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14
A near linear time constant factor approximation for Euclidean bichromatic matching (cost)
 IN PROC. 18TH SYMP. ON DISC. ALG
, 2007
"... We give an N log O(1) Ntime randomized O(1)approximation algorithm for computing the cost of minimum bichromatic matching between two planar pointsets of size N. ..."
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Cited by 19 (3 self)
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We give an N log O(1) Ntime randomized O(1)approximation algorithm for computing the cost of minimum bichromatic matching between two planar pointsets of size N.
On minimizing the maximum sensor movement for barrier coverage of a line segment
 IN PROCEEDINGS OF 8TH INTERNATIONAL CONFERENCE ON AD HOC NETWORKS AND WIRELESS
"... We consider n mobile sensors located on a line containing a barrier represented by a finite line segment. Sensors form a wireless sensor network and are able to move within the line. An intruder traversing the barrier can be detected only when it is within the sensing range of at least one sensor. ..."
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Cited by 12 (5 self)
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We consider n mobile sensors located on a line containing a barrier represented by a finite line segment. Sensors form a wireless sensor network and are able to move within the line. An intruder traversing the barrier can be detected only when it is within the sensing range of at least one sensor. The sensor network establishes barrier coverage of the segment if no intruder can penetrate the barrier from any direction in the plane without being detected. Starting from arbitrary initial positions of sensors on the line we are interested in finding final positions of sensors that establish barrier coverage and minimize the maximum distance traversed by any sensor. We distinguish several variants of the problem, based on (a) whether or not the sensors have identical ranges, (b) whether or not complete coverage is possible and (c) in the case when complete coverage is impossible, whether or not the maximal coverage is required to be contiguous. For the case of n sensors with identical range, when complete coverage is impossible, we give linear time optimal
Comparing distributions and shapes using the kernel distance
 In ACM SoCG
, 2011
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On Minimizing the Sum of Sensor Movements for Barrier Coverage of a Line Segment
"... Abstract. A set of sensors establishes barrier coverage of a given line segment if every point of the segment is within the sensing range of a sensor. Given a line segment I, n mobile sensors in arbitrary initial positions on the line (not necessarily inside I) and the sensing ranges of the sensors, ..."
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Abstract. A set of sensors establishes barrier coverage of a given line segment if every point of the segment is within the sensing range of a sensor. Given a line segment I, n mobile sensors in arbitrary initial positions on the line (not necessarily inside I) and the sensing ranges of the sensors, we are interested in finding final positions of sensors which establish a barrier coverage of I so that the sum of the distances traveled by all sensors from initial to final positions is minimized. It is shown that the problem is NP complete even to approximate up to constant factor when the sensors may have different sensing ranges. When the sensors have an identical sensing range we give several efficient algorithms to calculate the final destinations so that the sensors either establish a barrier coverage or maximize the coverage of the segment if complete coverage is not feasible while at the same time the sum of the distances traveled by all sensors is minimized. Some open problems are also mentioned. Key words and phrases: Mobile Sensor, Barrier Coverage, Line segment,
On the parameterized complexity of ddimensional point set pattern matching
 Inf. Process. Lett
"... Ljubljana, October 25, 2006On the parameterized complexity of ddimensional point set pattern matching ∗ ..."
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Cited by 8 (4 self)
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Ljubljana, October 25, 2006On the parameterized complexity of ddimensional point set pattern matching ∗
On Bipartite Matching under the RMS Distance
, 2006
"... Given two sets A and B of n points each in R 2, we study the problem of computing a matching between A and B that minimizes the root mean square (rms) distance of matched pairs. We can compute an optimal matching in O(n 2+δ) time, for any δ> 0, and an εapproximation in time O((n/ε) 3/2 log 6 n). ..."
