**Matrix: DIMACS10/delaunay_n13**

Description: DIMACS10 set: delaunay/delaunay_n13

(undirected graph drawing) |

Matrix properties | |

number of rows | 8,192 |

number of columns | 8,192 |

nonzeros | 49,094 |

# strongly connected comp. | 1 |

explicit zero entries | 0 |

nonzero pattern symmetry | symmetric |

numeric value symmetry | symmetric |

type | binary |

structure | symmetric |

Cholesky candidate? | no |

positive definite? | no |

author | M. Holtgrewe, P. Sanders, C. Schulz |

editor | C. Schulz |

date | 2011 |

kind | undirected graph |

2D/3D problem? | no |

Notes:

10th DIMACS Implementation Challenge: http://www.cc.gatech.edu/dimacs10/index.shtml As stated on their main website ( http://dimacs.rutgers.edu/Challenges/ ), the "DIMACS Implementation Challenges address questions of determining realistic algorithm performance where worst case analysis is overly pessimistic and probabilistic models are too unrealistic: experimentation can provide guides to realistic algorithm performance where analysis fails." For the 10th DIMACS Implementation Challenge, the two related problems of graph partitioning and graph clustering were chosen. Graph partitioning and graph clustering are among the aforementioned questions or problem areas where theoretical and practical results deviate significantly from each other, so that experimental outcomes are of particular interest. Problem Motivation Graph partitioning and graph clustering are ubiquitous subtasks in many application areas. Generally speaking, both techniques aim at the identification of vertex subsets with many internal and few external edges. To name only a few, problems addressed by graph partitioning and graph clustering algorithms are: * What are the communities within an (online) social network? * How do I speed up a numerical simulation by mapping it efficiently onto a parallel computer? * How must components be organized on a computer chip such that they can communicate efficiently with each other? * What are the segments of a digital image? * Which functions are certain genes (most likely) responsible for? Challenge Goals * One goal of this Challenge is to create a reproducible picture of the state-of-the-art in the area of graph partitioning (GP) and graph clustering (GC) algorithms. To this end we are identifying a standard set of benchmark instances and generators. * Moreover, after initiating a discussion with the community, we would like to establish the most appropriate problem formulations and objective functions for a variety of applications. * Another goal is to enable current researchers to compare their codes with each other, in hopes of identifying the most effective algorithmic innovations that have been proposed. * The final goal is to publish proceedings containing results presented at the Challenge workshop, and a book containing the best of the proceedings papers. Problems Addressed The precise problem formulations need to be established in the course of the Challenge. The descriptions below serve as a starting point. * Graph partitioning: The most common formulation of the graph partitioning problem for an undirected graph G = (V,E) asks for a division of V into k pairwise disjoint subsets (partitions) such that all partitions are of approximately equal size and the edge-cut, i.e., the total number of edges having their incident nodes in different subdomains, is minimized. The problem is known to be NP-hard. * Graph clustering: Clustering is an important tool for investigating the structural properties of data. Generally speaking, clustering refers to the grouping of objects such that objects in the same cluster are more similar to each other than to objects of different clusters. The similarity measure depends on the underlying application. Clustering graphs usually refers to the identification of vertex subsets (clusters) that have significantly more internal edges (to vertices of the same cluster) than external ones (to vertices of another cluster). There are 10 data sets in the DIMACS10 collection: Kronecker: synthetic graphs from the Graph500 benchmark dyn-frames: frames from a 2D dynamic simulation Delaunay: Delaunay triangulations of random points in the plane coauthor: citation and co-author networks streets: real-world street networks Walshaw: Chris Walshaw's graph partitioning archive matrix: graphs from the UF collection (not added here) random: random geometric graphs (random points in the unit square) clustering: real-world graphs commonly used as benchmarks numerical: graphs from numerical simulation Some of the graphs already exist in the UF Collection. In some cases, the original graph is unsymmetric, with values, whereas the DIMACS graph is the symmetrized pattern of A+A'. Rather than add duplicate patterns to the UF Collection, a MATLAB script is provided at http://www.cise.ufl.edu/research/sparse/dimacs10 which downloads each matrix from the UF Collection via UFget, and then performs whatever operation is required to convert the matrix to the DIMACS graph problem. Also posted at that page is a MATLAB code (metis_graph) for reading the DIMACS *.graph files into MATLAB. Delaunay: Delaunay Graphs These files have been generated as Delaunay triangulations of random points in the unit square. Engineering a scalable high quality graph partitioner, M. Holtgrewe, P. Sanders, C. Schulz, IPDPS 2010 http://dx.doi.org/10.1109/IPDPS.2010.5470485

SVD-based statistics: | |

norm(A) | 6.45642 |

min(svd(A)) | 0.000628993 |

cond(A) | 10264.7 |

rank(A) | 8,192 |

null space dimension | 0 |

full numerical rank? | yes |

singular values (MAT file): | click here |

SVD method used: | s = svd (full (A)) ; |

status: | ok |

For a description of the statistics displayed above, click here.

*Maintained by Tim Davis, last updated 12-Mar-2014.Matrix pictures by cspy, a MATLAB function in the CSparse package.
Matrix graphs by Yifan Hu, AT&T Labs Visualization Group.
*