Cyclic Equalizability Characterized by Parikh Vectors

2026-04-21

AI Summary

This study addresses the problem of determining whether two strings over an arbitrary finite alphabet are cyclically balanced—that is, whether they can be transformed into cyclic shifts of each other by inserting identical characters at identical positions. Extending prior work limited to binary alphabets, this paper provides the first generalization to arbitrary finite alphabets. By leveraging Parikh vectors—vectors capturing character frequencies—and algebraic properties of cyclic string structures, the authors establish that two strings are cyclically balanced if and only if their Parikh vectors are identical. This characterization yields a complete and efficient decision criterion, thereby resolving an open problem posed by Shinagawa and Nuida.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Game Theory and Economic Paradigms: Equilibrium

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Cyclic equalizability is a notion introduced by Shinagawa and Nuida in 2025, in the study of card-based cryptography. Informally, a collection of words is cyclically equalizable if, by inserting the same letters at the same positions in all words, they can be transformed into words that are cyclic shifts of one another. Shinagawa and Nuida showed that two binary words of equal length are cyclically equalizable if and only if they have the same Hamming weight. They also posed the problem of characterizing cyclic equalizability over larger alphabets. In this paper, we completely characterize cyclic equalizability for two words over an arbitrary finite alphabet by proving that two words are cyclically equalizable if and only if they have the same Parikh vector.

Problem

• cyclic equalizability
• Parikh vector
• card-based cryptography
• string transformation
• finite alphabet

Innovation

• cyclic equalizability
• Parikh vector
• card-based cryptography
• string transformation
• combinatorics on words