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Cited by 8 (3 self)
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Given two sets A and B of n points each in R 2, we study the problem of computing a matching between A and B that minimizes the root mean square (rms) distance of matched pairs. We can compute an optimal matching in O(n 2+δ) time, for any δ> 0, and an εapproximation in time O((n/ε) 3/2 log 6 n). If the set B is allowed to move rigidly to minimize the rms distance, we can compute a rigid motion of B and a matching in O((n 4 /ε 5/2) log 6 n) time whose cost is within (1 + ε) factor of the optimal one.
Small manhattan networks and algorithms for the earth mover’s distance
 IN PROC. 23RD EUROPEAN WORKSHOP COMPUT. GEOM. (EWCG’07
, 2007
"... Given a set S of n points in the plane, a Manhattan network on S is a (not necessarily planar) rectilinear network G with the property that for every pair of points in S the network G contains a path between them whose length is equal to the Manhattan distance between the points. A Manhattan network ..."
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Given a set S of n points in the plane, a Manhattan network on S is a (not necessarily planar) rectilinear network G with the property that for every pair of points in S the network G contains a path between them whose length is equal to the Manhattan distance between the points. A Manhattan network on S can be thought of as a graph G = (V, E) where the vertex set V corresponds to the points of S and a set of Steiner points S ′. The edges in E correspond to horizontal and vertical line segments connecting points in S ∪ S ′. A Manhattan network can also be thought of as a 1spanner (for the L1metric) for the points in S. We will show that there is a Manhattan network on S with O(n log n) vertices and edges which can be constructed in O(n log n) time. This allows us to to compute the L1Earth Mover’s Distance on weighted planar point sets in O(n² log³ n) time, which improves the currently best known result of O(n 4 log n). At the expense of a slightly higher time and space complexity we are able to extend our approach to any dimension d ≥ 3. We will further show that our construction is optimal in the sense that there are point sets in the plane where every Manhattan network needs Ω(n log n) vertices and edges.
Noncrossing Matchings of Points with Geometric Objects
, 2011
"... Given an ordered set of points and an ordered set of geometric objects in the plane, we are interested in finding a noncrossing matching between pointobject pairs. In this paper, we address the algorithmic problem of determining whether a noncrossing matching exists between a given pointobject p ..."
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Cited by 3 (0 self)
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Given an ordered set of points and an ordered set of geometric objects in the plane, we are interested in finding a noncrossing matching between pointobject pairs. In this paper, we address the algorithmic problem of determining whether a noncrossing matching exists between a given pointobject pair. We show that when the objects we match the points to are finite point sets, the problem is NPcomplete in general, and polynomial when the ob
Approximation algorithms for the earth mover’s distance under transformations using reference points
 Institute of Information and Computing Sciences, www.cs.uu.nl
, 2005
"... Reference points have been introduced in [2] and [1] to construct approximation algorithms for matching compact subsets of Rd under a given class of transformations. Also a general discussion of reference point ..."
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Cited by 2 (1 self)
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Reference points have been introduced in [2] and [1] to construct approximation algorithms for matching compact subsets of Rd under a given class of transformations. Also a general discussion of reference point
Matching Points with Things
, 2009
"... Representing a matching between pairs of planar objects as a set of noncrossing line segments is a natural problem in computational geometry. It is well known, for instance, that given two sets of n points in the plane, say n red points and n blue points, there always exists such a noncrossing matc ..."
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Representing a matching between pairs of planar objects as a set of noncrossing line segments is a natural problem in computational geometry. It is well known, for instance, that given two sets of n points in the plane, say n red points and n blue points, there always exists such a noncrossing matching between red and blue points. In particular, it is not difficult to show that the minimum Euclidean length matching is noncrossing. Kaneko and Kano [3] survey a number of related results. We investigate related questions for general planar objects instead of points. In this case, the matching is represented by line segments, the endpoints of which belong to the corresponding matched objects. Note that, from the above result, a noncrossing matching always exists between two sets of objects. However, in this paper, we consider the problem of finding a matching when we are given object pairs as input. Since pairs are enforced, the existence of a noncrossing