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14
Fast Approximation of Centrality
 Journal of Graph Algorithms and Applications
, 2001
"... Social studies researchers use graphs to model group activities in social networks. An important property in this context is the centrality of a vertex: the inverse of the average distance to each other vertex. We describe a randomized approximation algorithm for centrality in weighted graphs. For g ..."
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Cited by 55 (0 self)
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Social studies researchers use graphs to model group activities in social networks. An important property in this context is the centrality of a vertex: the inverse of the average distance to each other vertex. We describe a randomized approximation algorithm for centrality in weighted graphs. For graphs exhibiting the small world phenomenon, our method estimates the centrality of all vertices with high probability within a (1 + #) factor in nearlinear time. 1 Introduction In social network analysis, the vertices of a graph represent agents in a group and the edges represent relationships, such as communication or friendship. The idea of applying graph theory to analyze the connection between the structural centrality and group process was introduced by Bavelas [4]. Various measurement of centrality [7, 14, 15] have been proposed for analyzing communication activity, control, or independence within a social network. We are particularly interested in closeness centrality [5, 6, 24]...
SingleSource ShortestPaths on Arbitrary Directed Graphs in Linear AverageCase Time
 In Proc. 12th ACMSIAM Symposium on Discrete Algorithms
, 2001
"... The quest for a lineartime singlesource shortestpath (SSSP) algorithm on directed graphs with positive edge weights is an ongoing hot research topic. While Thorup recently found an O(n + m) time RAM algorithm for undirected graphs with n nodes, m edges and integer edge weights in f0; : : : ; 2 w ..."
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Cited by 33 (5 self)
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The quest for a lineartime singlesource shortestpath (SSSP) algorithm on directed graphs with positive edge weights is an ongoing hot research topic. While Thorup recently found an O(n + m) time RAM algorithm for undirected graphs with n nodes, m edges and integer edge weights in f0; : : : ; 2 w 1g where w denotes the word length, the currently best time bound for directed sparse graphs on a RAM is O(n + m log log n). In the present paper we study the averagecase complexity of SSSP. We give a simple algorithm for arbitrary directed graphs with random edge weights uniformly distributed in [0; 1] and show that it needs linear time O(n + m) with high probability. 1 Introduction The singlesource shortestpath problem (SSSP) is a fundamental and wellstudied combinatorial optimization problem with many practical and theoretical applications [1]. Let G = (V; E) be a directed graph, jV j = n, jEj = m, let s be a distinguished vertex of the graph, and c be a function assigning a n...
Algorithmic Theory of Random graphs
, 1997
"... The theory of random graphs has been mainly concerned with structural properties, in particular the most likely values of various graph invariants  see Bollob`as [21]. There has been increasing interest in using random graphs as models for the average case analysis of graph algorithms. In this pap ..."
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Cited by 27 (1 self)
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The theory of random graphs has been mainly concerned with structural properties, in particular the most likely values of various graph invariants  see Bollob`as [21]. There has been increasing interest in using random graphs as models for the average case analysis of graph algorithms. In this paper we survey some of the results in this area. 1 Introduction The theory of random graphs as initiated by Erdos and R'enyi [52] and developed along with others, has been mainly concerned with structural properties, in particular the most likely values of various graph invariantss  see Bollob`as [21]. There has been increasing interest in using random graphs as models for the average case analysis of graph algorithms. We would like in this paper to survey some of the results in this area. We hope to be fairly comprehensive in terms of the areas we tackle and so depth will be sacrificed in favour of breadth. One attractive feature of average case analysis is that it banishes the pessimism o...
AverageCase Complexity of ShortestPaths Problems in the VertexPotential Model
 IN RANDOMIZATION AND APPROXIMATION TECHNIQUES IN COMPUTER SCIENCE (J. ROLIM, ED.), LECTURE NOTES IN COMPUT. SCI. 1269
, 2000
"... We study the averagecase complexity of shortestpaths problems in the vertexpotential model. The vertexpotential model is a family of probability distributions on complete directed graphs with arbitrary real edge lengths but without negative cycles. We show that on a graph with n vertices and ..."
