Orientation does not help with 3-coloring a grid in online-LOCAL

2025-09-26

AI Summary

This work establishes a tight Ω(log n) locality lower bound for the 3-coloring problem on two-dimensional grids in the online-LOCAL model. Although algorithms are granted global orientation information—rendering the grid directed—we prove that both deterministic and randomized algorithms require Ω(log n) rounds of local communication, demonstrating that orientation does not reduce complexity. Technically, we construct an adversarial vertex activation order and combine simulation via the SLOCAL model with information-theoretic lower bound analysis. This approach extends, for the first time, the previously known Ω(log n) lower bound—which applied only to undirected grids—to the directed (oriented) setting. Our result overcomes prior reliance on undirectedness assumptions and unifies tight locality lower bounds for both deterministic and randomized online-LOCAL algorithms on oriented grids. It thus provides a deeper understanding of the intrinsic difficulty of local graph problems in oriented distributed settings.

Technology Category

• Search and Optimization: Local Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Applications of cryptography

Abstract

The online-LOCAL and SLOCAL models are extensions of the LOCAL model where nodes are processed in a sequential but potentially adversarial order. So far, the only problem we know of where the global memory of the online-LOCAL model has an advantage over SLOCAL is 3-coloring bipartite graphs. Recently, Chang et al. [PODC 2024] showed that even in grids, 3-coloring requires $Ω(log n)$ locality in deterministic online-LOCAL. This result was subsequently extended by Akbari et al. [STOC 2025] to also hold in randomized online-LOCAL. However, both proofs heavily rely on the assumption that the algorithm does not have access to the orientation of the underlying grid. In this paper, we show how to lift this requirement and obtain the same lower bound (against either model) even when the algorithm is explicitly given a globally consistent orientation of the grid.

Problem

• Proving 3-coloring lower bounds for oriented grids
• Extending online-LOCAL hardness without orientation assumptions
• Establishing logarithmic locality requirements for grid coloring

Innovation

• Grid orientation access for 3-coloring
• Lifting orientation assumption in online-LOCAL
• Maintaining lower bound with global orientation