Despite many empirical successes of spectral clustering methods— algorithms that cluster points using eigenvectors of matrices derived from the data—there are several unresolved issues. First. there are a wide variety of algorithms that use the eigenvectors in slightly different ways. Second, many of these algorithms have no proof that they will actually compute a reasonable clustering. In this paper, we present a simple spectral clustering algorithm that can be implemented using a few lines of Matlab. Using tools from matrix perturbation theory, we analyze the algorithm, and give conditions under which it can be expected to do well. We also show surprisingly good experimental results on a number of challenging clustering problems.

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On Spectral Clustering: Analysis and an algorithm

Data analysis plays an indispensable role for understanding various phenomena. Cluster analysis, primitive exploration with little or no prior knowledge, consists of research developed across a wide variety of communities. The diversity, on one hand, equips us with many tools. On the other hand, the profusion of options causes confusion. We survey clustering algorithms for data sets appearing in statistics, computer science, and machine learning, and illustrate their applications in some benchmark data sets, the traveling salesman problem, and bioinformatics, a new field attracting intensive efforts. Several tightly related topics, proximity measure, and cluster validation, are also discussed.

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Survey of clustering algorithms

We consider the problem of detecting communities or modules in networks, groups of vertices with a higher-than-average density of edges connecting them. Previous work indicates that a robust approach to this problem is the maximization of the benefit function known as ``modularity'' over possible divisions of a network. Here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix, which plays a role in community detection similar to that played by the graph Laplacian in graph partitioning calculations. This result leads us to a number of possible algorithms for detecting community structure, as well as several other results, including a spectral measure of bipartite structure in networks and a centrality measure that identifies vertices that occupy central positions within the communities to which they belong. The algorithms and measures proposed are illustrated with applications to a variety of real-world complex networks.

/pdf/finding-community-structure-in-networks-using-the-4em5so8glh.pdf

Finding community structure in networks using the eigenvectors of matrices

This paper introduces the problem of combining multiple partitionings of a set of objects into a single consolidated clustering without accessing the features or algorithms that determined these partitionings. We first identify several application scenarios for the resultant 'knowledge reuse' framework that we call cluster ensembles. The cluster ensemble problem is then formalized as a combinatorial optimization problem in terms of shared mutual information. In addition to a direct maximization approach, we propose three effective and efficient techniques for obtaining high-quality combiners (consensus functions). The first combiner induces a similarity measure from the partitionings and then reclusters the objects. The second combiner is based on hypergraph partitioning. The third one collapses groups of clusters into meta-clusters which then compete for each object to determine the combined clustering. Due to the low computational costs of our techniques, it is quite feasible to use a supra-consensus function that evaluates all three approaches against the objective function and picks the best solution for a given situation. We evaluate the effectiveness of cluster ensembles in three qualitatively different application scenarios: (i) where the original clusters were formed based on non-identical sets of features, (ii) where the original clustering algorithms worked on non-identical sets of objects, and (iii) where a common data-set is used and the main purpose of combining multiple clusterings is to improve the quality and robustness of the solution. Promising results are obtained in all three situations for synthetic as well as real data-sets.

/pdf/cluster-ensembles-a-knowledge-reuse-framework-for-combining-3qjcysq4q0.pdf

Cluster ensembles --- a knowledge reuse framework for combining multiple partitions

This is a survey on graph visualization and navigation techniques, as used in information visualization. Graphs appear in numerous applications such as Web browsing, state-transition diagrams, and data structures. The ability to visualize and to navigate in these potentially large, abstract graphs is often a crucial part of an application. Information visualization has specific requirements, which means that this survey approaches the results of traditional graph drawing from a different perspective.

/pdf/graph-visualization-and-navigation-in-information-24tisrjfna.pdf

Graph visualization and navigation in information visualization: A survey

A method, system, and computer program product for timing driven routing in a design of an integrated circuit (IC) are provided in the illustrative embodiments. A router application executing in a data processing system performs a pre-global routing optimization of the design. A plurality of wirelength target constraints are set on a plurality of subsets of a set of nets in the design. Global routing is performed on the design. The design is adjusted using wires placed in the design during the global routing. A priority is assigned to each net in the set of nets. Detailed routing is performed on the design.

Timing driven routing in integrated circuit design

Analysis of Elmore delay in distributed RC tree structures shows the influence of both tree cost and tree radius on signal delay in VLSI interconnects. We give new and efficient interconnection tree constructions that smoothly combine the minimum cost and the minimum radius objectives, by combining respectively optimnl algorithms due to Prim and Dijkstra. Previous "shallow-light" techniques (Z), (31, (81, (131 are both less direct and less effective: in practice, our methods achieve uniformly superior cost-radius tradeoffs. Timing simulations for a range of IC and MCM interconnect technologies show that our wirelength savings yield reduced signal delays when compared to shallow-light or standard minimum spanning tree and Steiner tree routing.

/pdf/performance-driven-routing-tree-design-4xkw9l8zaw.pdf

Performance-Driven Routing Tree Design

A design tool with a direct current (DC) transformation analysis unit determines combinations of candidate sink locations for sector buffers within a sector of a clock network design. For each of the combination of candidate sink locations, the design tool transforms resistances of the sector with the combination of candidate sink locations into resistances of an electrical circuit. The design tool transforms capacitances of the sector with the combination of candidate sink locations into current sources of an electrical circuit. The design tool performs a DC circuit analysis, wherein results of the DC circuit analysis include a variance of voltage at nodes of the sector and a maximum value of current from currents flowing between pairs of the nodes of the sector. The design tool determines which of the combination of candidate sink locations has the minimum variance of voltage with the results of the DC circuit analysis.

Direct current circuit analysis based clock network design

A method for cell placement in an integrated circuit design that uses a calculated diffusion velocity determined from a density value in order to relocate the cells until the cell placement reduces the density below a predetermined threshold. The method acts to control the movement of different cells to reduce the density of the cells prior to legalization of the cell placement.

Method for diffusion based cell placement migration

A method of determining a circuit response (such as delay or slew) from a ramp input of an RC circuit calculates two circuit response parameters using a given circuit response metric based on a step input for the RC circuit, and extends the circuit response metric to a ramp input of the RC circuit by combining the first and second circuit response parameters to yield an estimated ramp response. The novel technique is based on the use of probability distribution functions and cumulative distribution functions to characterize the impulse response of the RC circuit, and the calculating steps derive the first and second circuit response parameters from such statistical distribution functions. In particular, the calculating steps may use a standard deviation or a mean of a probability distribution function corresponding to the circuit response parameter. In one application, the invention is used to estimate delay response for the ramp input of the RC circuit. In another application, the invention is used to estimate output slew for the ramp input of the RC circuit. The invention may be used to extend any valid delay or slew metric.

Charles J. Alpert

Papers

Timing driven routing in integrated circuit design

Performance-Driven Routing Tree Design

Direct current circuit analysis based clock network design

Method for diffusion based cell placement migration

Method and system for extending delay and slew metrics to ramp inputs