Abstract
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 π, to find a sequence of 〈L1,L2,…,Lk〉 of optimal ladder lotteries of π such that (1) L1=L and Lk=L′ and (2) Li 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 π have a reconfiguration sequence of length O(n3), where n is the number of elements in π. 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.
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