On the Quality of Partitions Based on Space-Filling Curves

Hungershöfer, Jan; Wierum, Jens-Michael

doi:10.1007/3-540-47789-6_4

Cited by 40 publications

(16 citation statements)

References 11 publications

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“…The constant C depends on the type of the curve. Some constants have been determined for 2D regular grids in worst case [14]. The quality of a partition based on the Lebesgue curve is bounded by Upper bounds for the Hilbert curve (5.56) and the βΩ-indexing (5.48) can be extracted from experimental results [11].…”

Section: Analytical Resultsmentioning

confidence: 99%

Partitioning finite element meshes using space-filling curves

Schamberger

Wierum

2005

Future Generation Computer Systems

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Section: Analytical Resultsmentioning

confidence: 99%

Partitioning finite element meshes using space-filling curves

Schamberger

Wierum

2005

Future Generation Computer Systems

Self Cite

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“…Yet, efficient hash codes exist [57,69]. Notably DFS/SFC codes yield high quality hash codes or preimages to a hash function due to their Hölder continuity [4,19,34,37,38,40]. If a mesh remains stationary or remeshing cost are negligible compared to the compute load, it finally is convenient to make the "hash code" store all user data continuously in memory [68].…”

Section: Application Data Managementmentioning

confidence: 99%

The Peano Software—Parallel, Automaton-based, Dynamically Adaptive Grid Traversals

Weinzierl

2019

ACM Trans. Math. Softw.

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We discuss the design decisions, design alternatives and rationale behind the third generation of Peano, a framework for dynamically adaptive Cartesian meshes derived from spacetrees. Peano ties the mesh traversal to the mesh storage and supports only one element-wise traversal order resulting from space-filling curves. The user is not free to choose a traversal order herself. The traversal can exploit regular grid subregions and shared memory as well as distributed memory systems with almost no modifications to a serial application code. We formalize the software design by means of two interacting automata-one automaton for the multiscale grid traversal and one for the applicationspecific algorithmic steps. This yields a callback-based programming paradigm. We further sketch the supported application types and the two data storage schemes realized, before we detail high-performance computing aspects and lessons learned. Special emphasis is put on observations regarding the used programming idioms and algorithmic concepts. This transforms our report from a "one way to implement things" code description into a generic discussion and summary of some alternatives, rationale and design decisions to be made for any tree-based adaptive mesh refinement software.

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“…ese techniques are also fast and scalable and rely on the fact that two points whose indices on the curve are close, are also o en close in the original space. While load balance is fairly easy to maintain, the quality of the computed partitions in terms of graphbased methods is relatively poor for non-trivial meshes [24]. Implementations of partitioning algorithms using space-lling curves are available in the ParMetis and Zoltan packages.…”

Section: Geometric Techniquesmentioning

confidence: 99%

Balanced k-means for Parallel Geometric Partitioning

Looz

Tzovas

Meyerhenke

2018

Proceedings of the 47th International Conference on Parallel Processing

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Mesh partitioning is an indispensable tool for e cient parallel numerical simulations. Its goal is to minimize communication between the processes of a simulation while achieving load balance. Established graph-based partitioning tools yield a high solution quality; however, their scalability is limited. Geometric approaches usually scale be er, but their solution quality may be unsatisfactory for "non-trivial" mesh topologies.In this paper, we present a scalable version of k-means that is adapted to yield balanced clusters. Balanced k-means constitutes the core of our new partitioning algorithm Geographer. Bootstrapping of initial centers is performed with space-lling curves, leading to fast convergence of the subsequent balanced k-means algorithm.Our experiments with up to 16 384 MPI processes on numerous benchmark meshes show the following: (i) Geographer produces partitions with a lower communication volume than state-of-the-art geometric partitioners from the Zoltan package; (ii) Geographer scales well on large inputs; (iii) a Delaunay mesh with a few billion vertices and edges can be partitioned in a few seconds. RCB 0.088 234 2 1 537 980 12 558 791 086 48 0.000 297 71

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On the Quality of Partitions Based on Space-Filling Curves

Cited by 40 publications

References 11 publications

Partitioning finite element meshes using space-filling curves

Partitioning finite element meshes using space-filling curves

The Peano Software—Parallel, Automaton-based, Dynamically Adaptive Grid Traversals

Balanced k-means for Parallel Geometric Partitioning

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