Matrix: DIMACS10/delaunay_n21

Description: DIMACS10 set: delaunay/delaunay_n21

DIMACS10/delaunay_n21 graph
(undirected graph drawing)


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  • Matrix group: DIMACS10
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  • download as a MATLAB mat-file, file size: 21 MB. Use UFget(2476) or UFget('DIMACS10/delaunay_n21') in MATLAB.
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  • download in Rutherford/Boeing format, file size: 16 MB.

    Matrix properties
    number of rows2,097,152
    number of columns2,097,152
    # strongly connected comp.1
    explicit zero entries0
    nonzero pattern symmetrysymmetric
    numeric value symmetrysymmetric
    Cholesky candidate?no
    positive definite?no

    authorM. Holtgrewe, P. Sanders, C. Schulz
    editorC. Schulz
    kindundirected graph
    2D/3D problem?no


    10th DIMACS Implementation Challenge:                                   
    As stated on their main website (                                   ), 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       
    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       
        * 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           
        * 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    
        * 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       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                                                

    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.