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PARTITION REFINEMENT TECHNIQUES: AN INTERESTING ALGORITHMIC TOOL KIT
 INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE
, 1999
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On the Power of BFS to Determine a Graph's Diameter
 Networks
, 2003
"... this paper, we show that, in some cases, the full power of LBFS is not required and that other variations of Breadth First Search (BFS) suffice. The restricted graph classes that are amenable to this approach all have a small constant upper bound on the maximumsized cycle that may appear as an indu ..."
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this paper, we show that, in some cases, the full power of LBFS is not required and that other variations of Breadth First Search (BFS) suffice. The restricted graph classes that are amenable to this approach all have a small constant upper bound on the maximumsized cycle that may appear as an induced subgraph. We show that, on graphs that have no induced cycle of size greater than k, BFS finds an estimate of the diameter that is no worse than diam(G) # #k/2#. 2003 Wiley Periodicals, Inc
Fast Computation of Empirically Tight Bounds for the Diameter of Massive Graphs
"... The diameter of a graph is among its most basic parameters. Since a few years, it moreover became a key issue to compute it for massive graphs in the context of complex network analysis. However, known algorithms, including the ones producing approximate values, have too high a time and/or space com ..."
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The diameter of a graph is among its most basic parameters. Since a few years, it moreover became a key issue to compute it for massive graphs in the context of complex network analysis. However, known algorithms, including the ones producing approximate values, have too high a time and/or space complexity to be used in such cases. We propose here a new approach relying on very simple and fast algorithms that compute (upper and lower) bounds for the diameter. We show empirically that, on various realworld cases representative of complex networks studied in the literature, the obtained bounds are very tight (and even equal in some cases). This leads to rigorous and very accurate estimations of the actual diameter in cases which were previously untractable in practice. 1 Context. Throughout the paper, we consider a connected undirected unweighted graph G = (V, E) with n = V  vertices and m = E  edges. We denote by d(u, v) the distance between u and v in G, by ecc(v) = maxu d(v, u) the eccentricity of v in G, and by D = maxu,v d(u, v) = maxvecc(v) the diameter of G.
Measuring Fundamental Properties of RealWorld Complex Networks
"... Abstract — Complex networks (internet maps, web graphs, data exchanges, etc) received much attention during these last years. However data on such networks are only available through intricate measurement procedures. Until recently, most studies assumed that these procedures eventually lead to sampl ..."
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Abstract — Complex networks (internet maps, web graphs, data exchanges, etc) received much attention during these last years. However data on such networks are only available through intricate measurement procedures. Until recently, most studies assumed that these procedures eventually lead to sample large enough to be representative of the whole, at least concerning some key properties. This has crucial impact on network modeling and simulation. We propose here a new way to investigate the relevance of this approach: we put together data on complex network measurements that are representative of data commonly used, but significantly larger. Then we study how the observed properties evolve when the sample grows. The obtained results are in sharp contrast with usual assumptions, with important consequences that we discuss.
LexBFSordering in Asteroidal Triplefree Graphs
 SpringerVerlag Lecture Notes in Computer Science 1741
, 1999
"... . In this paper, we study the metric property of LexBFSordering on ATfree graphs. Based on a 2sweep LexBFS algorithm, we show that every ATfree graph admits a vertex ordering, called the strong 2cocomparability ordering, that for any three vertices u OE v OE w in the ordering, if d(u; w) 2 ..."
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. In this paper, we study the metric property of LexBFSordering on ATfree graphs. Based on a 2sweep LexBFS algorithm, we show that every ATfree graph admits a vertex ordering, called the strong 2cocomparability ordering, that for any three vertices u OE v OE w in the ordering, if d(u; w) 2 then d(u; v) = 1 or d(v; w) 2. As an application of this ordering, we provide a simple linear time recognition algorithm for bipartite permutation graphs, which form a subclass of ATfree graphs. 1 Introduction In past years, specific orderings of vertices characterizing certain graph classes are studied by many researchers. Usually, these ordering can be described from a metric point of view and the metric associated with a connected graph is, of course, the distance function d, giving the length of a shortest path between two vertices. One of the first results is due to Rose [19] for recognizing chordal graphs. A graph is chordal if every cycle of length at least four has a chord. A p...
Almost diameter of a householefree graph in linear time via LexBFS
 DISCRETE APPL. MATH
, 1999
"... We show that the vertex visited last by a LexBFS has eccentricity at least diam(G)  2 for householefree graphs, at least diam(G)  1 for householedominofree graphs, and equal to diam(G) for householedominofree and ATfree graphs. To prove these results we use special metric properties o ..."
