High-Quality Hypergraph Partitioning

This dissertation focuses on computing high-quality solutions for the NP-hard balanced hypergraph partitioning problem: Given a hypergraph and an integer $k$, partition its vertex set into $k$ disjoint blocks of bounded size, while minimizing an objective function over the hyperedges. Here, we consider the two most commonly used objectives: the cut-net metric and the connectivity metric. Since the problem is computationally intractable, heuristics are used in practice - the most prominent being the three-phase multi-level paradigm: During coarsening, the hypergraph is successively contracted to obtain a hierarchy of smaller instances. After applying an initial partitioning algorithm to the smallest hypergraph, contraction is undone and, at each level, refinement algorithms try to improve the current solution. With this work, we give a brief overview of the field and present several algorithmic improvements to the multi-level paradigm. Instead of using a logarithmic number of levels like traditional algorithms, we present two coarsening algorithms that create a hierarchy of (nearly) $n$ levels, where $n$ is the number of vertices. This makes consecutive levels as similar as possible and provides many opportunities for refinement algorithms to improve the partition. This approach is made feasible in practice by tailoring all algorithms and data structures to the $n$-level paradigm, and developing lazy-evaluation techniques, caching mechanisms and early stopping criteria to speed up the partitioning process. Furthermore, we propose a sparsification algorithm based on locality-sensitive hashing that improves the running time for hypergraphs with large hyperedges, and show that incorporating global information about the community structure into the coarsening process improves quality. Moreover, we present a portfolio-based initial partitioning approach, and propose three refinement algorithms. Two are based on the Fiduccia-Mattheyses (FM) heuristic, but perform a highly localized search at each level. While one is designed for two-way partitioning, the other is the first FM-style algorithm that can be efficiently employed in the multi-level setting to directly improve $k$-way partitions. The third algorithm uses max-flow computations on pairs of blocks to refine $k$-way partitions. Finally, we present the first memetic multi-level hypergraph partitioning algorithm for an extensive exploration of the global solution space. All contributions are made available through our open-source framework KaHyPar. In a comprehensive experimental study, we compare KaHyPar with hMETIS, PaToH, Mondriaan, Zoltan-AlgD, and HYPE on a wide range of hypergraphs from several application areas. Our results indicate that KaHyPar, already without the memetic component, computes better solutions than all competing algorithms for both the cut-net and the connectivity metric, while being faster than Zoltan-AlgD and equally fast as hMETIS. Moreover, KaHyPar compares favorably with the current best graph partitioning system KaFFPa - both in terms of solution quality and running time

Distributed Duplicate Removal

Ziel der verteilten Duplikaterkennung ist die Identifikation von Elementen, welche mehrfach in einer großen, über mehrere Rechenknoten verteilten Datenmenge vorkommen. Sanders et al. [48] präsentieren einen verteilten Algorithmus, welcher dieses Problem in einer besonders kommunikationseffizienten Art und Weise löst. In einer Vorverarbeitungsphase werden mit Hilfe eines verteilten, platz-effizienten Bloom Filters zunächst möglichst viele distinkte Elemente als solche identifiziert und somit die Gesamtmenge der noch zu betrachtenden Elemente stark reduziert. Da hierbei jedoch auch falsch positive Ergebnisse auftreten, müssen alle als potentiell nicht distinkt erkannten Elemente in einer zweiten Phase noch einmal überprüft werden. Hierzu wird ein klassischer Hash-basierter Algorithmus zur verteilten Duplikaterkennung angewendet. Die vorliegende Arbeit ergänzt die theoretische Analyse durch eine praktische Evaluation. Wir erarbeiten hierzu eine effiziente Implementierung für Shared-Nothing Systeme. Besonders rechenintensive Schritte des Algorithmus werden zusätzlich durch Shared-Memory-Programmierung innerhalb eines Knotens parallelisiert. Die Ergebnisse unserer experimentellen Untersuchung untermauern die durch die Theorie vorhergesagten Vorteile des Algorithmus. Unsere Implementierung ist signifikant schneller als der am besten geeignete klassische Ansatz solange die Eingabedaten zu weniger als 50% aus Duplikaten bestehen. Wird der Algorithmus auf Datensätzen ausgeführt, die zu weniger als 10% aus Duplikaten bestehen, so ist das gesamte Kommunikationsvolumen zudem mehr als eine Größenordnung kleiner als das des klassischen Konkurrenten

Improving Coarsening Schemes for Hypergraph Partitioning by Exploiting Community Structure

We present an improved coarsening process for multilevel hypergraph partitioning that incorporates global information about the community structure. Community detection is performed via modularity maximization on a bipartite graph representation. The approach is made suitable for different classes of hypergraphs by defining weights for the graph edges that express structural properties of the hypergraph. We integrate our approach into a leading multilevel hypergraph partitioner with strong local search algorithms and perform extensive experiments on a large benchmark set of hypergraphs stemming from application areas such as VLSI design, SAT solving, and scientific computing. Our results indicate that respecting community structure during coarsening not only significantly improves the solutions found by the initial partitioning algorithm, but also consistently improves overall solution quality

Load-Balanced Bottleneck Objectives in Process Mapping

We propose a new problem formulation for graph partitioning that is tailored to the needs of time-critical simulations on modern heterogeneous supercomputers

Network Flow-Based Refinement for Multilevel Hypergraph Partitioning

We present a refinement framework for multilevel hypergraph partitioning that uses max-flow computations on pairs of blocks to improve the solution quality of a k-way partition. The framework generalizes the flow-based improvement algorithm of KaFFPa from graphs to hypergraphs and is integrated into the hypergraph partitioner KaHyPar. By reducing the size of hypergraph flow networks, improving the flow model used in KaFFPa, and developing techniques to improve the running time of our algorithm, we obtain a partitioner that computes the best solutions for a wide range of benchmark hypergraphs from different application areas while still having a running time comparable to that of hMetis

Advanced Flow-Based Multilevel Hypergraph Partitioning

The balanced hypergraph partitioning problem is to partition a hypergraph into k disjoint blocks of bounded size such that the sum of the number of blocks connected by each hyperedge is minimized. We present an improvement to the flow-based refinement framework of KaHyPar-MF, the current state-of-the-art multilevel k-way hypergraph partitioning algorithm for high-quality solutions. Our improvement is based on the recently proposed HyperFlowCutter algorithm for computing bipartitions of unweighted hypergraphs by solving a sequence of incremental maximum flow problems. Since vertices and hyperedges are aggregated during the coarsening phase, refinement algorithms employed in the multilevel setting must be able to handle both weighted hyperedges and weighted vertices - even if the initial input hypergraph is unweighted. We therefore enhance HyperFlowCutter to handle weighted instances and propose a technique for computing maximum flows directly on weighted hypergraphs. We compare the performance of two configurations of our new algorithm with KaHyPar-MF and seven other partitioning algorithms on a comprehensive benchmark set with instances from application areas such as VLSI design, scientific computing, and SAT solving. Our first configuration, KaHyPar-HFC, computes slightly better solutions than KaHyPar-MF using significantly less running time. The second configuration, KaHyPar-HFC*, computes solutions of significantly better quality and is still slightly faster than KaHyPar-MF. Furthermore, in terms of solution quality, both configurations also outperform all other competing partitioners