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200
Separators for spherepackings and nearest neighbor graphs
 J. ACM
, 1997
"... Abstract. A collection of n balls in d dimensions forms a kply system if no point in the space is covered by more than k balls. We show that for every kply system �, there is a sphere S that intersects at most O(k 1/d n 1�1/d) balls of � and divides the remainder of � into two parts: those in the ..."
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Cited by 100 (8 self)
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Abstract. A collection of n balls in d dimensions forms a kply system if no point in the space is covered by more than k balls. We show that for every kply system �, there is a sphere S that intersects at most O(k 1/d n 1�1/d) balls of � and divides the remainder of � into two parts: those in the interior and those in the exterior of the sphere S, respectively, so that the larger part contains at most (1 � 1/(d � 2))n balls. This bound of O(k 1/d n 1�1/d) is the best possible in both n and k. We also present a simple randomized algorithm to find such a sphere in O(n) time. Our result implies that every knearest neighbor graphs of n points in d dimensions has a separator of size O(k 1/d n 1�1/d). In conjunction with a result of Koebe that every triangulated planar graph is isomorphic to the intersection graph of a diskpacking, our result not only gives a new geometric proof of the planar separator theorem of Lipton and Tarjan, but also generalizes it to higher dimensions. The separator algorithm can be used for point location and geometric divide and conquer in a fixed dimensional space.
An Analysis of Recent Work on Clustering Algorithms
, 1999
"... This paper describes four recent papers on clustering, each of which approaches the clustering problem from a different perspective and with different goals. It analyzes the strengths and weaknesses of each approach and describes how a user could could decide which algorithm to use for a given clust ..."
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Cited by 86 (0 self)
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This paper describes four recent papers on clustering, each of which approaches the clustering problem from a different perspective and with different goals. It analyzes the strengths and weaknesses of each approach and describes how a user could could decide which algorithm to use for a given clustering application. Finally, it concludes with ideas that could make the selection and use of clustering algorithms for data analysis less difficult.
Learning Eigenfunctions Links Spectral Embedding And Kernel PCA
 NEURAL COMPUTATION
, 2004
"... In this paper, we show a direct relation between spectral embedding methods and kernel PCA, and how both are special cases of a more general learning problem, that of learning the principal eigenfunctions of an operator defined from a kernel and the unknown data generating density. Whereas ..."
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Cited by 85 (5 self)
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In this paper, we show a direct relation between spectral embedding methods and kernel PCA, and how both are special cases of a more general learning problem, that of learning the principal eigenfunctions of an operator defined from a kernel and the unknown data generating density. Whereas
Models and Approximation Algorithms for Channel Assignment in Radio Networks
, 2000
"... We consider the frequency assignment (broadcast scheduling) problem for packet radio networks. Such networks are naturally modeled by graphs with a certain geometric structure. The problem of broadcast scheduling can be cast as a variant of the vertex coloring problem (called the distance2 coloring ..."
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Cited by 84 (3 self)
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We consider the frequency assignment (broadcast scheduling) problem for packet radio networks. Such networks are naturally modeled by graphs with a certain geometric structure. The problem of broadcast scheduling can be cast as a variant of the vertex coloring problem (called the distance2 coloring problem) on the graph that models a given packet radio network. We present efficient approximation algorithms for the distance2 coloring problem for various geometric graphs including those that naturally model a large class of packet radio networks. The class of graphs considered include (r, s)civilized graphs, planar graphs, graphs with bounded genus, etc.
Mining Newsgroups Using Networks Arising From Social Behavior
, 2003
"... Recent advances in information retrieval over hyperlinked corpora have convincingly demonstrated that links carry less noisy information than text. We investigate the feasibility of applying linkbased methods in new applications domains. The specific application we consider is to partition authors ..."
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Cited by 81 (0 self)
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Recent advances in information retrieval over hyperlinked corpora have convincingly demonstrated that links carry less noisy information than text. We investigate the feasibility of applying linkbased methods in new applications domains. The specific application we consider is to partition authors into opposite camps within a given topic in the context of newsgroups. A typical newsgroup posting consists of one or more quoted lines from another posting followed by the opinion of the author. This social behavior gives rise to a network in which the vertices are individuals and the links represent "respondedto" relationships. An interesting characteristic of many newsgroups is that people more frequently respond to a message when they disagree than when they agree. This behavior is in sharp contrast to the WWW link graph, where linkage is an indicator of agreement or common interest. By analyzing the graph structure of the responses, we are able to effectively classify people into opposite camps. In contrast, methods based on statistical analysis of text yield low accuracy on such datasets because the vocabulary used by the two sides tends to be largely identical, and many newsgroup postings consist of relatively few words of text.
Spectral Partitioning: The More Eigenvectors, the Better
 PROC. ACM/IEEE DESIGN AUTOMATION CONF
, 1995
"... The graph partitioning problem is to divide the vertices of a graph into disjoint clusters to minimize the total cost of the edges cut by the clusters. A spectral partitioning heuristic uses the graph's eigenvectors to construct a geometric representation of the graph (e.g., linear orderings) w ..."
