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

The physical design flow of any project depends upon the size of the design, the technology, the number of designers, the clock frequency, and the time to do the design. As technology advances and design-styles change, physical design flows are constantly reinvented as traditional phases are removed and new ones are added to accommodate changes in technology. Includes a personal perspective from Ralph Otten as he looks back on the major technical milestones in the history of physical design automation. Explore State-of-the-Art Techniques and TrendsHandbook of Algorithms for Physical Design Automation provides a detailed overview of VLSI physical design automation, emphasizing state-of-the-art techniques, trends and improvements that have emerged during the previous decade. After a brief introduction to the modern physical design problem, basic algorithmic techniques, and partitioning, the book discusses significant advances in floorplanning representations and describes recent formulations of the floorplanning problem. The text also addresses issues of placement, net layout and optimization, routing multiple signal nets, manufacturability, physical synthesis, special nets, and designing for specialized technologies. Highly-Focused Information for Next Generation Design Problems Although several books on this topic are currently available, most are either too broad or out of date. Alternatively, proceedings and journal articles are valuable resources for researchers in this area, but the material is widely dispersed in the literature. This handbook pulls together a broad variety of perspectives on the most challenging problems in the field, and focuses on emerging problems and research results.

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Handbook of Algorithms for Physical Design Automation

Spectral partitioning with multiple eigenvectors

Recent work has illustrated the promise ofmultilevel approaches for partitioning large circuits. Multilevel partitioningrecursively clusters the instance until its size is smallerthan a given threshold, then unclusters the instance while applyinga partitioning refinement algorithm. Our multilevel partitioner usesa new technique to control the number of levels in the matching-basedclustering phase and also exploits recent innovations in classiciterative partitioning. Our heuristic outperforms numerousexisting bipartitioning heuristics, with improvements rangingfrom 6.9 to 27.9% for 100 runs and 3.0 to 20.6% for just 10 runs(while also using less CPU time).

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Multilevel circuit partitioning

Many previous works in partitioning have used some underlying clustering algorithm to improve performance. As problem sizes reach new levels of complexity, a single application of a clustering algorithm is insufficient to produce excellent solutions. Recent work has illustrated the promise of multilevel approaches. A multilevel partitioning algorithm recursively clusters the instance until its size is smaller than a given threshold, then unclusters the instance, while applying a partitioning refinement algorithm. In this paper, we propose a new multilevel partitioning algorithm that exploits some of the latest innovations of classical iterative partitioning approaches. Our method also uses a new technique to control the number of levels in our matching-based clustering algorithm. Experimental results show that our heuristic outperforms numerous existing bipartitioning heuristics with improvements ranging from 6.9 to 27.9% for 100 runs and 3.0 to 20.6% for just ten runs (while also using less CPU time). Further, our algorithm generates solutions better than the best known mincut bipartitionings for seven of the ACM/SIGDA benchmark circuits, including golem3 (which has over 100000 cells). We also present quadrisection results which compare favorably to the partitionings obtained by the GORDIAN cell placement tool. Our work in multilevel quadrisection has been used as the basis for an effective cell placement package.

Buffer insertion has successfully been applied to reduce delay in global interconnect paths: however, existing techniques only optimize delay and timing slack. With the increasing ratio of coupling to total capacitance and the use of aggressive dynamic logic circuit families, noise is becoming a major design bottleneck. We present comprehensive buffer insertion techniques for noise and delay optimisation. Our experiments on a microprocessor design show that our approach fixes all noise violations that were identified by a detailed, simulation-based noise analysis tool. Further, we show that the performance penalty induced by optimizing both delay and noise as opposed to only delay is 2%.

Charles J. Alpert

Papers

Handbook of Algorithms for Physical Design Automation

Spectral partitioning with multiple eigenvectors

Multilevel circuit partitioning

Multilevel circuit partitioning

Buffer insertion for noise and delay optimization