[fc417fc802]

What would it mean for a maze to be infinite? It seems to me that a key part of the concept is having a goal to reach.

Although I guess you could have an infinitely large map and an algorithm that guaranteed connectivity. Infinite ways to fail to reach the goal. But I doubt there would be much practical benefit.

To actually answer your question it should be fairly easy to convert nearly any existing algorithm to cover an infinite area by simply tiling it. A common method to avoid boundary issues is to overlap the tiles slightly.

[skulk]

I've always had a dream of creating an infinite braid maze algorithm:

The requirements I've come up with are:

1. Distribution of path length for any two points of a fixed taxicab distance should be some kind of long-tail distribution.

2. In general, the path between any two nearby points should often stray far outside the smallest box that contains both of those two points. Of course, this won't apply to nearby points in the same corridor. I'm not sure how best to state this formally.

3. It should be possible to calculate the exits for any cell in O(log(N)) time where N = abs(sum of the coordinates).

And the most hazy requirement of all: the maze should look decent.

AFAICT, no such algorithm exists.

[tasuki]

I'm looking for such an algorithm, with intermediate goals: you start at the bottom, go up, and as you reach the goal, more maze appears. I left out some unimportant details :)

[fjfaase]

I maze that grows in one direction can be generated with Eller's or the Sidewinder algorithms (as also mentioned by the user John Tromp in one of the other replies).

[fjfaase]

Please explain how to deal with slightly overlapping tiles and still create a maze that has no cycles and locations that cannot be reached from any other location. These are properties that go tiles.

I do know of an algorithm with 'nesting' that generate mazes but results in very long walls and thus does not feel random.

[fc417fc802]

I'm not sure how to address that within the scope of an HN comment (and I don't think I have the time). To start with the problem is severely underspecified.

What do you mean by "infinity" exactly? 2^64? 2^128? Any arbitrary bigint? Periodic boundary condition?

Why perfect mazes - does that really matter? Are these supposed to be human solvable or is this some sort of abstract art? Anything human solvable requires at least one path of (very) finite length from start to finish.

Overlapping tiles and doing nothing else produces multiple paths. It's the quick and easy solution if the goal is practicality of implementation and human oriented puzzles.

If you insist on a perfect maze an obvious solution is 2^64, a periodic boundary condition, and a space partitioning tree with many children per node (ie n >>> 2). Connectivity is determined at each level. Long uniform walls can be reduced or even eliminated by warping block boundaries in various ways; among other things, you could potentially use the maze solution at each level to assign node membership for the next level.