Amortized efficiency of generation, ranking and unranking left-child sequences in lexicographic order

Abstract A new type of sequences called left-child sequences (LC-sequences for short) was recently introduced by Wu et al. (2014) for representing binary trees. They pointed out that such sequences have a natural interpretation from the view point of data structure and gave a characterization of them. Based on this characterization, there is an easily implementing algorithm that uses generate-and-test approach to filter all LC-sequences of binary trees with n internal nodes in lexicographic order, while in general it is not efficient at all. In this paper, we first design two novel rotations that allow us to drastically alter the shape of binary trees (and thus their corresponding LC-sequences). As an application, these operations can be employed to generate all LC-sequences in lexicographic order. Accordingly, we present a more efficient algorithm associated with the new types of rotations for generating all LC-sequences and show that it takes only constant amortized running cost. Moreover, we extend our study to the ranking and unranking problems. By integrating a measure called “left distances” introduced by Makinen (1987) to represent binary trees, we develop efficient ranking and unranking algorithms for LC-sequences in lexicographic order. With the help of aggregate analysis, we show that both ranking and unranking algorithms can be run in amortized cost of O ( n ) time and space.