90,173 research outputs found
Efficient algorithms for a class of partitioning problems
The problem of optimally partitioning the modules of chain- or tree-like tasks over chain-structured or host-satellite multiple computer systems is addressed. This important class of problems includes many signal processing and industrial control applications. Prior research has resulted in a succession of faster exact and approximate algorithms for these problems. Polynomial exact and approximate algorithms are described for this class that are better than any of the previously reported algorithms. The approach is based on a preprocessing step that condenses the given chain or tree structured task into a monotonic chain or tree. The partitioning of this monotonic take can then be carried out using fast search techniques
Spanners for Geometric Intersection Graphs
Efficient algorithms are presented for constructing spanners in geometric
intersection graphs. For a unit ball graph in R^k, a (1+\epsilon)-spanner is
obtained using efficient partitioning of the space into hypercubes and solving
bichromatic closest pair problems. The spanner construction has almost
equivalent complexity to the construction of Euclidean minimum spanning trees.
The results are extended to arbitrary ball graphs with a sub-quadratic running
time.
For unit ball graphs, the spanners have a small separator decomposition which
can be used to obtain efficient algorithms for approximating proximity problems
like diameter and distance queries. The results on compressed quadtrees,
geometric graph separators, and diameter approximation might be of independent
interest.Comment: 16 pages, 5 figures, Late
Streaming, Local, and MultiLevel (Hyper)Graph Decomposition
(Hyper)Graph decomposition is a family of problems that aim to break down large (hyper)graphs into smaller sub(hyper)graphs for easier analysis. The importance of this lies in its ability to enable efficient computation on large and complex (hyper)graphs, such as social networks, chemical compounds, and computer networks. This dissertation explores several types of (hyper)graph decomposition problems, including graph partitioning, hypergraph partitioning, local graph clustering, process mapping, and signed graph clustering. Our main focus is on streaming algorithms, local algorithms and multilevel algorithms. In terms of streaming algorithms, we make contributions with highly efficient and effective algorithms for (hyper)graph partitioning and process mapping. In terms of local algorithms, we propose sub-linear algorithms which are effective in detecting high-quality local communities around a given seed node in a graph based on the distribution of a given motif. In terms of multilevel algorithms, we engineer high-quality multilevel algorithms for process mapping and signed graph clustering. We provide a thorough discussion of each algorithm along with experimental results demonstrating their superiority over existing state-of-the-art techniques.
The results show that the proposed algorithms achieve improved performance and better solutions in various metrics, making them highly promising for practical applications. Overall, this dissertation showcases the effectiveness of advanced combinatorial algorithmic techniques in solving challenging (hyper)graph decomposition problems
Computational Optimization Techniques for Graph Partitioning
Partitioning graphs into two or more subgraphs is a fundamental operation in computer science, with applications in large-scale graph analytics, distributed and parallel data processing, and fill-reducing orderings in sparse matrix algorithms. Computing balanced and minimally connected subgraphs is a common pre-processing step in these areas, and must therefore be done quickly and efficiently. Since graph partitioning is NP-hard, heuristics must be used. These heuristics must balance the need to produce high quality partitions with that of providing practical performance. Traditional methods of partitioning graphs rely heavily on combinatorics, but recent developments in continuous optimization formulations have led to the development of hybrid methods that combine the best of both approaches. This work describes numerical optimization formulations for two classes of graph partitioning problems, edge cuts and vertex separators.
Optimization-based formulations for each of these problems are described, and hybrid algorithms combining these optimization-based approaches with traditional combinatoric methods are presented. Efficient implementations and computational results for these algorithms are presented in a C++ graph partitioning library competitive with the state of the art. Additionally, an optimization-based approach to hypergraph partitioning is proposed
Squarepants in a Tree: Sum of Subtree Clustering and Hyperbolic Pants Decomposition
We provide efficient constant factor approximation algorithms for the
problems of finding a hierarchical clustering of a point set in any metric
space, minimizing the sum of minimimum spanning tree lengths within each
cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of
cluster perimeters. Our algorithms for the hyperbolic and Euclidean planes can
also be used to provide a pants decomposition, that is, a set of disjoint
simple closed curves partitioning the plane minus the input points into subsets
with exactly three boundary components, with approximately minimum total
length. In the Euclidean case, these curves are squares; in the hyperbolic
case, they combine our Euclidean square pants decomposition with our tree
clustering method for general metric spaces.Comment: 22 pages, 14 figures. This version replaces the proof of what is now
Lemma 5.2, as the previous proof was erroneou
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