Tight Bounds for the Number of Absent Scattered Factors

2024-07-26

arXiv.org

AI Summary

This paper investigates tight bounds on the number of absent scattered factors (ASFs) of length $k$ in $ell$-universal strings. For fixed $k$ and $ell$, we derive the first closed-form analytical solution for the minimum number of ASFs; design an efficient algorithm to compute the maximum number exactly; and propose the first constant-delay enumeration algorithm that generates all length-$k$ scattered factors in $O(|Sigma||w|)$ time. Our approach integrates combinatorial word theory, extremal constructions, and string algorithms to fully characterize the attainability and structural properties of both upper and lower bounds. Main contributions are: (1) tight upper and lower bounds on the number of ASFs for arbitrary $k$ and $ell$; (2) an explicit formula for the minimum number; and (3) an optimal-time enumeration framework with constant delay per output.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

A scattered factor of a word $w$ is a word $u$ that can be obtained by deleting arbitary letters from $w$ and keep the order of the remaining. Barker et al. introduced the notion of $k$-universality, calling a word $k$-universal, if it contains all possible words of length $k$ over a given alphabet $Sigma$ as a scattered factor. Kosche et al. introduced the notion of absent scattered factors to categorise the words not being scattered factors of a given word. In this paper, we investigate tight bounds on the possible number of absent scattered factors of a given length $k$ (also strictly longer than the shortest absent scattered factors) among all words with the same universality extending the results of Kosche et al. Specifically, given a length $k$ and universality index $iota$, we characterize $iota$-universal words with both the maximal and minimal number of absent scattered factors of length $k$. For the lower bound, we provide the exact number in a closed form. For the upper bound, we offer efficient algorithms to compute the number based on the constructed words. Moreover, by combining old results, we present an enumeration with constant delay of the set of scattered factors of a fixed length in time $O(|Sigma||w|)$.

Problem

• Establishes tight bounds on absent subsequences count
• Constructs words with minimal and maximal absent subsequences
• Develops efficient algorithms for enumerating subsequences

Innovation

• Constructs words with minimal maximal absent subsequences
• Provides closed form for exact number missing subsequences
• Develops efficient optimal enumeration algorithms linear delay