For a large class of irregular mesh applications, the structure of the mesh changes from one phase of the computation to the next. Eventually, as the mesh evolves, the adapted mesh has to be repartitioned to ensure good load balance. If this new graph is partitioned from scratch, it may lead to an excessive migration of data among processors. In this paper, we present schemes for computing repartitionings of adaptively refined meshes that perform diffusion of

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 . . . . . .

. We present a massively parallel algorithm for static and dynamic partitioning of unstructured FEM-meshes. The method consists of two parts. First a fast but inaccurate sequential clustering is determined which is used, together with a simple mapping heuristic, to map the mesh initially onto the processors of a massively parallel system. The second part of the method uses a massively parallel algorithm to remap and optimize the mesh decomposition taking several cost functions into account. It first calculates the amount of nodes that have to be migrated between pairs of clusters in order to obtain an optimal load balancing. In a second step, nodes to be migrated are chosen according to cost functions optimizing the amount and necessary communication and other measures which are important for the numerical solution method (like for example the aspect ratio of the resulting domains). The parallel parts of the method are implemented in C under Parix to run on the Parsytec GCel systems. R...

The problem of redistributing work load on parallel computers is considered. An optimal redistribution algorithm, which minimises the Euclidean norm of the migrating load, is derived. The problem is further studied by modelling with the unsteady heat conduction equation. Relationship between this algorithm and other dynamic load balancing algorithms is discussed. Convergence of the algorithm for special graphs is studied. Finally numerical results on randomly generated graphs are given to demonstrate the effectiveness of the algorithm. 1. Introduction To achieve good performance on a parallel computer, it is essential to maintain a balanced work load among all the processors of the computer. Sometimes the load can be balanced statically. However in many cases the load on each processor can not be predicted a priori. One example that demonstrates the need for both static and dynamic load balancing strategies, which is also the main motivation for this paper, is in the parallel finite e...

Massively parallel architectures require decentralized and scalable system policies. The paper presents and evaluates a set of local dynamic load balancing strategies inspired by diffusion and characterized by different scopes of locality: the goal is to compare their performance depending on the application dynamicity. The paper shows that only slowly dynamic applications can take advantage of the diffusion policies with larger scope, because they are likely to base their decisions on non obsolete information only for limited load variations. Highly

Load balancing in large parallel systems with distributed memory is a difficult task often influencing the overall efficiency of applications substantially. A number of efficient distributed load balancing strategies have been developed in the recent years. Although they are currently not generally available as part of parallel operating systems, it is often not difficult to integrate them into applications. This paper gives a classification of different load balancing problems based on application characteristics. For the case of applications out of the field of scientific computing, useful methods are described in more detail.

Parallel scheduling is a new approach for load balancing. In parallel scheduling, all processors cooperate to schedule work. Parallel scheduling is able to accurately balance the load by using global load information at compile-time or runtime. It provides high-quality load balancing. This paper presents an overview of the parallel scheduling technique. Scheduling algorithms for tree, hypercube, and mesh networks are presented. These algorithms can fully balance the load and maximize locality 1. Introduction Static scheduling balances the workload before runtime and can be applied to problems with a predictable structure, which are called static problems. Dynamic scheduling performs scheduling activities concurrently at runtime, which applies to problems with an unpredictable structure, which are called dynamic problems. Static scheduling utilizes the knowledge of problem characteristics to reach a well-balanced load [1, 2, 3, 4]. However, it is not able to balance the load for dynami...

With nearest neighbor load balancing algorithms, a processor makes balancing decisions based on localized workload information and manages workload migrations within its neighborhood. This paper compares a couple of fairly well-known nearest neighbor algorithms, the dimension-exchange (DE, for short) and the diffusion (DF, for short) methods and their several variants---the average dimension-exchange (ADE), the optimally-tuned dimension-exchange (ODE), the local average diffusion (ADF) and the optimally-tuned diffusion (ODF). The measures of interest are their efficiency in driving any initial workload distribution to a uniform distribution and their ability in controlling the growth of the variance among the processors' workloads. The comparison is made with respect to both one-port and all-port communication architectures and in consideration of various implementation strategies including synchronous/asynchronous invocation policies and static/dynamic random workload behaviors. It t...

Programming irregular and dynamic data-parallel algorithms requires to take data distribution into account. The implementation of a load balancing algorithm is a quite difficult task for the programmer. However, a load balancing strategy may be developed independently of the application. The integration of such a strategy in the data-parallel algorithm may be relevant to a library or a data-parallel compiler run-time. We propose load distribution data-parallel algorithms for a class of irregular data-parallel algorithms called stack algorithms. Our algorithms allow the use of regular and/or irregular communication patterns to exchange the works between processors. The results of theoretical analysis of these algorithms are presented. They allow a comparison of the different load balancing algorithms and the identification of criterion for the choice of a load balancing algorithm.

Existing state-of-the-art schemes for dynamic repartitioning of adaptive meshes can be classified as either diffusion-based schemes or scratch-remap schemes. We present a new scratch-remap scheme called Locally-Matched Multilevel Scratch-Remap (or simply LMSR). The LMSR scheme tries to compute a partitioning that has a high overlap with the existing partitioning. We show that LMSR decreases the amount of vertex migration required to balance the graph compared to current scratch-remap schemes, particularly for slightly imbalanced graphs. We describe a new diffusion-based scheme that we refer to as Wavefront Diffusion. In Wavefront Diffusion, the flow of vertices moves in a wavefront from overweight to underweight domains. We show that Wavefront Diffusion obtains significantly lower vertex migration requirements while maintaining similar or better edge-cut results compared to existing diffusion algorithms, especially for highly imbalanced graphs. Furthermore, we compare Wavefront Diffusi...