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, system, and computer program product for improved routing using layer ranges in the design of an integrated circuit (IC) are provided in the illustrative embodiments. Using an application executing in a data processing system, a score is computed for a net in a set of nets routed using a set of layers in the design. The set of nets is sorted according to scores associated with nets in the set of nets. A layer range from a set of layer ranges is assigned to a net in the sorted list such that a net with a higher than threshold score is assigned a high layer range.

Routing and timing using layer ranges

Routing congestion has become a critical layout challenge in nanoscale circuits since it is a critical factor in determining the routability of a design. An unroutable design is not useful even though it closes on all other design metrics. Fast design closure can only be achieved by accurately evaluating whether a design is routable or not early in the design cycle. Lately, it has become common to use a “light mode” version of a global router to quickly evaluate the routability of a given placement. This approach suffers from three weaknesses: (i) it does not adequately model local routing resources, which can cause incorrect routability predictions that are only detected late, during detailed routing; (ii) the congestion maps obtained by it tend to have isolated hotspots surrounded by noncongested spots, called “noisy hotspots”, which further affects the accuracy in routability evaluation; and (iii) the metrics used to represent congestion may yield numbers that do not provide sufficient intuition to the designer, and moreover, they may often fail to predict the routability accurately. This article presents solutions to these issues. First, we propose three approaches to model local routing resources. Second, we propose a smoothing technique to reduce the number of noisy hotspots and obtain a more accurate routability evaluation result. Finally, we develop a new metric which represents congestion maps with higher fidelity. We apply the proposed techniques to several industrial circuits and demonstrate that one can better predict and evaluate design routability and that congestion mitigation tools can perform much better to improve the design routability.

/pdf/techniques-for-scalable-and-effective-routability-evaluation-1w1oaf4548.pdf

Techniques for scalable and effective routability evaluation

As VLSI technology enters the nanoscale regime, the interconnect delay becomes the bottleneck of circuit performance. Compared with gate delays, wires are becoming increasingly resistive, making it more difficult to propagate signals across the chip. However, more advanced technologies (65 and 45 nm) provide relief as the number of metal layers continues to increase. The wires on the upper metal layers are much less resistive and can be used to drive further and faster than on thin metals. This provides an entirely new dimension to the traditional wire-sizing problem, namely, layer assignment for efficient timing closure. Assigning all wires to thick metals improves timing; however, the routability of the design may be hurt. The challenge is to assign a minimal amount of wires to thick metals to meet timing constraints. In this brief, the minimum cost layer assignment problem is proven to be NP-complete. As a theoretical solution for NP-complete problems, a fully polynomial-time approximation scheme is proposed. The new algorithm can approximate the optimal layer assignment solution by a factor of 1 + isin in O(m log log M ldrn 3 isin2) time for 0 < isin < 1, where n is the number of nodes in the tree, m is the number of routing layers, and M is the maximum cost ratio among layers. This work presents the first theoretical advance for the timing-driven minimum cost layer assignment problem. In addition to its theoretical guarantee, the new algorithm is highly practical. Our experiments on 500 industrial test cases demonstrate that the new algorithm can run 2times faster than the optimal dynamic programming algorithm, with only 2% additional wire.

A Fully Polynomial-Time Approximation Scheme for Timing-Constrained Minimum Cost Layer Assignment

The presence of fixed terminals in hypergraph partitioning instances arising in top-down standard-cell placement makes such instances qualitatively different from the free hypergraphs that have driven the past two decades of VLSI CAD partitioning research. In this paper we empirically show that with fixed terminals in the instance, less effort is needed to stably reach a given solution quality. We then develop new benchmark formats that flexibly capture the presence of terminals and any geometric embedding information associated with the partitioning instance. Our new formats not only allow modeling of top-down placement, but also enable study of placement-specific partitioning objectives, e.g., based on net bounding boxes and Steiner tree estimators. Finally, we develop a new suite of partitioning benchmarks with fixed terminals, based on the actual placement data from the IBM-internal circuits released in the ISPD-98 Benchmark Suite [1, 2]. A set of partitioning results is presented with runtime regimes appropriate to the placement use model, as a baseline for future efforts in the research community.

https://web.eecs.umich.edu/~imarkov/pubs/conf/c009.pdf

Partitioning with terminals: a “new” problem and new benchmarks

Along with the progress of very-large-scale-integration technology, buffer insertion plays an increasingly critical role on affecting circuit design and performance. Traditional buffer insertion algorithms are mostly net based and therefore often result in suboptimal delay or unnecessary buffer expense due to the lack of global view. In this paper, we propose a novel path-based-buffer-insertion (PBBI) scheme which can overcome the weakness of the net-based approaches. We also discuss some potential difficulties of the PBBI approach and propose solutions to them. A fast estimation on buffered delay is employed to improve the solution quality. Gate sizing is also considered at the same time. Experimental results show that our method can efficiently reduce buffer/gate cost significantly (by 71% on average) when compared to traditional net-based approaches. To the best of our knowledge, this is the first work on path based buffer insertion and simultaneous gate sizing.

Charles J. Alpert

Papers

Routing and timing using layer ranges

Techniques for scalable and effective routability evaluation

A Fully Polynomial-Time Approximation Scheme for Timing-Constrained Minimum Cost Layer Assignment

Partitioning with terminals: a “new” problem and new benchmarks

Path-Based Buffer Insertion