A Tight Lower Bound for Cycle Detection in Grid Graphs

Andrew Au

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arXiv · 2604.23894

A Tight Lower Bound for Cycle Detection in Grid Graphs

Andrew Au

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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.

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Andrew Au. 2026-04-26. A Tight Lower Bound for Cycle Detection in Grid Graphs. https://arxiv.org/abs/2604.23894

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