Optimal Reconfiguration Of Optimal Ladder Lotteries

A ladder lottery, known as "Amidakuji" in Japan, is a common way to decide an assignment at random. A ladder lottery L of a given permutation is optimal if L has the minimum number of horizontal lines. In this paper, we investigate a reconfiguration problem of optimal ladder lotteries. The reconfiguration problem on a set of optimal ladder lotteries asks, given two optimal ladder lotteries L, L' of a permutation pi, to find a sequence of (L-1, L-2, ... , L-k) of optimal ladder lotteries of pi such that (1) L-1 = L and L-k = L' and (2) L-i for i = 2, 3, ... , k is obtained from Li-1 by moving a bar in Li-1 locally. An existing result implies that any two optimal ladder lotteries of a permutation pi have a reconfiguration sequence of length O(n3), where n is the number of elements in pi. In this paper, we characterize the minimum length of reconfiguration sequences between two optimal ladder lotteries. Moreover, we present a linear-time algorithm that computes the minimum length. (C) 2021 Elsevier B.V. All rights reserved.