Core Idea - Cycle Position Index

Assign each cycle node an index from $0$ to $\lambda - 1$ where $\lambda$ is cycle length. Start from any cycle node and assign indices in order. Distance from cycle node $i$ to cycle node $j$ is $(j - i) \mod \lambda$.

This works because the cycle wraps around. The modulo handles the wraparound. Storing indices makes distance computation a single modulo operation instead of simulation. This is much faster for large cycles.