A Tight Lower Bound for Cycle Detection in Grid Graphs

TL;DR

This paper establishes a fundamental lower bound showing that detecting cycles in grid graphs requires reading all cells in the worst case, using an adversary argument to prove the necessity of full inspection.

Contribution

It introduces a tight lower bound for cycle detection in grid graphs and employs a novel adversary construction to prove the bound.

Findings

Any cycle detection algorithm must read all cells in the worst case.

The proof uses an adversary that maintains ambiguity until the last cell.

The construction involves tiling the grid with adversarially controlled blocks.

Abstract

We prove that any algorithm for detecting cycles in an $m \times n$ grid graph, where cells are colored and adjacency is defined by matching colors, must read all $mn$ cells in the worst case for all grids with $m \geq 2$ and $n \geq 2$. The proof is by adversary argument: we construct an adaptive adversary that maintains ambiguity -- one completion containing a cycle and one without -- until the final cell is read. The construction proceeds by tiling the grid with $2 \times 2$, $2 \times 3$, $3 \times 2$, and $3 \times 3$ blocks, each equipped with an independent block adversary, composed via a checkerboard isolation scheme.