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29 March 2024
 
  » arxiv » math/0512612

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Berge Sorting
Antoine Deza ; William Hua ;
Date 28 Dec 2005
Subject Combinatorics
AbstractIn 1966, Claude Berge proposed the following sorting problem. Given a string of $n$ 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 $lceilfrac{n}{2} ceil$ white pegs followed immediately by $lfloorfrac{n}{2} floor$ 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 $lceilfrac{n}{2} ceil$ such {em Berge 2-moves} for $ngeq 5$. Extending Berge’s original problem, we consider the same sorting problem using {em Berge $k$-moves}, i.e., moves which take $k$ adjacent pegs to $k$ vacant adjacent holes. We prove that the alternating string can be sorted in $lceilfrac{n}{2} ceil$ Berge 3-moves for $n otequiv 0pmod{4}$ and in $lceilfrac{n}{2} ceil+1$ Berge 3-moves for $nequiv 0pmod{4}$, for $ngeq 5$. In general, we conjecture that, for any $k$ and large enough $n$, the alternating string can be sorted in $lceilfrac{n}{2} ceil$ Berge $k$-moves. This estimate is tight as $lceilfrac{n}{2} ceil$ is a lower bound for the minimum number of required Berge $k$-moves for $kgeq 2$ and $ngeq 5$.
Source arXiv, math/0512612
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