, Given an n × m binary matrix M , it is possible to approximate it by a totally balanced one with the following algorithm: 1. Reorder the lines and the columns of M such that M is doubly lexically ordered, 2. Check and approximate if necessary the doubly lexically ordered matrix M

, Each step can be done in O(nm) operations: Step 1 is an algorithm of Spinrad, 1987.

, our aim is to build a dissimilarity d such that: d(x, y) = min{? j | M x,j = M y,j = 1}

, We suppose without loss of generality that the last column of M is full of 1's. Computing this dissimilarity can be done in O(n 3 ) operations using Algorithm 5, as shown in Proposition 15, In addition, since M is ? -free, d is totally balanced

, Given an n × m ? -free binary matrix M with last column full of 1's and m real positive numbers, Algorithm 5 computes a dissimilarity d such that d(x, y) = min{? j | M x,j = M y,j =

, Notice that for any i, j, N i [j] = k?i M i,j and that there is no repetition in P i . Thus, |P i | ? n for any i ? n

M i ,j = 1}, the lists N i and the lists P i are such that: ? If M i,j = M i,j = 1, N i [j] < N i [j ] ?? {i} ? C ij C ij (for all i, {C i,j : 1 ? j ? m, M i,j = 1} is a chain); this allows to check wether C ij ? C ij by comparing N i ,

So after Line 14, P i contains all indices of the interesting columns (and only them): -Indices not in P i are useless: if, for j / ? P i , M i,j = M i ,j = 1, then there exists j ? P i with M i,j = M i ,j = 1 and ? j < ? j . -All indices in P i are useful: ?j < j ? P i ,

, Totally-Balanced-Dissimilarity-Approximation Data: A dissimilarity d on a n-set V Result: A totally balanced dissimilarity d on V, vol.6

, Approximate d into a chordal dissimilarity d admitting v 1 <

, (d )) defined by d (x, y) = min{? j | M x,j = M y,j = 1}

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