Efficient Enumeration of All Ladder Lotteries

A ladder lottery, known as “Amidakuji” in Japan, is a common way to choose one winner or to make an assignment randomly in Japan. Formally, a ladder lottery L of a permutation π = (x1, x2, . . . , xn) is a network with n vertical lines (lines for short) and many horizontal lines (bars for short) connecting two consecutive vertical lines. The top ends of lines correspond to π. See Fig. 1. Each number xi in π goes to down along the corresponding line but at each bar xi jump to the other end of the bar. Finally, L outputs the sorted sequence (1, 2, . . . , n) at the bottom ends of lines. See Fig. 1 for an example. A ladder lottery of a permutation π = (x1, x2, . . . , xn) is called optimal if it contains no redundant bars. Let L be an optimal ladder lottery of π and m be the number of bars in L. Then we can observe that m is equal to the number of “reverse pairs” in π. For example, a ladder lottery in Fig. 1 is optimal, since a permutation (5,1,4,3,2) has seven reverse pairs: (5,1), (5,4), (5,3), (5,2), (4,3), (4,2), (3,2). The ladder lotteries are strongly related to primitive sorting networks, which are deeply investigated by Knuth [2]. While a comparator in a primitive sorting network replaces xi and xi+1 by min (xi, xi+1) and max (xi, xi+1), a bar in a ladder lottery always exchanges them. In 1936 Wagner [6] showed that for any two maximal planar graphs G1 and G2 having the same number of vertices, one can transform G1 and G2 by a sequence of diagonal flip operations. In this paper we show a similar result on ladder lotteries. Let Sπ be the set of all optimal ladder lotteries of a given permutation π. A local swap operation is a local modification of a ladder lottery as shown in Fig. 2.