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98
Voronoi diagrams  a survey of a fundamental geometric data structure
 ACM COMPUTING SURVEYS
, 1991
"... This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. ..."
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Cited by 743 (5 self)
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This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. The paper puts particular emphasis on the unified exposition of its mathematical and algorithmic properties. Finally, the paper provides the first comprehensive bibliography on Voronoi diagrams and related structures.
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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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 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 to Christofides) achieves a 3/2approximation in polynomial time. We also give similar approximation schemes for some other NPhard Euclidean problems: Minimum Steiner Tree, kTSP, and kMST. (The running times of the algorithm for kTSP and kMST involve an additional multiplicative factor k.) The previous best approximation algorithms for all these problems achieved a constantfactor approximation. We also give efficient approximation schemes for Euclidean MinCost Matching, a problem that can be solved exactly in polynomial time. All our algorithms also work, with almost no modification, when distance is measured using any geometric norm (such as ᐉ p for p Ն 1 or other Minkowski norms). They also have simple parallel (i.e., NC) implementations.
ON THE COMPLEXITY OF SOME COMMON GEOMETRIC LOCATION PROBLEMS
 SIAM J. COMPUTING
, 1984
"... Given n demand points in the plane, the pcenter problem is to find p supply points (anywhere in the plane) so as to minimize the maximum distance from a demo & point to its respective nearest supply point. The pmedian problem is to minimize the sum of distances from demand points to their resp ..."
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Cited by 140 (1 self)
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Given n demand points in the plane, the pcenter problem is to find p supply points (anywhere in the plane) so as to minimize the maximum distance from a demo & point to its respective nearest supply point. The pmedian problem is to minimize the sum of distances from demand points to their respective nearest supply points. We prove that the pcenter and the pmedia problems relative to both the Euclidean and the rectilinear metrics are NPhard. In fact, we prove that it is NPhard even to approximate the pcenter problems sufficiently closely. The reductions are from 3satisfiability.
Recursive Markov chains, stochastic grammars, and monotone systems of nonlinear equations
 IN STACS
, 2005
"... We define Recursive Markov Chains (RMCs), a class of finitely presented denumerable Markov chains, and we study algorithms for their analysis. Informally, an RMC consists of a collection of finitestate Markov chains with the ability to invoke each other in a potentially recursive manner. RMCs offer ..."
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Cited by 95 (13 self)
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We define Recursive Markov Chains (RMCs), a class of finitely presented denumerable Markov chains, and we study algorithms for their analysis. Informally, an RMC consists of a collection of finitestate Markov chains with the ability to invoke each other in a potentially recursive manner. RMCs offer a natural abstract model for probabilistic programs with procedures. They generalize, in a precise sense, a number of well studied stochastic models, including Stochastic ContextFree Grammars (SCFG) and MultiType Branching Processes (MTBP). We focus on algorithms for reachability and termination analysis for RMCs: what is the probability that an RMC started from a given state reaches another target state, or that it terminates? These probabilities are in general irrational, and they arise as (least) fixed point solutions to certain (monotone) systems of nonlinear equations associated with RMCs. We address both the qualitative problem of determining whether the probabilities are 0, 1 or inbetween, and
Nearly Linear Time Approximation Schemes for Euclidean TSP and other Geometric Problems
, 1997
"... We present a randomized polynomial time approximation scheme for Euclidean TSP in ! 2 that is substantially more efficient than our earlier scheme in [2] (and the scheme of Mitchell [21]). For any fixed c ? 1 and any set of n nodes in the plane, the new scheme finds a (1+ 1 c )approximation to ..."
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Cited by 92 (3 self)
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We present a randomized polynomial time approximation scheme for Euclidean TSP in ! 2 that is substantially more efficient than our earlier scheme in [2] (and the scheme of Mitchell [21]). For any fixed c ? 1 and any set of n nodes in the plane, the new scheme finds a (1+ 1 c )approximation to the optimum traveling salesman tour in O(n(log n) O(c) ) time. (Our earlier scheme ran in n O(c) time.) For points in ! d the algorithm runs in O(n(log n) (O( p dc)) d\Gamma1 ) time. This time is polynomial (actually nearly linear) for every fixed c; d. Designing such a polynomialtime algorithm was an open problem (our earlier algorithm in [2] ran in superpolynomial time for d 3). The algorithm generalizes to the same set of Euclidean problems handled by the previous algorithm, including Steiner Tree, kTSP, kMST, etc, although for kTSP and kMST the running time gets multiplied by k. We also use our ideas to design nearlylinear time approximation schemes for Euclidean vers...
Approximation Algorithms For The Geometric Covering Salesman Problem
 Discrete Applied Mathematics
, 1995
"... We introduce a geometric version of the Covering Salesman Problem: Each of the n salesman's clients specifies a neighborhood in which they are willing to meet the salesman. Identifying a tour of minimum length that visits all neighborhoods is an NPhard problem, since it is a generalization of ..."