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Cited by 75 (3 self)
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The graph partitioning problem is to divide the vertices of a graph into disjoint clusters to minimize the total cost of the edges cut by the clusters. A spectral partitioning heuristic uses the graph's eigenvectors to construct a geometric representation of the graph (e.g., linear orderings) which are subsequently partitioned. Our main result shows that when all the eigenvectors are used, graph partitioning reduces to a new vector partitioning problem. This result implies that as many eigenvectors as are practically possible should be used to construct a solution. This philosophy isincontrast to that of the widelyused spectral bipartitioning (SB) heuristic (which uses a single eigenvector to construct a 2way partitioning) and several previous multiway partitioning heuristics [7][10][16][26][37] (which usek eigenvectors to construct a kway partitioning). Our result motivates a simple ordering heuristic that is a multipleeigenvector extension of SB. This heuristic not only signi cantly outperforms SB, but can also yield excellent multiway VLSI circuit partitionings as compared to [1] [10]. Our experiments suggest that the vector partitioning perspective opens the door to new and effective heuristics.
Isoperimetric graph partitioning for image segmentation
 IEEE TRANS. ON PAT. ANAL. AND MACH. INT
, 2006
"... Spectral graph partitioning provides a powerful approach to image segmentation. We introduce an alternate idea that finds partitions with a small isoperimetric constant, requiring solution to a linear system rather than an eigenvector problem. This approach produces the high quality segmentations o ..."
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Cited by 74 (12 self)
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Spectral graph partitioning provides a powerful approach to image segmentation. We introduce an alternate idea that finds partitions with a small isoperimetric constant, requiring solution to a linear system rather than an eigenvector problem. This approach produces the high quality segmentations of spectral methods, but with improved speed and stability.
Experiments on Graph Clustering Algorithms
 In 11th Europ. Symp. Algorithms
, 2003
"... A promising approach to graph clustering is based on the intuitive notion of intracluster density vs. intercluster sparsity. While both formalizations and algorithms focusing on particular aspects of this rather vague concept have been proposed no conclusive argument on their appropriateness h ..."
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Cited by 71 (9 self)
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A promising approach to graph clustering is based on the intuitive notion of intracluster density vs. intercluster sparsity. While both formalizations and algorithms focusing on particular aspects of this rather vague concept have been proposed no conclusive argument on their appropriateness has been given.
A Graph Theoretic Approach to Software Watermarking
, 2001
"... We present a graph theoretic approach for watermarking software in a robust fashion. While watermarking software that are small in size (e.g. a few kilobytes) may be infeasible through this approach, it seems to be a viable scheme for large applications. Our approach works with control/data flow gra ..."
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Cited by 64 (0 self)
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We present a graph theoretic approach for watermarking software in a robust fashion. While watermarking software that are small in size (e.g. a few kilobytes) may be infeasible through this approach, it seems to be a viable scheme for large applications. Our approach works with control/data flow graphs and uses abstractions,approximate kpartitions,and a random walk method to embed the watermark,with the goal of minimizing and controlling the additions to be made for embedding,while keeping the estimated effort to undo the watermark (WM) as high as possible. The watermarks are so embedded that small changes to the software or flow graph are unlikely to disable detection by a probabilistic algorithm that has a secret. This is done by using some relatively robust graph properties and error correcting codes. Under some natural assumptions about the code added to embed the WM,locating the WM by an attacker is related to some graph approximation problems. Since little theoretical foundation exists for hardness of typical instances of graph approximation problems,we present heuristics to generate such hard instances and,in a limited case,present a heuristic analysis of how hard it is to separate the WM in an information theoretic model. We describe some related experimental work. The approach and methods described here also suitable for solving the problem of software tamper resistance.
A local clustering algorithm for massive graphs and its application to nearlylinear time graph partitioning
, 2013
"... We study the design of local algorithms for massive graphs. A local graph algorithm is one that finds a solution containing or near a given vertex without looking at the whole graph. We present a local clustering algorithm. Our algorithm finds a good cluster—a subset of vertices whose internal conn ..."
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Cited by 60 (9 self)
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We study the design of local algorithms for massive graphs. A local graph algorithm is one that finds a solution containing or near a given vertex without looking at the whole graph. We present a local clustering algorithm. Our algorithm finds a good cluster—a subset of vertices whose internal connections are significantly richer than its external connections—near a given vertex. The running time of our algorithm, when it finds a nonempty local cluster, is nearly linear in the size of the cluster it outputs. The running time of our algorithm also depends polylogarithmically on the size of the graph and polynomially on the conductance of the cluster it produces. Our clustering algorithm could be a useful primitive for handling massive graphs, such as social networks and webgraphs. As an application of this clustering algorithm, we present a partitioning algorithm that finds an approximate sparsest cut with nearly optimal balance. Our algorithm takes time nearly linear in the number edges of the graph. Using the partitioning algorithm of this paper, we have designed a nearly linear time algorithm for constructing spectral sparsifiers of graphs, which we in turn use in a nearly linear time algorithm for solving linear systems in symmetric, diagonally dominant matrices. The linear system solver also leads to a nearly linear time algorithm for approximating the secondsmallest eigenvalue and corresponding eigenvector of the Laplacian matrix of a graph. These other results are presented in two companion papers.