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Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems
 Journal of the ACM
, 1998
"... Abstract. We present a polynomial time approximation scheme for Euclidean TSP in fixed dimensions. For every fixed c Ͼ 1 and given any n nodes in 2 , a randomized version of the scheme finds a (1 ϩ 1/c)approximation to the optimum traveling salesman tour in O(n(log n) O(c) ) time. When the nodes ..."
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Cited by 397 (2 self)
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are in d , the running time increases to O(n(log n) ). For every fixed c, d the running time is n ⅐ poly(log n), that is nearly linear in n. The algorithm can be derandomized, but this increases the running time by a factor O(n d ). The previous best approximation algorithm for the problem (due
Solving geometric problems with the rotating calipers
, 1983
"... Shamos [1] recently showed that the diameter of a convex nsided polygon could be computed in O(n) time using a very elegant and simple procedure which resembles rotating a set of calipers around the polygon once. In this paper we show that this simple idea can be generalized in two ways: several se ..."
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Cited by 147 (11 self)
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sets of calipers can be used simultaneously on one convex polygon, or one set of calipers can be used on several convex polygons simultaneously. We then show that these generalizations allow us to obtain simple O(n) algorithms for solving a variety of problems defined on convex polygons. Such problems
Pcomplete geometric problems
, 1990
"... In this paper we show that it is impossible to solve a number of "natural" 2dimensional geometric problems in polyIog time with a polynomial number of processors (unless P = NC). Thus, we disprove a popular belief that there are no natural Pcomplete geometric problems in the plane. The p ..."
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Cited by 6 (0 self)
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In this paper we show that it is impossible to solve a number of "natural" 2dimensional geometric problems in polyIog time with a polynomial number of processors (unless P = NC). Thus, we disprove a popular belief that there are no natural Pcomplete geometric problems in the plane
DavenportSchinzel Sequences and Their Geometric Applications
, 1998
"... An (n; s) DavenportSchinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly noncontiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \ ..."
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Cited by 439 (105 self)
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of geometric problems can be formulated in terms of lower envelopes. A nearlinear bound on the maximum length of DavenportSchinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems.
Some NPcomplete Geometric Problems
"... We show that the STEINER TREE problem and TRAVELING SALESMAN problem for points in the plane are NPcomplete when distances are measured either by the rectilinear (Manhattan) metric or by a natural discretized version of the Euclidean metric. Our proofs also indicate that the problems are NPhard i ..."
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Cited by 98 (1 self)
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We show that the STEINER TREE problem and TRAVELING SALESMAN problem for points in the plane are NPcomplete when distances are measured either by the rectilinear (Manhattan) metric or by a natural discretized version of the Euclidean metric. Our proofs also indicate that the problems are NP
and Problems—Geometrical problems and computations
, 2007
"... We present CRFGradient, a selfhealing gradient algorithm that provably reconfigures in O(diameter) time. Selfhealing gradients are a frequently used building block for distributed selfhealing systems, but previous algorithms either have a healing rate limited by the shortest link in the network o ..."
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We present CRFGradient, a selfhealing gradient algorithm that provably reconfigures in O(diameter) time. Selfhealing gradients are a frequently used building block for distributed selfhealing systems, but previous algorithms either have a healing rate limited by the shortest link in the network or must rebuild invalid regions from scratch. We have verified CRFGradient in simulation and on a network of Mica2 motes. Our approach can also be generalized and applied to create other selfhealing calculations, such as cumulative probability fields.
Parameterized Complexity of Geometric Problems
, 2007
"... This paper surveys parameterized complexity results for hard geometric algorithmic problems. It includes fixedparameter tractable problems in graph drawing, geometric graphs, geometric covering and several other areas, together with an overview of the algorithmic techniques used. Fixedparameter in ..."
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Cited by 13 (5 self)
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This paper surveys parameterized complexity results for hard geometric algorithmic problems. It includes fixedparameter tractable problems in graph drawing, geometric graphs, geometric covering and several other areas, together with an overview of the algorithmic techniques used. Fixed
INHERENTLY PARALLEL GEOMETRIC PROBLEMS
, 2004
"... A new computational paradigm is described which o ers the possibility of superlinear (and sometimes unbounded) speedup, when parallel computation is used. The computations involved are subject only to given mathematical constraints and hence do not depend on external circumstances to achieve superli ..."
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Cited by 3 (3 self)
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superlinear performance. The focus here is on geometric transformations. Given a geometric object A with some property, it is required to transform A into another object B which enjoys the same property. If the transformation requires several steps, each resulting in an intermediate object, then each
Results 1  10
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