# grPartition.m

Status: Pretty stable with a nice technical report to explain the algorithm. MATLAB’s Statistics Toolbox is no longer required.

MATLAB function to partition very large graphs very fast. This function implements a graph partitioning algorithm based on spectral factorization. This algorithm is described in the following technical report:

João Hespanha. An efficient MATLAB Algorithm for Graph Partitioning. Technical Report, University of California, Oct. 2004. Available at http://www.ece.ucsb.edu/~hespanha/techrep.html.

This algorithm was extensively used in this paper:

C. Lim, S. Bohacek, J. Hespanha, K. Obraczka. Hierarchical Max-Flow Routing. In Proc. of the IEEE GLOBECOM, Nov. 2005.

The script is used as follows:

% function [ndx,Pi,cost]= grPartition(C,k,nrep);

%

% Partitions the n-node undirected graph G defined by the matrix C

%

% Inputs:

% C - n by n edge-weights matrix. In particular, c(i,j)=c(j,i) is equal

%     to the cost associated with cuting the edge between nodes i and j.

%     This matrix should be symmetric and doubly stochastic. If this

%     is not the case, this matrix will be normalized to

%     satisfy these properties (with a warning).

% k - desired number of partitions

% nrep - number of repetion for the clustering algorithm

%       (optional input, defaults to 1)

%

% Outputs:

% ndx  - n-vector with the cluster index for every node

%       (indices from 1 to k)

% Pi   - Projection matrix [see Technical report

% cost - cost of the partition (sum of broken edges)

%

% By Joao Pedro Hespanha, Copyright 2004

%

% Example:

%

% X=rand(200,2);               % place random points in the plane

% C=pdist(X,'euclidean');      % compute distance between points

% C=exp(-.1*squareform(C));    % edge cost is a negative exponential of distance

%

% k=6;                         % # of partitions

% [ndx,Pi,cost]= grPartition(C,k,30);

%

% colors=hsv(k);               % plots points with appropriate colors

% colormap(colors)

% cla

% line(X(:,1),X(:,2),'MarkerEdgeColor',[0,0,0],'linestyle','none','marker','.');

% for i=1:k

%   line(X(find(ndx==i),1),X(find(ndx==i),2),...

%       'MarkerEdgeColor',colors(i,:),'linestyle','none','marker','.');

% end

% title(sprintf('Cost %g',cost))

% colorbar

The following figures shows the output of the demo script ‘demo_grPartition.m’, which is included to illustrate the creation of several graphs. The top-left figure corresponds to the example above. The code that generated the remaining figures requires the @graph class available at http://www.ece.ucsb.edu/~hespanha/software.

The bottom-right figure demonstrates the speed of the algorithms for a large graph (10,000 nodes and about 60,000 edges). In my 2GHz PC, it took less than 1.5secc to create the graph and less than one minute to partition it. Since the costs are monotone functions of the Euclidean distance, one should not be too surprised to get a Voronoi-like partition of the space. ## Bibliographic citation

When used in research, please acknowledge the use of this software with the following reference:

grPartition — a MATLAB function for graph partitioning. Available at http://www.ece.ucsb.edu/~hespanha, Oct. 2004.

or if you use latex/bibtex:

@Misc{HespanhaOct04,

key =          {software},

author =       {Jo{\~a}o Pedro Hespanha},

title =        {\texttt{grPartition} a {MATLAB} function for graph partitioning},

howpublished = {Available at \url{http://www.ece.ucsb.edu/~hespanha}},

month =  oct,

year =   2004

}

## Disclaimer

This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License for more details (http://www.gnu.org/copyleft/gpl.html)