A "permutation vector" shall mean an N-vector whose entries are some permutation of (0,1,2,3,...,N-1). For example (4,2,1,3,0). Let V be an N-dimensional real vector. I want "the natural method" for converting V into a weighted sum of permutation vectors plus some multiple of the all-1 vector. What is "natural"? Well, you may have your own ideas about that, and I'm open to revisions, but here are mine: 1. few nonzero weights. And they tend to be positive. 2. The weights are minimal in some norm (e.g. L1 or L2). 3. Fast algorithm to perform the conversion, i.e. compute the weights and the used permutations. E.g: I have a simple way to convert V into a weighted sum of at most 2N-2 perm-vectors plus some multiple of the all-1-vector, which takes O(N) operations if V is pre-sorted, and all weights automatically are nonnegative. But this is almost certainly improvable. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)