Berge Sorting


In 1966, Claude Berge proposed the following sorting problem. Given a string of nn alternating white and black pegs on a one-dimensional board consisting of an unlimited number of empty holes, rearrange the pegs into a string consisting of ⌈n2βŒ‰\lceil\frac{n}{2}\rceil white pegs followed immediately by ⌊n2βŒ‹\lfloor\frac{n}{2}\rfloor black pegs (or vice versa) using only moves which take 2 adjacent pegs to 2 vacant adjacent holes. Avis and Deza proved that the alternating string can be sorted in ⌈n2βŒ‰\lceil\frac{n}{2}\rceil such {\em Berge 2-moves} for nβ‰₯5n\geq 5. Extending Berge's original problem, we consider the same sorting problem using {\em Berge kk-moves}, i.e., moves which take kk adjacent pegs to kk vacant adjacent holes. We prove that the alternating string can be sorted in ⌈n2βŒ‰\lceil\frac{n}{2}\rceil Berge 3-moves for n≑̸0(mod4)n\not\equiv 0\pmod{4} and in ⌈n2βŒ‰+1\lceil\frac{n}{2}\rceil+1 Berge 3-moves for n≑0(mod4)n\equiv 0\pmod{4}, for nβ‰₯5n\geq 5. In general, we conjecture that, for any kk and large enough nn, the alternating string can be sorted in ⌈n2βŒ‰\lceil\frac{n}{2}\rceil Berge kk-moves. This estimate is tight as ⌈n2βŒ‰\lceil\frac{n}{2}\rceil is a lower bound for the minimum number of required Berge kk-moves for kβ‰₯2k\geq 2 and nβ‰₯5n\geq 5.Comment: 10 pages, 2 figure

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