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42
A Multilevel Algorithm for ForceDirected GraphDrawing
, 2003
"... We describe a heuristic method for drawing graphs which uses a multilevel framework combined with a forcedirected placement algorithm. ..."
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Cited by 88 (3 self)
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We describe a heuristic method for drawing graphs which uses a multilevel framework combined with a forcedirected placement algorithm.
Mesh Partitioning: a Multilevel Balancing and Refinement Algorithm
, 1998
"... Multilevel algorithms are a successful class of optimisation techniques which address the mesh partitioning problem. They usually combine a graph contraction algorithm together with a local optimisation method which refines the partition at each graph level. In this paper we present an enhancement o ..."
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Cited by 55 (22 self)
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Multilevel algorithms are a successful class of optimisation techniques which address the mesh partitioning problem. They usually combine a graph contraction algorithm together with a local optimisation method which refines the partition at each graph level. In this paper we present an enhancement of the technique which uses imbalance to achieve higher quality partitions. We also present a formulation of the KernighanLin partition optimisation algorithm which incorporates loadbalancing. The resulting algorithm is tested against a different but related stateofthe art partitioner and shown to provide improved results. Keywords: graphpartitioning, mesh partitioning, loadbalancing, multilevel algorithms. 1 Introduction The need for mesh partitioning arises naturally in many finite element (FE) and finite volume (FV) applications. Meshes composed of elements such as triangles or tetrahedra are often better suited than regularly structured grids for representing completely general ge...
Parallel Optimisation Algorithms for Multilevel Mesh Partitioning
 Parallel Comput
, 2000
"... Three parallel optimisation algorithms, for use in the context of multilevel graph partitioning of unstructured meshes, are described. The first, interface optimisation, reduces the computation to a set of independent optimisation problems in interface regions. The next, alternating optimisation, is ..."
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Cited by 45 (14 self)
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Three parallel optimisation algorithms, for use in the context of multilevel graph partitioning of unstructured meshes, are described. The first, interface optimisation, reduces the computation to a set of independent optimisation problems in interface regions. The next, alternating optimisation, is a restriction of this technique in which mesh entities are only allowed to migrate between subdomains in one direction. The third treats the gain as a potential field and uses the concept of relative gain for selecting appropriate vertices to migrate. The results are compared and seen to produce very high global quality partitions, very rapidly. The results are also compared with another partitioning tool and shown to be of higher quality although taking longer to compute. 2000 Elsevier Science B.V. All rights reserved.
A New Approach to Effective Circuit Clustering
 In Proceedings of IEEE International Conference on Computer Aided Design
, 1992
"... The complexity of nextgeneration VLSI systems will exceed the capabilities of topdown layout synthesis algorithms, particularly in netlist partitioning and module placement. Bottomup clustering is needed to "condense" the netlist so that the problem size becomes tractable to existing optimization ..."
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Cited by 43 (5 self)
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The complexity of nextgeneration VLSI systems will exceed the capabilities of topdown layout synthesis algorithms, particularly in netlist partitioning and module placement. Bottomup clustering is needed to "condense" the netlist so that the problem size becomes tractable to existing optimization methods. In this paper, we establish the DS quality measure, the first general metric for evaluation of clustering algorithms. The DS metric in turn motivates our RWST algorithm, a new selftuning clustering method based on random walks in the circuit netlist. RWST efficiently captures a globally good circuit clustering. When incorporated within a twophase iterative FiducciaMattheyses partitioning strategy, the RWST clustering method improves bisection width by an average of 17% over previous matchingbased methods. 1 Introduction Topdown approaches are widely used to cope with increasing problem complexity in layout synthesis. Recursive calls to a partitioning algorithm generate a ci...
Finding kcuts within Twice the Optimal
, 1995
"... Two simple approximation algorithms for the minimum kcut problem are presented. Each algorithm finds a kcut having weight within a factor of (2 \Gamma 2=k) of the optimal. One of our algorithms is particularly efficient  it requires a total of only n \Gamma 1 maximum flow computations for find ..."
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Cited by 42 (2 self)
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Two simple approximation algorithms for the minimum kcut problem are presented. Each algorithm finds a kcut having weight within a factor of (2 \Gamma 2=k) of the optimal. One of our algorithms is particularly efficient  it requires a total of only n \Gamma 1 maximum flow computations for finding a set of nearoptimal kcuts, one for each value of k between 2 and n. i 1 Introduction The minimum kcut problem is as follows: given an undirected graph G = (V; E) with nonnegative edge weights and a positive integer k, find a set S ` E of minimum weight whose removal leaves k connected components. This problem is of considerable practical significance, especially in the area of VLSI design. Solving this problem exactly is NPhard [GH], but no efficient approximation algorithms were known for it. In this paper we give two simple algorithms for finding kcuts. We prove a performance guarantee of (2 \Gamma 2=k) for each algorithm; however, neither algorithm dominates the other on a...
On the Intrinsic Rent Parameter and SpectraBased Partitioning Methodologies
 IEEE Trans. on Comput.Aided Des., Integrated Circuits & Syst
, 1994
"... The complexity of circuit designs has necessitated a topdown approach to layout synthesis. A large body of work shows that a good layout hierarchy, or partitioning tree, as measured by the associated Rent parameter, will correspond to an areaefficient layout. We define the intrinsic Rent parameter ..."