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We show that the vertex visited last by a LexBFS has eccentricity at least diam(G)  2 for householefree graphs, at least diam(G)  1 for householedominofree graphs, and equal to diam(G) for householedominofree and ATfree graphs. To prove these results we use special metric properties of householefree graphs with respect to LexBFS.
On Querying Historical Evolving Graph Sequences
"... In many applications, information is best represented as graphs. In a dynamic world, information changes and so the graphs representing the information evolve with time. We propose that historical graphstructured data be maintained for analytical processing. We call a historical evolving graph sequ ..."
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In many applications, information is best represented as graphs. In a dynamic world, information changes and so the graphs representing the information evolve with time. We propose that historical graphstructured data be maintained for analytical processing. We call a historical evolving graph sequence an EGS. We observe that in many applications, graphs of an EGS are large and numerous, and they often exhibit much redundancy among them. We study the problem of efficient query processing on an EGS and put forward a solution framework called FVF. Through extensive experiments on both real and synthetic datasets, we show that our FVF framework is highly efficient in EGS query processing. 1.
Fast Approximation Algorithms for the Diameter and Radius of Sparse Graphs
"... The diameter and the radius of a graph are fundamental topological parameters that have many important practical applications in real world networks. The fastest combinatorial algorithm for both parameters works by solving the allpairs shortest paths problem (APSP) and has a running time of Õ(mn) i ..."
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The diameter and the radius of a graph are fundamental topological parameters that have many important practical applications in real world networks. The fastest combinatorial algorithm for both parameters works by solving the allpairs shortest paths problem (APSP) and has a running time of Õ(mn) in medge, nnode graphs. In a seminal paper, Aingworth, Chekuri, Indyk and Motwani [SODA’96 and SICOMP’99] presented an algorithm that computes in Õ(m √ n+n 2) time an estimate ˆ D for the diameter D, such that ⌊2/3D ⌋ ≤ ˆ D ≤ D. Their paper spawned a long line of research on approximate APSP. For the specific problem of diameter approximation, however, no improvement has been achieved in over 15 years. Our paper presents the first improvement over the diameter approximation algorithm of Aingworth et al., producing an algorithm with the same estimate but with an expected running time of Õ(m √ n). We thus show that for all sparse enough graphs, the diameter can be 3/2approximated in o(n 2) time. Our algorithm is obtained using a surprisingly simple method of neighborhood depth estimation that is strong enough to also approximate, in the same running time, the radius and more generally, all of the eccentricities, i.e. for every node the distance to its furthest node. Wealsoprovidestrongevidencethatourdiameterapproximation result may be hard to improve. We show that if for some constant ε> 0 there is an O(m 2−ε) time (3/2 − ε)approximation algorithm for the diameter of undirected unweighted graphs, then there is an O ∗ ((2 − δ) n) time algorithm for CNFSAT on n variables for constant δ> 0, and the strong exponential time hypothesis of [Impagliazzo, Paturi, Zane JCSS’01] is false.
1 Complex Network Measurements: Estimating the Relevance of Observed Properties
"... Abstract—Complex networks, modeled as large graphs, received much attention during these last years. However, data on such networks is only available through intricate measurement procedures. Until recently, most studies assumed that these procedures eventually lead to samples large enough to be rep ..."
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Abstract—Complex networks, modeled as large graphs, received much attention during these last years. However, data on such networks is only available through intricate measurement procedures. Until recently, most studies assumed that these procedures eventually lead to samples large enough to be representative of the whole, at least concerning some key properties. This has crucial impact on network modeling and simulation, which rely on these properties. Recent contributions proved that this approach may be misleading, but no solution has been proposed. We provide here the first practical way to distinguish between cases where it is indeed misleading, and cases where the observed properties may be trusted. It consists in studying how the properties of interest evolve when the sample grows, and in particular whether they reach a steady state or not.
1 Measuring Fundamental Properties of RealWorld Complex Networks
, 2006
"... Abstract — Complex networks (internet maps, web graphs, data exchanges, etc) received much attention during these last years. However data on such networks are only available through intricate measurement procedures. Until recently, most studies assumed that these procedures eventually lead to sampl ..."
Abstract
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Abstract — Complex networks (internet maps, web graphs, data exchanges, etc) received much attention during these last years. However data on such networks are only available through intricate measurement procedures. Until recently, most studies assumed that these procedures eventually lead to sample large enough to be representative of the whole, at least concerning some key properties. This has crucial impact on network modeling and simulation. We propose here a new way to investigate the relevance of this approach: we put together data on complex network measurements that are representative of data commonly used, but significantly larger. Then we study how the observed properties evolve when the sample grows. The obtained results are in sharp contrast with usual assumptions, with important consequences that we discuss.