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11
A Fast MultiScale Method for Drawing Large Graphs
 JOURNAL OF GRAPH ALGORITHMS AND APPLICATIONS
, 2002
"... We present a multiscale layout algorithm for the aesthetic drawing of undirected graphs with straightline edges. The algorithm is extremely fast, and is capable of drawing graphs that are substantially larger than those we have encountered in prior work. For example, the paper contains a drawi ..."
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Cited by 79 (10 self)
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We present a multiscale layout algorithm for the aesthetic drawing of undirected graphs with straightline edges. The algorithm is extremely fast, and is capable of drawing graphs that are substantially larger than those we have encountered in prior work. For example, the paper contains a drawing of a graph with over 15,000 vertices. Also we achieve "nice" drawings of 1000 vertex graphs in about 1 second. The proposed algorithm embodies a new multiscale scheme for drawing graphs, which was motivated by the earlier multiscale algorithm of Hadany and Harel [HH99]. In principle, it could significantly improve the speed of essentially any forcedirected method (regardless of that method's ability of drawing weighted graphs or the continuity of its costfunction).
Clustering with Constraints: Feasibility Issues and the kMeans Algorithm
, 2005
"... Recent work has looked at extending the kMeans algorithm to incorporate background information in the form of instance level mustlink and cannotlink constraints. We introduce two ways of specifying additional background information in the form of # and # constraints that operate on all instances ..."
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Cited by 59 (7 self)
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Recent work has looked at extending the kMeans algorithm to incorporate background information in the form of instance level mustlink and cannotlink constraints. We introduce two ways of specifying additional background information in the form of # and # constraints that operate on all instances but which can be interpreted as conjunctions or disjunctions of instance level constraints and hence are easy to implement. We present complexity results for the feasibility of clustering under each type of constraint individually and several types together. A key finding is that determining whether there is a feasible solution satisfying all constraints is, in general, NPcomplete. Thus, an iterative algorithm such as kMeans should not try to find a feasible partitioning at each iteration. This motivates our derivation of a new version of the kMeans algorithm that minimizes the constrained vector quantization error but at each iteration does not attempt to satisfy all constraints. Using standard UCI datasets, we find that using constraints improves accuracy as others have reported, but we also show that our algorithm reduces the number of iterations until convergence. Finally, we illustrate these benefits and our new constraint types on a complex real world object identification problem using the infrared detector on an Aibo robot.
Buffered Steiner trees for difficult instances
 IEEE Transactions on ComputerAided Design
, 2002
"... Buffer insertion has become an increasingly critical optimization in high performance design. The problem of finding a delayoptimal buffered Steiner tree has been an active area of research, and excellent solutions exist for most instances. However, current approaches fail to adequately solve a par ..."
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Cited by 20 (5 self)
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Buffer insertion has become an increasingly critical optimization in high performance design. The problem of finding a delayoptimal buffered Steiner tree has been an active area of research, and excellent solutions exist for most instances. However, current approaches fail to adequately solve a particular class of realworld “difficult ” instances which are characterized by a large number of sinks, variations in sink criticalities, and varying polarity requirements. We propose a new Steiner tree construction called CTree for these instance types. When combined with van Ginneken style buffer insertion, CTree achieves higher quality solutions with fewer resources compared to traditional approaches. 1.
Continuous Clustering of Moving Objects
 IEEE TKDE
, 2007
"... Abstract—This paper considers the problem of efficiently maintaining a clustering of a dynamic set of data points that move continuously in twodimensional euclidean space. This problem has received little attention and introduces new challenges to clustering. The paper proposes a new scheme that is ..."
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Cited by 17 (0 self)
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Abstract—This paper considers the problem of efficiently maintaining a clustering of a dynamic set of data points that move continuously in twodimensional euclidean space. This problem has received little attention and introduces new challenges to clustering. The paper proposes a new scheme that is capable of incrementally clustering moving objects. This proposal employs a notion of object dissimilarity that considers object movement across a period of time, and it employs clustering features that can be maintained efficiently in incremental fashion. In the proposed scheme, a quality measure for incremental clusters is used for identifying clusters that are not compact enough after certain insertions and deletions. An extensive experimental study shows that the new scheme performs significantly faster than traditional ones that frequently rebuild clusters. The study also shows that the new scheme is effective in preserving the quality of movingobject clusters. Index Terms—Spatial databases, temporal databases, clustering. Ç 1
Design and Analysis of Physical Design Algorithms
 IN PROC. INTERNATIONAL SYMPOSIUM ON PHYSICAL DESIGN
, 2001
"... We will review a few key algorithmic and analysis concepts with application to physical design problems. We argue that design and detailed analysis of algorithms is of fundamental importance in developing better physical design tools and to cope with the complexity of presentday designs. ..."
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Cited by 2 (2 self)
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We will review a few key algorithmic and analysis concepts with application to physical design problems. We argue that design and detailed analysis of algorithms is of fundamental importance in developing better physical design tools and to cope with the complexity of presentday designs.
Clustering With Constraints: Feasibility Issues and the kMeans Algorithm
"... Recent work has looked at extending the kMeans algorithm to incorporate background information in the form of instance level mustlink and cannotlink constraints. We introduce two ways of specifying additional background information in the form of δ and ɛ constraints that operate on all instances ..."
