Contents 0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0.2 Modeling Mesh-based Computations as Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 3 0.3 Static Graph Partitioning Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 0.3.1 Geometric Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 0.3.2 Combinatorial Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 0.3.3 Spectral Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 0.3.4 Multilevel Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 0.3.5 Combined Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 0.3.6 Qualitative Comparison of Graph Partitioning Schemes . . . . . . . . . . . . . . . . . 16 0.4 Load Balancing of Adaptive Computations . . . . . .

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 Kernighan-Lin partition optimisation algorithm which incorporates load-balancing. The resulting algorithm is tested against a different but related state-ofthe -art partitioner and shown to provide improved results. Keywords: graph-partitioning, mesh partitioning, load-balancing, 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...

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.

Abstract. Several different diffusion schemes have previously been developed for load balancing on homogeneous processor networks. We generalize existing schemes, in order to deal with heterogeneous networks. Generalized schemes may operate efficiently on networks where each processor can have arbitrary computing power, i.e., the load will be balanced proportionally to these powers. The balancing flow that is calculated by schemes for homogeneous networks is minimal with regard to the l2-norm and we prove this to hold true for generalized schemes, too. We demonstrate the usability of generalized schemes by a number of experiments on several heterogeneous networks. 1.

In the classical load balancing or multiprocessor scheduling problem, we are given a sequence of jobs of varying sizes and are asked to assign each job to one of the m empty processors. A typical objective is to minimize makespan, the load on the heaviest loaded processor. Since in most real world scenarios the load is a dynamic measure, the initial assignment may be not remain optimal with time. Motivated by such considerations in a variety of systems, we formulate the problem of load rebalancing --- given a possibly suboptimal assignment of jobs to processors, relocate a set of the jobs so as to decrease the makespan. Specifically, the goal is to achieve the best possible makespan under the constraint that no more than k jobs are relocated. We also consider a generalization of this problem where there is an arbitrary cost function associated with each job relocation. Since the problem is clearly NP-hard, we focus on approximation algorithms. We construct a sophisticated algorithm which achieves a 1.5-approximation, with near linear running time. We also show that the problem has a PTAS, resolving the complexity issue. Finally, we investigate the approximability of several extensions of the rebalancing model.

Data partitioning and load balancing are important components of parallel computations. Many different partitioning strategies have been developed, with great effectiveness in parallel applications. But the load-balancing problem is not yet solved completely; new applications and architectures require new partitioning features. Existing algorithms must be enhanced to support more complex applications. New models are needed for non-square, non-symmetric, and highly connected systems arising from applications in biology, circuits, and materials simulations. Increased use of heterogeneous computing architectures requires partitioners that account for non-uniform computing, network, and memory resources. And, for greatest impact, these new capabilities must be delivered in toolkits that are robust, easy-to-use, and applicable to a wide range of applications. In this paper, we discuss our approaches to addressing these issues within the Zoltan Parallel Data Services toolkit.

. We discuss iterative nearest neighbor load balancing schemes on processor networks which are represented by a cartesian product of graphs like e.g. tori or hypercubes. By the use of the AlternatingDirection Loadbalancing scheme, the number of load balance iterations decreases by a factor of 2 for this type of graphs. The resulting flow is analyzed theoretically and it can be very high for certain cases. Therefore, we furthermore present the Mixed-Direction scheme which needs the same number of iterations but results in a much smaller flow. Apart from that, we present a simple optimal diffusion scheme for general graphs which calculates a minimal balancing flow in the l 2 norm. The scheme is based on the spectrum of the graph representing the network and needs only m \Gamma 1 iterations in order to balance the load with m being the number of distinct eigenvalues. 1 Introduction We consider the load balancing problem in a synchronous, distributed processor network. Each node of the ne...

Adaptive scientific computations require that periodic repartitioning (load balancing) occur dynamically to maintain load balance. Hypergraph partitioning is a successful model for minimizing communication volume in scientific computations, and partitioning software for the static case is widely available. In this paper, we present a new hypergraph model for the dynamic case, where we minimize the sum of communication in the application plus the migration cost to move data, thereby reducing total execution time. The new model can be solved using hypergraph partitioning with fixed vertices. We describe an implementation of a parallel multilevel repartitioning algorithm within the Zoltan load-balancing toolkit, which to our knowledge is the first code for dynamic load balancing based on hypergraph partitioning. Finally, we present experimental results that demonstrate the effectiveness of our approach on a Linux cluster with up to 64 processors. Our new algorithm compares favorably to the widely used ParMETIS partitioning software in terms of quality, and would have reduced total execution time in most of our test cases. ∗ Sandia is a multiprogram laboratory operated by Sandia Corporation,

. Many load balancing algorithms balance the load according to a topology. Its choice can significantly influence the performance of the algorithm. We consider the two phase balancing model. The first phase calculates a balancing flow with respect to this topology by a diffusion scheme. Its time requirement depends on the maximum node degree and on the number of eigenvalues of the network. The second phase migrates the load according to this flow. A small flow volume and a small diameter of the graph keeps the time requirement of this phase low. We compare and propose several network topologies based on these measurements. Several experiments on a Cray T3E and on a cluster of PCs confirm our cost functions for both balancing phases. 1 Introduction Load balancing algorithms are typically based on a fixed topology which defines the load balancing partners in the system. Only processors that are neighbors in the topology exchange information and load items during the load balan...

Up to now, diffusive load balancing schemes have only been developed for homogeneous networks. We generalize existing diffusion schemes, in order to deal with heterogeneous networks. In these networks, every processor can have arbitrary computing power, and the load has to be balanced proportionally to these weights. The balancing flow that is calculated by the schemes for homogeneous networks is minimal with regard to the l 2 -norm and we prove this to hold true for the generalized schemes, too. By means of a number of experiments we demonstrate the usability of the generalized schemes on heterogeneous networks.