(Sets of ) Complement Scattered Factors

2026-03-21

AI Summary

This work investigates how to characterize the set of scattered factors that can be generated from a word after removing all embeddings of a given scattered factor. It introduces, for the first time, the notion of a “complementary scattered factor,” defining \( C(w, u) \) as the set of scattered factors obtainable from word \( w \) upon deleting all embeddings of scattered factor \( u \). The study uncovers a combinatorial connection between this construct and the shuffle operation. Leveraging combinatorial analysis, string embedding enumeration, and efficient enumeration techniques, the authors devise an algorithm to compute \( C(w, u) \) in \( O(|w|\cdot|u|\cdot\binom{|w|}{|u|}) \) time. Furthermore, they present polynomial- or exponential-time algorithms to reconstruct the original word or factor from \( C(w, u) \) together with \( u \) (or \( w \)), thereby offering a novel perspective on scattered factor theory.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Data Mining & Knowledge Management: Data Compression
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

Abstract

Starting in the 1970s with the fundamental work of Imre Simon, \emph{scattered factors} (also known as subsequences or scattered subwords) have remained a consistently and heavily studied object. The majority of work on scattered factors can be split into two broad classes of problems: given a word, what information, in the form of scattered factors, are contained, and which are not. In this paper, we consider an intermediary problem, introducing the notion of \emph{complement scattered factors}. Given a word $w$ and a scattered factor $u$ of $w$, the complement scattered factors of $w$ with regards to $u$, $C(w, u)$, is the set of scattered factors in $w$ that can be formed by removing any embedding of $u$ from $w$. This is closely related to the \emph{shuffle} operation in which two words are intertwined, i.e., we extend previous work relating to the shuffle operator, using knowledge about scattered factors. Alongside introducing these sets, we provide combinatorial results on the size of the set $C(w, u)$, an algorithm to compute the set $C(w, u)$ from $w$ and $u$ in $O(\vert w \vert \cdot \vert u \vert \binom{w}{u})$ time, where $\binom{w}{u}$ denotes the number of embeddings of $u$ into $w$, an algorithm to construct $u$ from $w$ and $C(w, u)$ in $O(\vert w \vert^2 \binom{\vert w \vert}{\vert w \vert - \vert u \vert})$ time, and an algorithm to construct $w$ from $u$ and $C(w, u)$ in $O(\vert u \vert \cdot \vert w \vert^{\vert u \vert + 1})$ time.

Problem

• scattered factors
• complement scattered factors
• subsequences
• shuffle
• embeddings

Innovation

• complement scattered factors
• subsequence reconstruction
• shuffle operation
• combinatorics on words
• embedding enumeration