Efficient enumeration of all ladder lotteries and its application

Authors:
Katsuhisa Yamanaka;Shin-ichi Nakano;Yasuko Matsui;Ryuhei Uehara;Kento Nakada
Affiliations:
Graduate School of Information Systems, University of Electro-Communications, Chofugaoka 1-5-1, Chofu, Tokyo 182-8585, Japan;Department of Computer Science, Gunma University, Tenjin-cho 1-5-1, Kiryu, Gunma 376-8515, Japan;Department of Mathematical Sciences, Tokai University, Kitakaname 1117, Hiratsuka, Kanagawa, 259-1292, Japan;School of Information Science, Japan Advanced Institute of Science and Technology, Asahidai 1-1, Nomi, Ishikawa 923-1292, Japan;Faculty of Integrated Media, Wakkanai Hokusei Gakuen University, Wakabadai 1-2290-28, Wakkanai, Hokkaido 097-0013, Japan
Venue:
Theoretical Computer Science
Year:
2010

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Abstract

A ladder lottery, known as ''Amidakuji'' in Japan, is a common way to choose a permutation randomly. A ladder lottery L corresponding to a given permutation @p is optimal if L has the minimum number of horizontal lines among the ladder lotteries corresponding to @p. In this paper we show that for any two optimal ladder lotteries L"1 and L"2 of a permutation, there exists a sequence of local modifications which transforms L"1 into L"2. We also give an algorithm to enumerate all optimal ladder lotteries of a given permutation. By setting @p=(n,n-1,...,1), the algorithm enumerates all arrangements of n pseudolines efficiently. By implementing the algorithm we compute the number of arrangements of n pseudolines for each n@?11.