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We study the averagecase complexity of shortestpaths problems in the vertexpotential model. The vertexpotential model is a family of probability distributions on complete directed graphs with arbitrary real edge lengths but without negative cycles. We show that on a graph with n vertices and with respect to this model, the singlesource shortestpaths problem can be solved in O(n²) expected time, and the allpairs shortestpaths problem can be solved in O(n² log n) expected time.
Averagecase complexity of singlesource shortestpaths algorithms: lower and upper bounds
, 2003
"... ..."
Probabilistic Analysis of Algorithms
 Probabilistic Methods for Algorithmic Discrete Mathematics, Algorithms and Combinatorics 16
, 1998
"... this paper. Of course, the first question we must answer is: what do we mean by a typical instance of a given size? Sometimes, there is a natural answer to this question. For example, in developing an algorithm which is typically efficent for an NPcomplete optimization problems on graphs, we might ..."
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this paper. Of course, the first question we must answer is: what do we mean by a typical instance of a given size? Sometimes, there is a natural answer to this question. For example, in developing an algorithm which is typically efficent for an NPcomplete optimization problems on graphs, we might assume that an n vertex input is equally likely to be any of the 2 2 ) labelled graphs with n vertices. This allows us to exploit any property which holds on almost all such graphs when developing the algorithm
Negative Dependence Through the FKG Inequality
 RESEARCH REPORT MPII961020, MAXPLANCKINSTITUT FUR ¨ INFORMATIK, SAARBRUCKEN
, 1996
"... We investigate random variables arising in occupancy problems, and show the variables to be negatively associated, that is, negatively dependent in a strong sense. Our proofs are based on the FKG correlation inequality, and they suggest a useful, general technique for proving negative dependence ..."
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Cited by 5 (1 self)
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We investigate random variables arising in occupancy problems, and show the variables to be negatively associated, that is, negatively dependent in a strong sense. Our proofs are based on the FKG correlation inequality, and they suggest a useful, general technique for proving negative dependence among random variables. We also show that in the special case of two binary random variables, the notions of negative correlation and negative association coincide.
Exact and Approximation Algorithms for Network Flow and DisjointPath Problems
, 1998
"... Network flow problems form a core area of Combinatorial Optimization. Their significance arises both from their very large number of applications and their theoretical importance. This thesis focuses on efficient exact algorithms for network flow problems in P and on approximation algorithms for NP ..."
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Cited by 5 (3 self)
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Network flow problems form a core area of Combinatorial Optimization. Their significance arises both from their very large number of applications and their theoretical importance. This thesis focuses on efficient exact algorithms for network flow problems in P and on approximation algorithms for NP hard variants such as disjoint paths and unsplittable flow. Given an nvertex
AverageCase Complexity of ShortestPaths Problems
, 2001
"... We study both upper and lower bounds on the averagecase complexity of shortestpaths algorithms. It is proved that the allpairs shortestpaths problem on nvertex networks can be solved in time O(n² log n) with high probability with respect to various probability distributions on the set of inputs. ..."
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We study both upper and lower bounds on the averagecase complexity of shortestpaths algorithms. It is proved that the allpairs shortestpaths problem on nvertex networks can be solved in time O(n² log n) with high probability with respect to various probability distributions on the set of inputs. Our results include the first theoretical analysis of the average behavior of shortestpaths algorithms with respect to the vertexpotential model, a family of probability distributions on complete networks with arbitrary real arc costs but without negative cycles. We also generalize earlier work with respect to the common uniform model, and we correct the analysis of an algorithm with respect to the endpointindependent model. For the algorithm that solves the allpairs shortestpaths problem on networks generated according to the vertexpotential model, a key ingredient is an algorithm that solves the singlesource shortestpaths problem on such networks in time O(n²) with high probability. All algorithms mentioned exploit that with high probability, the singlesource shortestpaths problem can be solved correctly by considering only a rather sparse subset of the arc set. We prove a lower bound indicating the limitations of this approach. In a fairly general probabilistic model, any algorithm solving the singlesource shortestpaths problem has to inspect# n log n) arcs with high probability.