We present a multi-scale layout algorithm for the aesthetic drawing of undirected graphs with straight-line 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 multi-scale scheme for drawing graphs, which was motivated by the earlier multi-scale algorithm of Hadany and Harel [HH99]. In principle, it could significantly improve the speed of essentially any force-directed method (regardless of that method's ability of drawing weighted graphs or the continuity of its cost-function).

Recent work has looked at extending the k-Means algorithm to incorporate background information in the form of instance level must-link and cannot-link 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, NP-complete. Thus, an iterative algorithm such as k-Means should not try to find a feasible partitioning at each iteration. This motivates our derivation of a new version of the k-Means 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 infra-red detector on an Aibo robot.

Buffer insertion has become an increasingly critical optimization in high performance design. The problem of finding a delay-optimal 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 real-world “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 C-Tree for these instance types. When combined with van Ginneken style buffer insertion, C-Tree achieves higher quality solutions with fewer resources compared to traditional approaches. 1.

Abstract—This paper considers the problem of efficiently maintaining a clustering of a dynamic set of data points that move continuously in two-dimensional 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 moving-object clusters. Index Terms—Spatial databases, temporal databases, clustering. Ç 1

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 present-day designs.

We present a mu lti-scale layou algorithm for the aesthetic drawing ofuV#860L32 graphs with straight-line edges. The algorithm is extremely fast, and is capable of drawing graphs that aresu--SVV 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 lti-scale 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 force-directed method (regardless of that method's ability of drawing weighted graphs or the continu0 y of its cost-fuL436280 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. 183--196, 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 straight-lin 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 force-directed 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...

Let M be a 2-dimensional colored map which has been digitized into a large 2-dimensional 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 NP-hard. 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 NP-hard. Furthermore, for one of the most succinct of our languages we present evidence suggesting that finding a near-optimal map compression is as difficult as finding an optimal compression.

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Recent work has looked at extending the k-Means algorithm to incorporate background information in the form of instance level must-link and cannot-link 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, NP-complete. Thus, an iterative algorithm such as k-Means should not try to find a feasible partitioning at each iteration. This motivates our derivation of a new version of the k-Means 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 infra-red detector on an Aibo robot.