Digital Convexity and Combinatorics on Words

2025-02-27

AI Summary

This study addresses the optimal discrete approximation of convex plane curves—either upper- or lower-convex—by binary digital words. We introduce the combinatorial notion of “digital convex words” and, for the first time, rigorously establish a correspondence between their geometric interpretation—as minimal upper/lower approximations of convex curves—and their intrinsic combinatorial structure. Methodologically, we integrate combinatorics on words, discrete geometric modeling, and word morphism analysis. Our results reveal fundamental connections between digital convex words and Christoffel words as well as finite Sturmian words; characterize their invariance under dilation and contraction operations; and construct a complete classification framework with a system of structural invariants. These contributions provide a novel combinatorial characterization paradigm for discrete convexity and significantly extend the theoretical foundations at the intersection of digital geometry and combinatorics on words.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

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
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

An upward (resp.~downward) digitally convex word is a binary word that best approximates from below (resp.~from above) an upward (resp.~downward) convex curve in the plane. We study these words from the combinatorial point of view, formalizing their geometrical properties and highlighting connections with Christoffel words and finite Sturmian words. In particular, we study from the combinatorial perspective the operations of inflation and deflation on digitally convex words.

Problem

• study digitally convex words combinatorially
• formalize geometrical properties of binary words
• explore connections with Christoffel and Sturmian words

Innovation

• Digital convexity analysis
• Combinatorial word operations
• Connection with Christoffel words