time complexity, researchers have investigated special cases. For example, Vaidya [Vai89] showed that the minimum cost perfect matching among 2n points in the Euclidean plane can be computed in O(n 5=2 log 4 n) time. For the case where n = m, the sink and source points lie on a convex polygon (respectively, simple polygon) and the distance between two points is simply the ....

P.M. Vaidya. Geometry helps in matching. SIAM Journal on Computing 18: 1201--1225, 1989.

.... we can achieve 3ff TSP = 3(1 ffl) in a running time of O(N O( 1 ffl ) If we use the Christofides heuristic, we obtain an overall approximation factor of 3ff TSP = 4:5, at a running time of f(N ) O(N 2:5 log 4 N ) the bottleneck being the computation of the minimum weight matching [21]) Alternatively, we can apply the approximate matching result of Vaidya [22] within the Christofides heuristic, to achieve a factor of 4:5 ffl at a time complexity of f(N ) O(N 1:5 log 2:5 N ) Finally, by simply doubling the minimum spanning tree, we obtain a factor of 6, with a simple ....

P. M.Vaidya. Geometry helps in matching. SIAM J. Comput., 18, 1989 pp. 1201--1225.

....problem is super cubic in jV j, Anstee (1987) The bound is cubic when b v = 1 for all v 2 V , Lawler (1976) If, in addition, the edge weights fc e g equal to the Euclidean distances between points in the plane which represent the nodes of the graph, then some subcubic algorithms are known. Vaidya (1989) presented such an algorithm for finding a minimum weight geometric perfect matching. Marcotte and Suri (1991) considered the case where the representing points are on the boundary of a simple polygon in the plane. They gave an O(jV j log 2 jV j) time algorithm for the minimum weight perfect ....

P. Vaidya, "Geometry helps in matching," SIAM J. Computing 18 (1989) 1201-1225.

....algorithm; using this, the MCPM problem in the plane can be solved in O(n 3 ) time. The question that motivates us is whether we can exploit geometry to do much better. Note that the complete graph induced by the set of 2n points is entirely specified by the co ordinates of the points. Vaidya [8] was the first to show that geometry can be exploited to get a sub cubic algorithm; his O(n 5=2 log 4 n) time algorithm is the best known for Euclidean MCPM. For the bipartite version of this problem, Agarwal et al. 1] have given a near quadratic algorithm that improves over an earlier ....

....that geometry can be exploited to get a sub cubic algorithm; his O(n 5=2 log 4 n) time algorithm is the best known for Euclidean MCPM. For the bipartite version of this problem, Agarwal et al. 1] have given a near quadratic algorithm that improves over an earlier sub cubic algorithm of Vaidya [8]. Our results. We present an O(n 3=2 log 5 n) time algorithm for computing a min cost perfect matching in the plane, which is an improvement over the previous best algorithm of Vaidya [8] by nearly a factor of n. Vaidya s algorithm is an implementation of the algorithm of Edmonds [3] which ....

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P. M. Vaidya. Geometry helps in matching. SIAM J. Comput., 18:1201--1225, 1989.

.... Kuhn [9] and for min cost non bipartite matching is due to Edmonds [4] The fastest known implementations of these algorithms run in O(jV j 3 ) time on dense graphs (see Lawler [10] and roughly O(jEjjV j) time on sparse graphs [8] For the Euclidean (planar) versions of these problems, Vaidya [13] showed that geometry can be exploited to get al..gorithms running in O(n 5=2 log O(1) n) time for both the bipartite and nonbipartite versions. Agarwal et al. 1] improved the running time for the bipartite case to O(n 2 ffi ) for any ffi 0. Very recently, Varadarajan [14] gave a ....

....By Lemma 2.4, this bounds jE j. 2 For a fast implementation of one phase of the conquer algorithm for U , we need a mechanism to quickly compute when ffi i becomes zero. The following theorem results from a careful implementation of a phase such as the one described by Galil et al. 8] or Vaidya [13]. Theorem 2.6 Suppose the total time spent in detecting when ffi i becomes zero, over an entire phase for the subset U of sites, is O( Then, one phase can be implemented in O(jU j log jU j ) time. We describe below the main ideas of an implementation that detects when ffi i becomes zero ....

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P. M. Vaidya. Geometry helps in matching. SIAM J. Comput. 18 (1989), 1201--1225.

....The University of Memphis, Memphis, TN 38152, USA. E mail: dasg msci.memphis.edu. y Department of Computer Science, University of Magdeburg, D 39106 Magdeburg, Germany. E mail: michiel isg.cs.uni magdeburg.de. The best known algorithm that computes a minimum weight matching is due to Vaidya [6]; its running time is bounded by O(n 5=2 (log n) 4 ) if d = 2, and O(n 3 Gamma1=c d ) if d 2, for some constant c 1. Rao and Smith [5] considered the easier problem of approximating the minimum weight matching. Let r 1 be a real number. A perfect matching M of S is called an ....

P. M. Vaidya. Geometry helps in matching. SIAM J. Comput., 18:1201-- 1225, 1989.

....The subject of spanning trees and spanners is surveyed extensively in the chapter by Eppstein [150] in this handbook. Other well studied network optimization problems that we do not attempt to survey here include minimum cost matching (which has polynomial time exact and approximate solutions; see [36, 379, 380, 389]) and minimum weight triangulation (MWT) whose complexity status is still open, although constant factor approximation algorithms exist for both the Steiner and non Steiner versions; see Bern and Eppstein [64] and Levcopoulos and Krznaric [257] We also refer the reader to the article of Smith ....

P. M. Vaidya. Geometry helps in matching. SIAM J. Comput., 18:1201--1225, 1989.

....is much more inefficient since finding a minimum weight matching [38] requires time O(n 3 ) An interesting open problem is to find a simple construction of a class of algorithms which allows a smooth trade off between the running time and the performance ratio. The results of Vaidya [57, 58] on exact and approximate minimumweight matching (for points in the Euclidean plane) does give a tradeoff, but it would seem that better results should be possible. Of course, improving the bound of 1:5 would be a major breakthrough Another way of looking at the Euclidean TSP problem is: given n ....

P.M. Vaidya, Geometry helps in matching, SIAM Journal on Computing, 18 (1989), pp. 1201--1225.

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P. M. Vaidya, Geometry helps in matching, SIAM J. Comput., 18 (1989), pp. 1201--1225.

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P. M. Vaidya, Geometry helps in matching, SIAM J. Comput., 18 (1989), pp. 1201--1225.

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