Results 1  10
of
586
Approximating discrete probability distributions with dependence trees
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 1968
"... A method is presented to approximate optimally an ndimensional discrete probability distribution by a product of secondorder distributions, or the distribution of the firstorder tree dependence. The problem is to find an optimum set of n1 first order dependence relationship among the n variables ..."
Abstract

Cited by 801 (0 self)
 Add to MetaCart
A method is presented to approximate optimally an ndimensional discrete probability distribution by a product of secondorder distributions, or the distribution of the firstorder tree dependence. The problem is to find an optimum set of n1 first order dependence relationship among the n variables. It is shown that the procedure derived in this paper yields an approximation of a minimum difference in information. It is further shown that when this procedure is applied to empirical observations from an unknown distribution of tree dependence, the procedure is the maximumlikelihood estimate of the distribution.
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. ..."
Abstract

Cited by 707 (5 self)
 Add to MetaCart
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.
A general approximation technique for constrained forest problems
 SIAM J. COMPUT.
, 1995
"... We present a general approximation technique for a large class of graph problems. Our technique mostly applies to problems of covering, at minimum cost, the vertices of a graph with trees, cycles, or paths satisfying certain requirements. In particular, many basic combinatorial optimization proble ..."
Abstract

Cited by 405 (22 self)
 Add to MetaCart
We present a general approximation technique for a large class of graph problems. Our technique mostly applies to problems of covering, at minimum cost, the vertices of a graph with trees, cycles, or paths satisfying certain requirements. In particular, many basic combinatorial optimization problems fit in this framework, including the shortest path, minimumcost spanning tree, minimumweight perfect matching, traveling salesman, and Steiner tree problems. Our technique produces approximation algorithms that run in O(n log n) time and come within a factor of 2 of optimal for most of these problems. For instance, we obtain a 2approximation algorithm for the minimumweight perfect matching problem under the triangle inequality. Our running time of O(n log n) time compares favorably with the best strongly polynomial exact algorithms running in O(n 3) time for dense graphs. A similar result is obtained for the 2matching problem and its variants. We also derive the first approximation algorithms for many NPcomplete problems, including the nonfixed pointtopoint connection problem, the exact path partitioning problem, and complex locationdesign problems. Moreover, for the prizecollecting traveling salesman or Steiner tree problems, we obtain 2approximation algorithms, therefore improving the previously bestknown performance guarantees of 2.5 and 3, respectively [Math. Programming, 59 (1993), pp. 413420].
Approximate distance oracles
 J. ACM
"... Let G = (V, E) be an undirected weighted graph with V  = n and E  = m. Let k ≥ 1 be an integer. We show that G = (V, E) can be preprocessed in O(kmn 1/k) expected time, constructing a data structure of size O(kn 1+1/k), such that any subsequent distance query can be answered, approximately, in ..."
Abstract

Cited by 253 (9 self)
 Add to MetaCart
Let G = (V, E) be an undirected weighted graph with V  = n and E  = m. Let k ≥ 1 be an integer. We show that G = (V, E) can be preprocessed in O(kmn 1/k) expected time, constructing a data structure of size O(kn 1+1/k), such that any subsequent distance query can be answered, approximately, in O(k) time. The approximate distance returned is of stretch at most 2k − 1, i.e., the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k−1. A 1963 girth conjecture of Erdős, implies that Ω(n 1+1/k) space is needed in the worst case for any real stretch strictly smaller than 2k + 1. The space requirement of our algorithm is, therefore, essentially optimal. The most impressive feature of our data structure is its constant query time, hence the name “oracle”. Previously, data structures that used only O(n 1+1/k) space had a query time of Ω(n 1/k). Our algorithms are extremely simple and easy to implement efficiently. They also provide faster constructions of sparse spanners of weighted graphs, and improved tree covers and distance labelings of weighted or unweighted graphs. 1
On constructing minimum spanning trees in kdimensional space and related problems
 SIAM Journal on Computing
, 1982
"... Abstract. The problem of finding a minimum spanning tree connecting n points in a kdimensional space is discussed under three common distance metrics: Euclidean, rectilinear, and L. By employing a subroutine that solves the post office problem, we show that, for fixed k _> 3, such a minimum spa ..."
Abstract

Cited by 224 (0 self)
 Add to MetaCart
(Show Context)
Abstract. The problem of finding a minimum spanning tree connecting n points in a kdimensional space is discussed under three common distance metrics: Euclidean, rectilinear, and L. By employing a subroutine that solves the post office problem, we show that, for fixed k _> 3, such a minimum spanning tree can be found in time O(n2a<k)(1og n)la<k)), where a(k) = 2+1). The bound can be improved to O((n log n) 1"8) for points in 3dimensional Euclidean space. We also obtain o(n 2) algorithms for finding a farthest pair in a set of n points and for other related problems. Key words, algorithm, minimum spanning tree, nearest neighbor, post office problem 1. Introduction. Given
Learning from Labeled and Unlabeled Data with Label Propagation
, 2002
"... We investigate the use of unlabeled data to help labeled data in classification. We propose a simple iterative algorithm, label propagation, to propagate labels through the dataset along high density areas defined by unlabeled data. We give the analysis of the algorithm, show its solution, and its c ..."
Abstract

Cited by 175 (0 self)
 Add to MetaCart
(Show Context)
We investigate the use of unlabeled data to help labeled data in classification. We propose a simple iterative algorithm, label propagation, to propagate labels through the dataset along high density areas defined by unlabeled data. We give the analysis of the algorithm, show its solution, and its connection to several other algorithms. We also show how to learn parameters by minimum spanning tree heuristic and entropy minimization, and the algorithm's ability to do feature selection. Experiment results are promising.
THE PRIMALDUAL METHOD FOR APPROXIMATION ALGORITHMS AND ITS APPLICATION TO NETWORK DESIGN PROBLEMS
"... The primaldual method is a standard tool in the design of algorithms for combinatorial optimization problems. This chapter shows how the primaldual method can be modified to provide good approximation algorithms for a wide variety of NPhard problems. We concentrate on results from recent researc ..."
Abstract

Cited by 138 (5 self)
 Add to MetaCart
The primaldual method is a standard tool in the design of algorithms for combinatorial optimization problems. This chapter shows how the primaldual method can be modified to provide good approximation algorithms for a wide variety of NPhard problems. We concentrate on results from recent research applying the primaldual method to problems in network design.
A Randomized LinearTime Algorithm to Find Minimum Spanning Trees
, 1994
"... We present a randomized lineartime algorithm to find a minimum spanning tree in a connected graph with edge weights. The algorithm uses random sampling in combination with a recently discovered lineartime algorithm for verifying a minimum spanning tree. Our computational model is a unitcost ra ..."
Abstract

Cited by 134 (6 self)
 Add to MetaCart
We present a randomized lineartime algorithm to find a minimum spanning tree in a connected graph with edge weights. The algorithm uses random sampling in combination with a recently discovered lineartime algorithm for verifying a minimum spanning tree. Our computational model is a unitcost randomaccess machine with the restriction that the only operations allowed on edge weights are binary comparisons.