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.

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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.

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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.

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Graph visualization and navigation in information visualization: A survey

A method, data processing system and computer program product for optimizing the placement of logic gates of a subcircuit in a physical synthesis flow. A Path Smoothing utility identifies one or more movable gates based on at least one selection criteria. A set of legalized candidate locations corresponding to one or more identified movable gates is generated. A disjunctive timing graph based on the generated set of legalized candidate locations is then generated. An optimal location of one or more movable gate(s) is determined using a recursive branch-and-bound search and stored in the computing device.

Incremental timing-driven, physical-synthesis using discrete optimization

a four-layer datapath routing environment, we present an algorithm that considers all the nets simultaneously. Routing probabilities are calculated for potential routing regions and consolidated into a congestion metric. This is followed by an iterative diversion technique where the region with the maximum congestion metric is repetitively relaxed until the track probabilities crystallize into integer values of 1 and 0. We have run the algorithm on large test cases and achieved significant routability within a small number of available tracks.

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Datapath routing based on a decongestion metric

Modern physical-synthesis flows operate on very large designs and perform increasingly aggressive timing optimizations. Traditional incremental timing analysis now represents the single greatest bottleneck in such optimizations and lacks the features necessary to support them efficiently. This article describes a paradigm of transactional timing analysis, which, together with incremental updates, offers an efficient, nested undo functionality that avoids significant timing calculations.

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Speeding Up Physical Synthesis with Transactional Timing Analysis

The top-down “quadratic placement” methodology is rooted in such works as [36, 9, 32]and is reputedly the basis of commercial and in-house VLSI placement tools. Thismethodology iterates between two basic steps: solving sparse systems of linear equationsto achieve a continuous placement solution, and “legalization” of the placement bytransportation or partitioning. Our work, which extends [5], studies implementationchoices and underlying motivations for the quadratic placement methodology. We firstrecall some observations from [5], e.g., that (i) Krylov subspace engines for solvingsparse linear systems are more effective than traditional successive over-relaxation(SOR) engines [33] and (ii) that correlation convergence criteria can maintain solutionquality while using substantially fewer solver iterations. The focus of our investigation isthe coupling of numerical solvers to iterative partitioners that is a hallmark of thequadratic placement methodology. We provide evidence that this coupling may havehistorically been motivated by the pre-1990’s weakness of min-cut partitioners, i.e.,numerical engines were needed to provide helpful hints to weak min-cut partitioners. Inparticular, we show that a modern multilevel FM implementation [2] derives no benefitfrom such coupling. We also show that most insights obtained from study of individualmin-cut partitioning instances (within the top-down placement) also hold within theoverall context of a complete top-down placer implementation.

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Analytical Engines are Unnecessary in Top-down Partitioning-based Placement

Boundary timing in the design of an integrated circuit is facilitated by designating a subset of boundary latches in the circuit, and applying placement constraints to the boundary latches. Global placement is performed while maintaining the boundary latch placement constraints, and a timing driven placement is performed after implementing timing assertions. Boundary latches are designated using a depth-first search to identify the first latches along interconnection paths with the PI/PO, and filtering out ineligible latches according to designer rules. A latch can be filtered out if it is in a large cluster of latches driven by a primary input or driving a primary output, if it drives too many POs, or is a feed-through latch. Constraints include movebounds, preplacement, or attractive forces between boundary latches and other boundary fixed objects, i.e., a fixed gate or a PI/PO.

Charles J. Alpert

Papers

Incremental timing-driven, physical-synthesis using discrete optimization

Datapath routing based on a decongestion metric

Speeding Up Physical Synthesis with Transactional Timing Analysis

Analytical Engines are Unnecessary in Top-down Partitioning-based Placement

Boundary latch and logic placement to satisfy timing constraints