Abstract
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Recent work has looked at extending the kMeans algorithm to incorporate background information in the form of instance level mustlink and cannotlink constraints. We introduce two ways of specifying additional background information in the form of δ and ɛ constraints that operate on all instances but which can be interpreted as conjunctions or disjunctions of instance level constraints and hence are easy to implement. We present complexity results for the feasibility of clustering under each type of constraint individually and several types together. A key finding is that determining whether there is a feasible solution satisfying all constraints is, in general, NPcomplete. Thus, an iterative algorithm such as kMeans should not try to find a feasible partitioning at each iteration. This motivates our derivation of a new version of the kMeans algorithm that minimizes the constrained vector quantization error but at each iteration does not attempt to satisfy all constraints. Using standard UCI datasets, we find that using constraints improves accuracy as others have reported, but we also show that our algorithm reduces the number of iterations until convergence. Finally, we illustrate these benefits and our new constraint types on a complex real world object identification problem using the infrared detector on an Aibo robot.
A Fast Multi Method for Drawing Large Graphs
 Journal of Graph Algorithms and Applications
, 2001
"... We present a mu ltiscale layou algorithm for the aesthetic drawing ofuV#860L32 graphs with straightline edges. The algorithm is extremely fast, and is capable of drawing graphs that aresuSVV tially larger than those we haveencou tered in prior work. For example, the paper contains a drawing of ..."
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We present a mu ltiscale layou algorithm for the aesthetic drawing ofuV#860L32 graphs with straightline edges. The algorithm is extremely fast, and is capable of drawing graphs that aresuSVV tially larger than those we haveencou tered in prior work. For example, the paper contains a drawing of a graph with over 15,000 vertices. Also we achieve "optimal" drawings of 1000 vertex graphs in ab ou 1 second. The proposed algorithm embodies a new mu ltiscale scheme for drawing graphs, which was motivated by the earlier muS8820L4# algorithm of Hadany and Harel [HH99]. In principle, it cou# significantly improve the speed of essentially any forcedirected method (regardless of that method's ability of drawing weighted graphs or the continu0 y of its costfuL436280 1 Introducti7 AgraphG(V,E) is an abstract structure that is used to model a relation over a set V ofen tities. Graph drawin is acon ven tion# tool for the visualization of relationq in]4#]Rq]#] an its usefulnRq depenV on its readability, that is, the capability of con veyin the meanfi# of the diagram quickly # A shorter version appeared in Proc. GraphD awing 2000, LNCS 1984, pp. 183196, Springer Verlag, 2000. clearly. In recen t years, man y algorithms for drawin graphs automatically were proposed (the state of the art is surveyedcomprehen]8 ely in 99, KW01]). Wecon#E trateon the problem of drawin an unF8ERqfi4 graph with straightlin edges. In this case the problem reduces to that of positionRq the vertices bydeterminfiF a mappin L : V R . A populargenrR# approach to this problem is the forcedirected technRVEF which in troduces a heuristic costfunR]F9 (an energy) of themappin L, which (hopefully) achieves its min4 umwhen the layout isn ice. Varian ts of this approach di#er in the defin9fiFR of t...
Complexity Aspects of 2Dimensional Data Compression
, 1991
"... Let M be a 2dimensional colored map which has been digitized into a large 2dimensional array (M). We define a class of languages (called rectilinear) to describe our digitized maps and classify them based on their level of succinct representation. We also study the map compression problem, i.e. ..."
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Let M be a 2dimensional colored map which has been digitized into a large 2dimensional array (M). We define a class of languages (called rectilinear) to describe our digitized maps and classify them based on their level of succinct representation. We also study the map compression problem, i.e., the problem of finding for any given map a shortest description within a given language. For one dimensional maps we show that a shortest description can be generated quickly for some languages, but for other languages the problem is NPhard. We also show that a large number of linear time algorithms for our languages generate map descriptions whose length is at most twice the length of the minimum length description. For all our languages we show that the two dimensional map compression problem is NPhard. Furthermore, for one of the most succinct of our languages we present evidence suggesting that finding a nearoptimal map compression is as difficult as finding an optimal compression.
The Complexity of NonHierarchical Clustering With Instance and Cluster Level Constraints ∗
"... Recent work has looked at extending clustering algorithms with instance level mustlink (ML) and cannotlink (CL) background information. Our work introduces δ and ǫ cluster level constraints that influence intercluster distances and cluster composition. The addition of background information, thou ..."
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Recent work has looked at extending clustering algorithms with instance level mustlink (ML) and cannotlink (CL) background information. Our work introduces δ and ǫ cluster level constraints that influence intercluster distances and cluster composition. The addition of background information, though useful at providing better clustering results, raises the important feasibility question: Given a collection of constraints and a set of data, does there exist at least one partition of the data set satisfying all the constraints? We study the complexity of the feasibility problem for each of the above constraints separately and also for combinations of constraints. Our results clearly delineate combinations of constraints for which the feasibility problem is computationally intractable (i.e., NPcomplete) from those for which the problem is efficiently solvable (i.e., in the computational class P). We also consider the ML and CL constraints in conjunctive and disjunctive normal forms (CNF and DNF respectively). We show that for ML constraints, the feasibility problem is intractable for CNF but efficiently solvable for DNF. Unfortunately, for CL constraints, the feasibility problem is intractable for both CNF and DNF. This effectively means that CLconstraints in a nontrivial form cannot be efficiently incorporated into clustering algorithms. To overcome this, we introduce the notion of a choiceset of constraints and prove that the feasibility problem for choicesets is efficiently solvable for both ML and CL constraints. We also present empirical results which indicate that the feasibility problem occurs extensively in real world problems.