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204
Design Galleries: A General Approach to Setting Parameters for Computer Graphics and Animation
, 1997
"... Image rendering maps scene parameters to output pixel values; animation maps motioncontrol parameters to trajectory values. Because these mapping functions are usually multidimensional, nonlinear, and discontinuous, #nding input parameters that yield desirable output values is often a painful pr ..."
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Cited by 206 (3 self)
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Image rendering maps scene parameters to output pixel values; animation maps motioncontrol parameters to trajectory values. Because these mapping functions are usually multidimensional, nonlinear, and discontinuous, #nding input parameters that yield desirable output values is often a painful process of manual tweaking. Interactiveevolution and inverse design are two general methodologies for computerassisted parameter setting in which the computer plays a prominent role. In this paper we present another such methodology.
Spectral Partitioning Works: Planar graphs and finite element meshes
 In IEEE Symposium on Foundations of Computer Science
, 1996
"... Spectral partitioning methods use the Fiedler vectorthe eigenvector of the secondsmallest eigenvalue of the Laplacian matrixto find a small separator of a graph. These methods are important components of many scientific numerical algorithms and have been demonstrated by experiment to work extr ..."
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Cited by 153 (8 self)
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Spectral partitioning methods use the Fiedler vectorthe eigenvector of the secondsmallest eigenvalue of the Laplacian matrixto find a small separator of a graph. These methods are important components of many scientific numerical algorithms and have been demonstrated by experiment to work extremely well. In this paper, we show that spectral partitioning methods work well on boundeddegree planar graphs and finite element meshes the classes of graphs to which they are usually applied. While naive spectral bisection does not necessarily work, we prove that spectral partitioning techniques can be used to produce separators whose ratio of vertices removed to edges cut is O( p n) for boundeddegree planar graphs and twodimensional meshes and O i n 1=d j for wellshaped ddimensional meshes. The heart of our analysis is an upper bound on the secondsmallest eigenvalues of the Laplacian matrices of these graphs. 1. Introduction Spectral partitioning has become one of the mos...
The ISPD98 Circuit Benchmark Suite
 Proc. ACM/IEEE Int’l Symp. Physical Design (ISPD 99), ACM
, 1998
"... From 19851993, the MCNC regularly introduced and maintained circuit benchmarks for use by the Design Automation community. However, during the last five years, no new circuits have been introduced that can be used for developing fundamental physical design applications, such as partitioning and pla ..."
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Cited by 131 (1 self)
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From 19851993, the MCNC regularly introduced and maintained circuit benchmarks for use by the Design Automation community. However, during the last five years, no new circuits have been introduced that can be used for developing fundamental physical design applications, such as partitioning and placement. The largest circuit in the existing set of benchmark suites has over 100,000 modules, but the second largest has just over 25,000 modules, which is small by today’s standards. This paper introduces the ISPD98 benchmark suite which consists of 18 circuits with sizes ranging from 13,000 to 210,000 modules. Experimental results for three existing partitioners are presented so that future researchers in partitioning can more easily evaluate their heuristics. 1
Multilevel Circuit Partitioning
 IN PROC. OF THE 34TH ACM/IEEE DESIGN AUTOMATION CONFERENCE
, 1998
"... 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 multi ..."
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Cited by 86 (8 self)
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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 matchingbased 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 100 000 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.
Spectral Partitioning: The More Eigenvectors, the Better
 PROC. ACM/IEEE DESIGN AUTOMATION CONF
, 1995
"... The graph partitioning problem is to divide the vertices of a graph into disjoint clusters to minimize the total cost of the edges cut by the clusters. A spectral partitioning heuristic uses the graph's eigenvectors to construct a geometric representation of the graph (e.g., linear orderings) w ..."
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Cited by 72 (3 self)
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The graph partitioning problem is to divide the vertices of a graph into disjoint clusters to minimize the total cost of the edges cut by the clusters. A spectral partitioning heuristic uses the graph's eigenvectors to construct a geometric representation of the graph (e.g., linear orderings) which are subsequently partitioned. Our main result shows that when all the eigenvectors are used, graph partitioning reduces to a new vector partitioning problem. This result implies that as many eigenvectors as are practically possible should be used to construct a solution. This philosophy isincontrast to that of the widelyused spectral bipartitioning (SB) heuristic (which uses a single eigenvector to construct a 2way partitioning) and several previous multiway partitioning heuristics [7][10][16][26][37] (which usek eigenvectors to construct a kway partitioning). Our result motivates a simple ordering heuristic that is a multipleeigenvector extension of SB. This heuristic not only signi cantly outperforms SB, but can also yield excellent multiway VLSI circuit partitionings as compared to [1] [10]. Our experiments suggest that the vector partitioning perspective opens the door to new and effective heuristics.
