Online and Offline Algorithms for Counting Distinct Closed Factors via Sliding Suffix Trees

TL;DR

This work addresses counting distinct closed factors in a string via sliding suffix trees, presenting both online and offline approaches. The online method achieves $O(n\log\sigma)$ time using $O(n)$ space by leveraging Ukkonen's suffix tree and sliding-window structures, while the offline method attains $O(n)$ time and space for linearly sortable alphabets by simulating sliding trees on a static suffix tree with WAQ. A border-based characterization using $t_j=\mathrm{lrs}(T[1..j])$ and $z_j=\mathrm{lrs}^2(T[1..j])$ underpins the counting, enabling linear-time offline processing and a pathway to enumeration via geometric range data structures. The paper also explores enumeration trade-offs, showing subquadratic enumeration under certain conditions but leaving an open question whether $O(n\mathrm{polylog}(n) + \mathrm{output})$ time can be achieved, i.e., linear in the output size up to polylog factors. These results advance efficient analysis of repetitive structures in strings and contribute techniques for sliding-window string processing.

Abstract

A string is said to be closed if its length is one, or if it has a non-empty factor that occurs both as a prefix and as a suffix of the string, but does not occur elsewhere. The notion of closed words was introduced by [Fici, WORDS 2011]. Recently, the maximum number of distinct closed factors occurring in a string was investigated by [Parshina and Puzynina, Theor. Comput. Sci. 2024], and an asymptotic tight bound was proved. In this paper, we propose two algorithms to count the distinct closed factors in a string T of length n over an alphabet of size σ. The first algorithm runs in O(n log σ) time using O(n) space for string T given in an online manner. The second algorithm runs in O(n) time using O(n) space for string T given in an offline manner. Both algorithms utilize suffix trees for sliding windows.

Figures (2)

• Figure 1: The suffix tree of string $T = \mathtt{babcab}$. Each leaf of the tree represents a suffix of $T$, with the integer inside each leaf indicating the starting position of the suffix. Namely, the leaf labeled with number $i$ corresponds to $\mathsf{leaf}_T(i)$. In this suffix tree, $\mathsf{str}_T(u) = \mathtt{b}$, $\mathsf{strlen}_T(u) = 1$, $\mathsf{str}_T(\mathsf{leaf}_T(3)) = \mathtt{bcab}$, and $\mathsf{strlen}_T(\mathsf{leaf}_T(3)) = 4$. The star symbol indicates the locus of the active point, which represents the longest repeating suffix $\mathtt{ab}$ of $T$. The dotted arrows represent suffix links.
• Figure 2: The suffix tree of string $T = \mathtt{ababcabcac\$}$ is shown on the left. The suffix tree of $T[2.. 7] = \mathtt{babcab}$, which is the same as the one in Fig. \ref{['fig:stree']}, is shown on the right. Suffix links are omitted in this figure. In the tree on the left, each node and edge enclosed in a bold line is connected to a corresponding one in the tree on the right. For clarity, those connections are not drawn.

Theorems & Definitions (10)

• Theorem 1: Ukkonen95
• Theorem 2: FialaG89Larsson96Senft2005LeonardarXiv
• Lemma 1
• proof
• Corollary 1
• Theorem 3
• Theorem 4
• Corollary 2
• Proposition 1
• Proposition 2