🤖 AI Summary
This work investigates the query complexity lower bound for cycle detection in colored grid graphs, where edges are defined by adjacent cells of the same color. For any $m \times n$ grid with $m, n \geq 2$, we establish a tight $\Omega(mn)$ lower bound by proving that every deterministic algorithm must query all $mn$ cells in the worst case. The core technical contribution is a novel adversarial construction based on block-wise opposition and checkerboard isolation. By composing local adversaries of sizes $2\times2$, $2\times3$, $3\times2$, and $3\times3$, an adaptive adversary can maintain ambiguity—preserving both cyclic and acyclic colorings—until the algorithm queries the very last cell.
📝 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.