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Christian Schulz

Publications and source records attributed to Christian Schulz.

118 records · Page 7Linked to original sources
arXiv2014

(Semi-)External Algorithms for Graph Partitioning and Clustering

In this paper, we develop semi-external and external memory algorithms for graph partitioning and clustering problems. Graph partitioning and clustering are key tools for processing and analyzing large complex networks. We address both problems in the (semi-)external model by adapting the size-constrained label propagation technique. Our (semi-)external size-constrained label propagation algorithm can be used to compute graph clusterings and is a prerequisite for the (semi-)external graph partitioning algorithm. The algorithm is then used for both the coarsening and the refinement phase of a multilevel algorithm to compute graph partitions. Our algorithm is able to partition and cluster huge complex networks with billions of edges on cheap commodity machines. Experiments demonstrate that the semi-external graph partitioning algorithm is scalable and can compute high quality partitions in time that is comparable to the running time of an efficient internal memory implementation. A parallelization of the algorithm in the semi-external model further reduces running time.

cs.DS↗
arXiv2014

Partitioning Complex Networks via Size-constrained Clustering

The most commonly used method to tackle the graph partitioning problem in practice is the multilevel approach. During a coarsening phase, a multilevel graph partitioning algorithm reduces the graph size by iteratively contracting nodes and edges until the graph is small enough to be partitioned by some other algorithm. A partition of the input graph is then constructed by successively transferring the solution to the next finer graph and applying a local search algorithm to improve the current solution. In this paper, we describe a novel approach to partition graphs effectively especially if the networks have a highly irregular structure. More precisely, our algorithm provides graph coarsening by iteratively contracting size-constrained clusterings that are computed using a label propagation algorithm. The same algorithm that provides the size-constrained clusterings can also be used during uncoarsening as a fast and simple local search algorithm. Depending on the algorithm's configuration, we are able to compute partitions of very high quality outperforming all competitors, or partitions that are comparable to the best competitor in terms of quality, hMetis, while being nearly an order of magnitude faster on average. The fastest configuration partitions the largest graph available to us with 3.3 billion edges using a single machine in about ten minutes while cutting less than half of the edges than the fastest competitor, kMetis.

cs.DC↗
arXiv2014

Tree-based Coarsening and Partitioning of Complex Networks

Many applications produce massive complex networks whose analysis would benefit from parallel processing. Parallel algorithms, in turn, often require a suitable network partition. For solving optimization tasks such as graph partitioning on large networks, multilevel methods are preferred in practice. Yet, complex networks pose challenges to established multilevel algorithms, in particular to their coarsening phase. One way to specify a (recursive) coarsening of a graph is to rate its edges and then contract the edges as prioritized by the rating. In this paper we (i) define weights for the edges of a network that express the edges' importance for connectivity, (ii) compute a minimum weight spanning tree $T^m$ with respect to these weights, and (iii) rate the network edges based on the conductance values of $T^m$'s fundamental cuts. To this end, we also (iv) develop the first optimal linear-time algorithm to compute the conductance values of \emph{all} fundamental cuts of a given spanning tree. We integrate the new edge rating into a leading multilevel graph partitioner and equip the latter with a new greedy postprocessing for optimizing the maximum communication volume (MCV). Experiments on bipartitioning frequently used benchmark networks show that the postprocessing already reduces MCV by 11.3%. Our new edge rating further reduces MCV by 10.3% compared to the previously best rating with the postprocessing in place for both ratings. In total, with a modest increase in running time, our new approach reduces the MCV of complex network partitions by 20.4%.

cs.DS↗
arXiv2014

Efficient Parallel and External Matching

We show that a simple algorithm for computing a matching on a graph runs in a logarithmic number of phases incurring work linear in the input size. The algorithm can be adapted to provide efficient algorithms in several models of computation, such as PRAM, External Memory, MapReduce and distributed memory models. Our CREW PRAM algorithm is the first O(log^2 n) time, linear work algorithm. Our experimental results indicate the algorithm's high speed and efficiency combined with good solution quality.

cs.DS↗
arXiv2012

Think Locally, Act Globally: Perfectly Balanced Graph Partitioning

We present a novel local improvement scheme for the perfectly balanced graph partitioning problem. This scheme encodes local searches that are not restricted to a balance constraint into a model allowing us to find combinations of these searches maintaining balance by applying a negative cycle detection algorithm. We combine this technique with an algorithm to balance unbalanced solutions and integrate it into a parallel multi-level evolutionary algorithm, KaFFPaE, to tackle the problem. Overall, we obtain a system that is fast on the one hand and on the other hand is able to improve or reproduce most of the best known perfectly balanced partitioning results ever reported in the literature.

cs.DS↗
arXiv2012

Advanced Coarsening Schemes for Graph Partitioning

The graph partitioning problem is widely used and studied in many practical and theoretical applications. The multilevel strategies represent today one of the most effective and efficient generic frameworks for solving this problem on large-scale graphs. Most of the attention in designing the multilevel partitioning frameworks has been on the refinement phase. In this work we focus on the coarsening phase, which is responsible for creating structurally similar to the original but smaller graphs. We compare different matching- and AMG-based coarsening schemes, experiment with the algebraic distance between nodes, and demonstrate computational results on several classes of graphs that emphasize the running time and quality advantages of different coarsenings.

cs.DS↗
arXiv2011

Distributed Evolutionary Graph Partitioning

We present a novel distributed evolutionary algorithm, KaFFPaE, to solve the Graph Partitioning Problem, which makes use of KaFFPa (Karlsruhe Fast Flow Partitioner). The use of our multilevel graph partitioner KaFFPa provides new effective crossover and mutation operators. By combining these with a scalable communication protocol we obtain a system that is able to improve the best known partitioning results for many inputs in a very short amount of time. For example, in Walshaw's well known benchmark tables we are able to improve or recompute 76% of entries for the tables with 1%, 3% and 5% imbalance.

cs.NE↗
arXiv2011

Engineering Multilevel Graph Partitioning Algorithms

We present a multi-level graph partitioning algorithm using novel local improvement algorithms and global search strategies transferred from the multi-grid community. Local improvement algorithms are based max-flow min-cut computations and more localized FM searches. By combining these techniques, we obtain an algorithm that is fast on the one hand and on the other hand is able to improve the best known partitioning results for many inputs. For example, in Walshaw's well known benchmark tables we achieve 317 improvements for the tables 1%, 3% and 5% imbalance. Moreover, in 118 additional cases we have been able to reproduce the best cut in this benchmark.

cs.DS↗
arXiv2010

Engineering a Scalable High Quality Graph Partitioner

We describe an approach to parallel graph partitioning that scales to hundreds of processors and produces a high solution quality. For example, for many instances from Walshaw's benchmark collection we improve the best known partitioning. We use the well known framework of multi-level graph partitioning. All components are implemented by scalable parallel algorithms. Quality improvements compared to previous systems are due to better prioritization of edges to be contracted, better approximation algorithms for identifying matchings, better local search heuristics, and perhaps most notably, a parallelization of the FM local search algorithm that works more locally than previous approaches.

cs.DC↗