MinCut (and also the
MinimumBandwidth command in the
GraphUtilities package) may be useful in many situations, this is a very general problem and might also benefit from more general solutions. For example, a while ago I had some data about the relationship between 120 items in the form of a distance matrix, basically a measurement of the distance between each object. With the data labelled arbitrarily, the matrix looks like this:
Black means zero distance (which you see along the diagonal because each item is zero distance from itself) while white represents the maximum possible distance between the items. Obviously, the matrix is completely scrambled. Laboriously using prior knowledge of the relationships between the various items, we were able to manually reorder the distance matrix into the form
which nicely shows the clustering along the diagonal. Though we were quite happy with this, it nagged at me: shouldn't there be some way to automatically reorder the data and hence rearrange the matrix? I tried two commands from Matlab (symamd -- a variant of amd, and symrcm which implements the minimum bandwidth algorithm of Cuthill-McKee). I tried both
MinimumBandwidth ordering in the graph utilities in Mathematica. None came close to what we could do manually -- that is -- the reordered matrices, while more organized than the first matrix above, were still quite random looking.
At this point, someone suggested that what we really needed was a clustering algorithm because what we were trying for was to cluster the elements into small groups. Accordingly, I tried
clusters = FindClusters[scrambled -> Range[Length[scrambled]], 10,
DistanceFunction -> (Norm[#1 - #2, 1] &)];
ind = Flatten[clusters];
scrambled is the unsorted (original) distance matrix. This was by far the best automated results we were able to obtain. The beauty of it is that by changing the
DistanceFunction and the number of requested clusters, we could get many different plausible clusterings.
MinCutto permute rows/columns? $\endgroup$
MinCutthat applies to permuting rows/columns of a matrix, for the purpose of decreasing fill-in. Which is to say, it is a precondition-by-permutation example. As such, it could be an alternative to AMD. $\endgroup$