The flood and the arrow.

This is a real slice of Barcelona — the Eixample — drawn from 1,814 real street segments. To your eyes, a drawing. To a router, neither.

A router keeps only the places where streets meet or end: 1,281 intersections. Each becomes a vertex — a dot with a position and nothing else.

Each stretch of street between two intersections becomes an edge. This one — Carrer de Còrsega leaving our origin — is 133 m long, and one-way.

Then every edge gets a price in seconds: length ÷ speed by street class. Avenues 50, side streets 40, residential 30, living streets 20 km/h — illustrative free-flow prices, ours, stated.

The question is set: the cheapest total price from Còrsega, bottom left, to Sardenya, top right — 2,459 m apart as the crow flies. Everything from here on is this one question.

Start at Còrsega with price zero; every other intersection starts at infinity. The rule never changes: settle the cheapest unsettled vertex.

Each settlement is permanent, and it re-prices that intersection's neighbors. Growth follows one law: by price, in every direction.

Watch what the search does not know: where you are going. The flood spends real work behind you and sideways from you — streets no route to Sardenya could ever use.

Before Sardenya settles, the search has settled 1,123 of the 1,281 intersections — 88% of the neighborhood, to answer one question.

Sardenya settles at 3,464 m, 4.4 minutes — provably the cheapest. Now scale the same flood to an 18-million-vertex Europe: it reads 9.3 million vertices per query. Correct, and far too slow.

Change nothing about the search. Add one number per intersection: a lower bound on time-to-go — straight-line distance ÷ 50 km/h, the fastest price on this map. This is A*, from 1968.

The bound is admissible: it can never overestimate, because no street beats the fastest street. So the queue may prefer intersections that make progress — it just can't lie.

The flood collapses into a corridor. Behind it, the ghost: everything Dijkstra read that A* never touches.

311 settled instead of 1,123 — 3.6× less work — and the route is identical: 3,464 m, 4.4 min. The optimality proof survives.

But a corridor still widens with distance. Across a continent, straight-line bounds leave millions of intersections in the search. The next idea stops aiming at the map — and rewrites the map itself.

Our route already knows about the skeleton: 85% of its length is arterial. Long answers don't live on side streets.

Open engines bake this in: Valhalla keeps motorways and primaries at level 0, secondaries at 1, residential at 2 — search coarse first, descend at the ends.

Model it here: local streets exist only within 400 m of origin and destination. The search settles 428 instead of 1,123 — same optimal route, in this demo. But pruning by class has no guarantee in general.

Contraction Hierarchies earn the guarantee: importance is computed from the map's own shortest paths, then unimportant intersections are contracted away — neighbors bridged by shortcuts.

A query then runs up, across, down: locals to the skeleton, skeleton to the destination's exit. Continental queries settle hundreds, not millions — the scoreboard is next.

Same graph, same request. But the spine of our route — Carrer del Rosselló, 1,104 m of it — jams. We price it ×2.5: an illustrative jam, ours, stated.

Nothing reruns the world. The search reads the same streets with new numbers — and the old route now costs 6.4 minutes instead of 4.4.

So the line moves. Rosselló drops from 1,104 m of route to 135 m; Carrer de València, one block south, carries 2,276 m. Same origin, same destination, new best answer.

Notice the search itself worked harder: 1,037 settled instead of 311. When prices defy distance, aiming gets weaker — the bound still holds, but it says less.

In production the new price list arrives continuously — crowdsourced conditions, learned patterns, driver reports — and the blue line quietly steps around a jam that hasn't started yet.