Rho

Floyd’s cycle detection, walked step by step on a straight tail feeding into a loop — the ρ shape the algorithm is named for. A tortoise steps one node per tick, a hare steps two; phase one runs until they land on the same node, which is guaranteed once both are inside the loop. Phase two resets the hare to the head and both walkers now step one node per tick — and they meet again exactly at the loop’s entrance.

That second meeting is the part every explanation asserts and few show: the distance from the head to the entrance (μ) is provably the same as the distance from the phase-one meeting point forward to the entrance. Once the walk finishes, a ribbon draws both spans under the track so the equality is visible instead of taken on faith.

Tune tail length (μ) and loop length (λ), or hit randomize. Step through one tick at a time or let it autoplay at a fixed rate.

Reuse

src/rho.js is a framework-free ES module with two exports:

• makeRho(mu, lambda) — builds the next pointer array for a rho-shaped sequence: mu tail nodes then lambda loop nodes, wired so the last loop node points back to the first.
• floyd(next, start) — runs Floyd’s algorithm over any next array (not just one from makeRho) and returns { mu, lambda, entrance, seekMeet, trace }. trace is a frame per algorithm tick — { tag: 'seek' | 'locate', tortoise, hare } — which is what makes it animatable; step through it on any renderer, same shape as astar-grid’s visitedOrder.

No dependencies, no rendering or timers in the module — canvas drawing and the play/step controls in demo/ are reference code, not the reusable piece.

Gotchas

• floyd works over any next mapping, not just rho shapes — pass a pointer array built some other way (e.g. x -> f(x) mod n) and it still detects the cycle. makeRho is just the generator used for this demo’s controllable, guaranteed-cyclic graphs.
• The demo bundles its own copy of rho.js (self-contained by contract); re-copy after editing src/.
• μ = 0 is a valid, slightly odd case: the walk starts already inside the loop, so phase two ends immediately at node 0 — the ribbon’s tail span collapses to a point, which is correct, not a bug.