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Cited by 37 (6 self)
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The complexity of circuit designs has necessitated a topdown approach to layout synthesis. A large body of work shows that a good layout hierarchy, or partitioning tree, as measured by the associated Rent parameter, will correspond to an areaefficient layout. We define the intrinsic Rent parameter of a netlist to be the minimum possible Rent parameter of any partitioning tree for the netlist. Experimental results show that spectrabased ratio cut partitioning algorithms yield partitioning trees with the lowest observed Rent parameter over all benchmarks and over all algorithms tested. For examples where the intrinsic Rent parameter is known, spectral ratio cut partitioning yields a partitioning tree with Rent parameter essentially identical to this theoretical optimum. These results have deep implications withrespect to both the choice of partitioning algorithms for topdown layout, as well as new approaches to layout area estimation. The paper concludes with directions for future research, including several promising techniques for fast estimation of the (intrinsic) Rent parameter.
A General Framework for Vertex Orderings, with Applications to Netlist Clustering
, 1996
"... We present a general framework for the construction of vertex orderings for netlist clustering. Our WINDOW algorithm constructs an ordering by iteratively adding the vertex with highest attraction to the existing ordering. Variant choices for the attraction function allow our framework to subsume ma ..."
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Cited by 34 (12 self)
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We present a general framework for the construction of vertex orderings for netlist clustering. Our WINDOW algorithm constructs an ordering by iteratively adding the vertex with highest attraction to the existing ordering. Variant choices for the attraction function allow our framework to subsume many graph traversals and clustering objectives from the literature. The DPRP method of [3] is then applied to optimally split the ordering into a kway clustering. Our approach is adaptable to userspecified cluster size constraints. Experimental results for clustering and multiway partitioning are encouraging. 1 Introduction A netlist hypergraph H(V; E) consists of a set of modules (vertices) V = fv 1 ; v 2 ; : : : ; v n g and a set of nets (hyperedges) E = fe 1 ; e 2 ; : : : ; e m g. A cluster C i is a nonempty subset of V , and a kway clustering P k is a set of k clusters such that every v i 2 V belongs to exactly one cluster in P k . We study the following problem: The kWay Cl...
A Combined Evolutionary Search and Multilevel Approach to Graph Partitioning
 London SE10 9LS
, 2000
"... Graph partitioning divides a graph into several pieces by cutting edges. The graph partitioning problem is to divide so that the number of vertices in each piece is the same within some defined tolerance and the number of cut edges separating these pieces is minimised. Important examples of th ..."
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Cited by 29 (3 self)
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Graph partitioning divides a graph into several pieces by cutting edges. The graph partitioning problem is to divide so that the number of vertices in each piece is the same within some defined tolerance and the number of cut edges separating these pieces is minimised. Important examples of the problem arise in partitioning graphs known as meshes for the parallel execution of computational mechanics codes.
Multilevel Refinement for Combinatorial Optimisation Problems
 SE10 9LS
, 2001
"... Abstract. We consider the multilevel paradigm and its potential to aid the solution of combinatorial optimisation problems. The multilevel paradigm is a simple one, which involves recursive coarsening to create a hierarchy of approximations to the original problem. An initial solution is found (some ..."
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Cited by 27 (5 self)
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Abstract. We consider the multilevel paradigm and its potential to aid the solution of combinatorial optimisation problems. The multilevel paradigm is a simple one, which involves recursive coarsening to create a hierarchy of approximations to the original problem. An initial solution is found (sometimes for the original problem, sometimes the coarsest) and then iteratively refined at each level. As a general solution strategy, the multilevel paradigm has been in use for many years and has been applied to many problem areas (most notably in the form of multigrid techniques). However, with the exception of the graph partitioning problem, multilevel techniques have not been widely applied to combinatorial optimisation problems. In this paper we address the issue of multilevel refinement for such problems and, with the aid of examples and results in graph partitioning, graph colouring and the travelling salesman problem, make a case for its use as a metaheuristic. The results provide compelling evidence that, although the multilevel framework cannot be considered as a panacea for combinatorial problems, it can provide an extremely useful addition to the combinatorial optimisation toolkit. We also give a possible explanation for the underlying process and extract some generic guidelines for its future use on other combinatorial problems.
Multilevel Mesh Partitioning for Heterogeneous Communication Networks
 Future Generation Comput. Syst
, 2001
"... Multilevel algorithms are a successful class of optimisation techniques which address the mesh partitioning problem for distributing unstructured meshes onto parallel computers. They usually combine a graph contraction algorithm together with a local optimisation method which refines the partition a ..."
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Cited by 24 (9 self)
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Multilevel algorithms are a successful class of optimisation techniques which address the mesh partitioning problem for distributing unstructured meshes onto parallel computers. They usually combine a graph contraction algorithm together with a local optimisation method which refines the partition at each graph level. To date these algorithms have been used almost exclusively to minimise the cut edge weight in the graph with the aim of minimising the parallel communication overhead, but recently there has been a perceived need to take into account the communications network of the parallel machine. For example the increasing use of SMP clusters (systems of multiprocessor compute nodes with very fast intranode communications but relatively slow internode networks) suggest the use of hierarchical network models. Indeed this requirement is exacerbated in the early experiments with metacomputers (multiple supercomputers combined together, in extreme cases over intercontinental networks). In this paper therefore, we modify a multilevel algorithm in order to minimise a cost function based on a model of the communications network. Several network models and variants of the algorithm are tested and we establish that it is possible to successfully guide the optimisation to reflect the chosen architecture. 2001 Elsevier Science B.V. All rights reserved.