We describe a heuristic method for drawing graphs which uses a multilevel framework combined with a force-directed placement algorithm. The multilevel technique matches and coalesces pairs of adjacent vertices to define a new graph and is repeated recursively to create a hierarchy of increasingly coarse graphs, G0, G1, . . . , GL. The coarsest graph, GL, is then given an initial layout and the layout is refined and extended to all the graphs starting with the coarsest and ending with the original. At each successive change of level, l, the initial layout for G l is taken from its coarser and smaller child graph, G l+1 , and refined using force-directed placement. In this way the multilevel framework both accelerates and appears to give a more global quality to the drawing. The algorithm can compute both 2 & 3 dimensional layouts and we demonstrate it on examples ranging in size from 10 to 225,000 vertices. It is also very fast and can compute a 2D layout of a sparse graph in around 12 seconds for a 10,000 vertex graph to around 5-7 minutes for the largest graphs. This is an order of magnitude faster than recent implementations of force-directed placement algorithms.
Multilevel algorithms are a successful class of optimization techniques which addresses the mesh partitioning problem. They usually combine a graph contraction algorithm together with a local optimization 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 Kernighan-Lin partition optimization algorithm which incorporates load-balancing. The resulting algorithm is tested against a different but related state-of-the-art partitioner and shown to provide improved results.
AMS subject classifications. 05C85, 65Y05PII. S1064827598337373 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 geometries and resolving wide variations in behavior via variable mesh densities. Meanwhile, the modeling of complex behavior patterns means that the problems are often too large to fit onto serial computers, either because of memory limitations or computational demands, or both. Distributing the mesh across a parallel computer so that the computational load is evenly balanced and the data locality maximized is known as mesh partitioning. It is well known that this problem is NP-complete [7], so in recent years much attention has been focused on developing suitable heuristics, and some powerful methods, many based on a graph corresponding to the communication requirements of the mesh, have been devised, e.g., [5].A particularly popular and successful class of algorithms which addresses this mesh partitioning problem is known as multilevel algorithms. They usually combine a graph contraction algorithm which creates a series of progressively smaller and coarser graphs together with a local optimization method which, starting with the coarsest graph, 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 Kernighan-Lin (KL) partition optimization algorithm which incorporates load-balancing.We focus here on serial partitioning. Although emphasis in the field is switching to parallel partitioning methods, we aim here to address one of the fundamental mechanisms of the multilevel paradigm and thus a serial implementation provides a clear and relatively parameter-free environment for establishing how imbalance can affect the overall performance of the strategy. However, elsewhere we have provided a different but related parallel formulation [21], and indeed the algorithms described here are used directly as part of that parallel strategy.
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