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Cited by 78 (3 self)
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We introduce a geometric version of the Covering Salesman Problem: Each of the n salesman's clients specifies a neighborhood in which they are willing to meet the salesman. Identifying a tour of minimum length that visits all neighborhoods is an NPhard problem, since it is a generalization of the Traveling Salesman Problem. We present simple heuristic procedures for constructing tours, for a variety of neighborhood types, whose length is guaranteed to be within a constant factor of the length of an optimal tour. The neighborhoods we consider include, parallel unit segments, translates of a polygonal region, and circles. y Partially supported by NSF Grants DMS 8903304 and ECSE8857642. 1 Introduction A salesman wants to meet a set of potential buyers. Each buyer specifies a compact set in the plane, his neighborhood, within which he is willing to meet. For example, the neighborhoods may be disks centered at the buyers locations, and each disk's radius specifies the distance that a ...
On the complexity of numerical analysis
 IN PROC. 21ST ANN. IEEE CONF. ON COMPUTATIONAL COMPLEXITY (CCC ’06
, 2006
"... We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis: • The BlumShubSmale model of computation over the reals. • A problem we call the “Generic Task of Numerical Computation, ” which captures an aspect of doing numerical computation ..."
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Cited by 73 (5 self)
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We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis: • The BlumShubSmale model of computation over the reals. • A problem we call the “Generic Task of Numerical Computation, ” which captures an aspect of doing numerical computation in floating point, similar to the “long exponent model ” that has been studied in the numerical computing community. We show that both of these approaches hinge on the question of understanding the complexity of the following problem, which we call PosSLP: Given a divisionfree straightline program producing an integer N, decide whether N> 0. • In the BlumShubSmale model, polynomial time computation over the reals (on discrete inputs) is polynomialtime equivalent to PosSLP, when there are only algebraic constants. We conjecture that using transcendental constants provides no additional power, beyond nonuniform reductions to PosSLP, and we present some preliminary results supporting this conjecture. • The Generic Task of Numerical Computation is also polynomialtime equivalent to PosSLP. We prove that PosSLP lies in the counting hierarchy. Combining this with work of Tiwari, we obtain that the Euclidean Traveling Salesman Problem lies in the counting hierarchy – the previous best upper bound for this important problem (in terms of classical complexity classes) being PSPACE. In the course of developing the context for our results on arithmetic circuits, we present some new observations on the complexity of ACIT: the Arithmetic Circuit Identity Testing problem. In particular, we show that if n! is not ultimately easy, then ACIT has subexponential complexity.
Distance Measures for Point Sets and Their Computation
 Acta Informatica
, 1997
"... We consider the problem of measuring the similarity or distance between two finite sets of points in a metric space, and computing the measure. This problem has applications in, e.g., computational geometry, philosophy of science, updating or changing theories, and machine learning. We review some o ..."
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Cited by 70 (2 self)
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We consider the problem of measuring the similarity or distance between two finite sets of points in a metric space, and computing the measure. This problem has applications in, e.g., computational geometry, philosophy of science, updating or changing theories, and machine learning. We review some of the distance functions proposed in the literature, among them the minimum distance link measure, the surjection measure, and the fair surjection measure, and supply polynomial time algorithms for the computation of these measures. Furthermore, we introduce the minimum link measure, a new distance function which is more appealing than the other distance functions mentioned. We also present a polynomial time algorithm for computing this new measure. We further address the issue of defining a metric on point sets. We present the metric infimum method that constructs a metric from any distance functions on point sets. In particular, the metric infimum of the minimum link measure is a quite int...
On the Complexity of Nash Equilibria and Other Fixed Points (Extended Abstract)
 IN PROC. FOCS
, 2007
"... We reexamine what it means to compute Nash equilibria and, more generally, what it means to compute a fixed point of a given Brouwer function, and we investigate the complexity of the associated problems. Specifically, we study the complexity of the following problem: given a finite game, Γ, with 3 ..."
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Cited by 68 (8 self)
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We reexamine what it means to compute Nash equilibria and, more generally, what it means to compute a fixed point of a given Brouwer function, and we investigate the complexity of the associated problems. Specifically, we study the complexity of the following problem: given a finite game, Γ, with 3 or more players, and given ɛ> 0, compute an approximation within ɛ of some (actual) Nash equilibrium. We show that approximation of an actual Nash Equilibrium, even to within any nontrivial constant additive factor ɛ < 1/2 in just one desired coordinate, is at least as hard as the long standing squareroot sum problem, as well as a more general arithmetic circuit decision problem that characterizes Ptime in a unitcost model of computation with arbitrary precision rational arithmetic; thus placing the approximation problem in P, or even NP, would resolve major open problems in the complexity of numerical computation. We show similar results for market equilibria: it is hard to estimate with any nontrivial accuracy the equilibrium prices in an exchange economy with a unique equilibrium, where the economy is given by explicit algebraic formulas for the excess demand functions. We define a class, FIXP, which captures search problems that can be cast as fixed point
NPcomplete problems and physical reality
 ACM SIGACT News Complexity Theory Column, March. ECCC
, 2005
"... Can NPcomplete problems be solved efficiently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adiabatic algorithms, quantummechanical nonlinearities, hidden variables, relativistic time dilation, analog computing, Mal ..."
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Cited by 55 (6 self)
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Can NPcomplete problems be solved efficiently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adiabatic algorithms, quantummechanical nonlinearities, hidden variables, relativistic time dilation, analog computing, MalamentHogarth spacetimes, quantum gravity, closed timelike curves, and “anthropic computing. ” The section on soap bubbles even includes some “experimental ” results. While I do not believe that any of the proposals will let us solve NPcomplete problems efficiently, I argue that by studying them, we can learn something not only about computation but also about physics. 1