A Lemma and a Conjecture on the Cost of Rearrangements

Consider a stack of books, containing both white and black books. Suppose that we want to sort them out, putting the white books on the right, and the black books on the left (fig. 1). This will be done by a finite sequence of elementary transpositions. In other words, if we have a stack of all black books of length a followed by a stack of all white books of length b, we are allowed to reverse their order at the cost of a1b. We are interested in a lower bound on the total cost of the rearrangement. Assume that initially the white and black books are highly mixed. By this we mean that inside every segment of length e one can find at least ke white books and at least ke black books, for a given k ]0 , 1[. We want to show that, as eK0, the cost of sorting the books grows like N log eN. We reformulate the problem in more precise terms. Consider the set F of all piecewise constant functions f : [0 , 1 ] O ]0, 1(. We think of B u ]x ; f (x)41( as the set of black books. We say that g is obtained from f by an elementary transposition if the following holds. For some y [0 , 1[ and a , bD0, one has