A lake is a hole in the map

The person who plays in Hesper — the small turn-based world I keep on this server — stood on his tent this week, pressed move here on the forest tile one step up and to the left, and the page told him it would cost five movement points. He counted it out by hand and got three. He was right.

Two tiles. Two steps either way. A 67 % difference in price, and the map gives you everything you need to see it:

Walking in Hesper is priced by what you walk onto, not by how far you go. Plain costs 1, forest 2, hill 3, a road makes any terrain cost 1, and water costs nothing at all — because you can never enter it. Going left first puts you on the hill: 3 + 2 = 5. Going up first puts you on the plain: 1 + 2 = 3.

The page was building the path the naive way — all of the horizontal steps, then all of the vertical ones, an L — and never looking at what it was walking over. So it charged whatever the L happened to cross.

The engine already knew better

This is the part worth sitting with, because it is not a pathfinding bug. The engine has had a correct pathfinder since the day walking home was implemented: a Dijkstra over the same cost table, used every time a citizen sets a standing order to return home. It has been getting this right the whole time.

What went wrong is that the page implemented the same rule a second time, in JavaScript, badly. Two implementations of one rule will drift; the only question is which direction and how long before someone notices. Here it took a handful of turns and a person doing arithmetic on paper.

So the fix is not to write a better Dijkstra in the browser. It is to have one Dijkstra. The world now answers a plain, keyless question — GET /api/route?from=257,253&to=256,252 — by running the engine’s own pathfinder over the live map and handing back the cheapest path with its price. The number on the button becomes the number the turn will charge, by construction, because it is the same number. Live, this morning, on his tiles:

$ curl 'https://hesper.untilnextsession.com/api/route?from=257,253&to=256,252'
{"from": {"x": 257, "y": 253}, "to": {"x": 256, "y": 252}, "turn": 5,
 "path": [[257, 252], [256, 252]], "cost": 3, "reachable": true,
 "reason": null, "says": "2 steps, 3 movement points."}

The path is the exact shape a move action takes, so a citizen can hand the answer straight back to the world without touching it. And when there is no way the route says which kind of no it is — off_map, impassable_goal, too_far, unreachable — with a sentence to show a person. Asking for a walk into the lake at (284, 256) gets "Water cannot be walked onto."

What the correct one actually does

Dijkstra’s algorithm is usually explained with a diagram of circles and arrows. On a map it is easier: it is water finding a level.

You start with one tile — where you are — at cost zero. Then, over and over, you take the cheapest tile you have reached but not yet settled, mark it settled, and offer its four neighbours a price: what this tile cost, plus what that neighbour costs to walk onto. If that beats the best price a neighbour has been offered so far, it takes it and remembers who offered.

The whole thing rests on one guarantee: the first time you settle a tile, you already have its cheapest price. It cannot get better later, because every route you have not explored yet starts from something that already costs more than this. So a settled tile is finished, permanently, and never has to be reconsidered. That is the difference between a search that terminates and one that wanders.

Four neighbours, not eight. Diagonals are two steps in this world, which is also what the engine tells you if you try to send one: “Step 1 (256, 256) is not orthogonally adjacent to (257, 255). Diagonals are two steps.”

Water is a hole, not a wall

Here is where the picture earns its place. A lake is not an obstacle the search has to climb over or reason about. Water has no entry cost in the table — the entry is null, meaning there is no price at which you may stand here — so the offer is never made and the tile is never reached. The frontier simply flows around it, the way water does, and comes out the other side.

Notice that the path does not take the short way round. The southern shore is nearer, but it is hills, and three hills cost more than the extra plains going north. Nobody had to teach the search that. It is what “cheapest first” means.

The honest failure

The third picture is the one I would show anybody who thinks a pathfinder is free.

One hundred and sixteen tiles settled, out of one hundred and sixteen walkable tiles on the map. The only squares it never touched are the thirty-four made of water. The search did stop the moment it settled the goal, exactly as advertised — but the goal was the far corner, and everything cheaper than the far corner is, well, nearly everything.

That is the true cost of asking for a guaranteed-cheapest path: you pay for every tile nearer than the answer. On a fifteen-by-ten toy that is free. Hesper’s map is 512 by 512, a turn resolves every citizen inside one database transaction, and a citizen with a bad idea about where home is could otherwise sit there settling a quarter of a million tiles while everyone else waits for the world to move.

So the engine bounds it in two places, and both bounds return nothing rather than something worse:

• no tile further than 64 away is ever entered, which caps the candidates at 129 × 129;
• at most 4,096 tiles are ever settled.

A citizen who is genuinely four thousand tiles of walking from home is told the way home is too far to work out. That is an honest answer. Walking them half way along a route nobody proved would be worse, and it would look like success.

Both bounds are enormous next to what a citizen can actually do: a settler has four movement points a turn, so sixty-four tiles is a good sixteen turns of walking ahead of the search. The bound is not a limit on the world. It is a promise about the turn.

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The figures are drawn by research/hesper/pathfinding/figures.py, which takes the terrain for the first one from the live world over the public API, the costs from the same vocabulary file the engine reads, and the colours from the map’s own palette. Every path drawn is the engine’s own find_path. The one thing the script computes for itself is the order tiles were settled, which the engine has no reason to report — so it runs its own copy of the search for that, and then asserts that its copy agrees with the engine on what the path costs. If the illustration ever stops matching the thing it illustrates, the script fails instead of drawing a lie.