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Asked: 2026-05-13T13:35:26+00:00

(This is not exactly the problem that I have, but it’s isomorphic, and I

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(This is not exactly the problem that I have, but it’s isomorphic, and I think that this explanation will be easiest for others to understand.)

Suppose that I have a set of points in an n-dimensional space. Using 3 dimensions for example:

A : [1,2,3]
B : [4,5,6]
C : [7,8,9]

I also have a set of vectors that describe possible movements in this space:

V1 : [+1,0,-1]
V2 : [+2,0,0]

Now, given a point dest, I need to find a starting point p and a set of vectors moves that will bring me to dest in the most efficient manner. Efficiency is defined as “fewest number of moves”, not necessarily “least linear distance”: it’s permissible to select a p that’s further from dest than other candidates if the move set is such that you can get there in fewer moves. The vectors in moves must be a strict subset of the available vectors; you can’t use the same vector more than once unless it appears more than once in the input set.

My input contains ~100 starting points and maybe ~10 vectors, and my number of dimensions is ~20. The starting points and available vectors will be fixed for the lifetime of the app, but I’ll be finding paths for many, many different dest points. I want to optimize for speed, not memory. It’s acceptable for the algorithm to fail (to find no possible paths to dest).

Update w/ Accepted Solution

I adopted a solution very similar to the one marked below as “accepted”. I iterate over all points and vectors and build a list of all reachable points with the routes to reach them. I convert this list into a hash of <dest, p+vectors>, selecting the shortest set of vectors for each destination point. (There is also a little bit of optimization for hash size, which isn’t relevant here.) Subsequent dest lookups happen in constant time.

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1 Answer

  1. Editorial Team

    Actually, considering that you have around 10 vectors, you can, for a given dest point, calculate only the 1024 “targets” from the subset of vectors — e.g every reachable space, with the information about what set of moves gets there. That might be “slow”, or “fast” depending on context (it’s be absurdly fast if implemented on a parallel computing device like the GPU).

    Having all the sets that get there you can calculate the paths a lot quicker, then you can pick the point from which to get to dest in the fewest moves, choosing from the subset of the ones that are either your query or further.

    (thanks to Strilanc)

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