On Modularity Clustering
, 2008
"... Modularity is a recently introduced quality measure for graph clusterings. It has immediately received considerable attention in several disciplines, and in particular in the complex systems literature, although its properties are not well understood. We study the problem of finding clusterings wit ..."
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Cited by 71 (12 self)
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Modularity is a recently introduced quality measure for graph clusterings. It has immediately received considerable attention in several disciplines, and in particular in the complex systems literature, although its properties are not well understood. We study the problem of finding clusterings with maximum modularity, thus providing theoretical foundations for past and present work based on this measure. More precisely, we prove the conjectured hardness of maximizing modularity both in the general case and with the restriction to cuts, and give an Integer Linear Programming formulation. This is complemented by first insights into the behavior and performance of the commonly applied greedy agglomerative approach.
HypergraphPartitioning Based Decomposition for Parallel SparseMatrix Vector Multiplication
 IEEE Trans. on Parallel and Distributed Computing
"... In this work, we show that the standard graphpartitioning based decomposition of sparse matrices does not reflect the actual communication volume requirement for parallel matrixvector multiplication. We propose two computational hypergraph models which avoid this crucial deficiency of the graph mo ..."
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Cited by 63 (34 self)
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In this work, we show that the standard graphpartitioning based decomposition of sparse matrices does not reflect the actual communication volume requirement for parallel matrixvector multiplication. We propose two computational hypergraph models which avoid this crucial deficiency of the graph model. The proposed models reduce the decomposition problem to the wellknown hypergraph partitioning problem. The recently proposed successful multilevel framework is exploited to develop a multilevel hypergraph partitioning tool PaToH for the experimental verification of our proposed hypergraph models. Experimental results on a wide range of realistic sparse test matrices confirm the validity of the proposed hypergraph models. In the decomposition of the test matrices, the hypergraph models using PaToH and hMeTiS result in up to 63% less communication volume (30%38% less on the average) than the graph model using MeTiS, while PaToH is only 1.32.3 times slower than MeTiS on the average. ...
Permuting Sparse Rectangular Matrices into BlockDiagonal Form
 SIAM Journal on Scientific Computing
, 2002
"... We investigate the problem of permuting a sparse rectangular matrix into block diagonal form. Block diagonal form of a matrix grants an inherent parallelism for solving the deriving problem, as recently investigated in the context of mathematical programming, LU factorization and QR factorization. W ..."
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Cited by 57 (18 self)
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We investigate the problem of permuting a sparse rectangular matrix into block diagonal form. Block diagonal form of a matrix grants an inherent parallelism for solving the deriving problem, as recently investigated in the context of mathematical programming, LU factorization and QR factorization. We propose bipartite graph and hypergraph models to represent the nonzero structure of a matrix, which reduce the permutation problem to those of graph partitioning by vertex separator and hypergraph partitioning, respectively. Our experiments on a wide range of matrices, using stateoftheart graph and hypergraph partitioning tools MeTiS and PaToH, revealed that the proposed methods yield very effective solutions both in terms of solution quality and runtime.
Optimal Partitioners and Endcase Placers for Standardcell Layout
 IEEE TRANS. ON CAD
, 2000
"... We study alternatives to classic FMbased partitioning algorithms in the context of endcase processing for topdown standardcell placement. While the divide step in the topdown divide and conquer is usually performed heuristically, we observe that optimal solutions can be found for many su cientl ..."
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Cited by 57 (21 self)
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We study alternatives to classic FMbased partitioning algorithms in the context of endcase processing for topdown standardcell placement. While the divide step in the topdown divide and conquer is usually performed heuristically, we observe that optimal solutions can be found for many su ciently small partitioning instances. Our main motivation is that small partitioning instances frequently contain multiple cells that are larger than the prescribed partitioning tolerance, and that cannot be moved iteratively while preserving the legality ofa solution. To sample the suboptimality of FMbased partitioning algorithms, we focus on optimal partitioning and placement algorithms based on either enumeration or branchandbound that are invoked for instances below prescribed size thresholds,