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POAlgorithm to find most efficient moves to arrive at a given point
 

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    <p>(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.)</p> <p>Suppose that I have a set of points in an <em>n</em>-dimensional space. Using 3 dimensions for example:</p> <pre><code>A : [1,2,3] B : [4,5,6] C : [7,8,9] </code></pre> <p>I also have a set of vectors that describe possible movements in this space:</p> <pre><code>V1 : [+1,0,-1] V2 : [+2,0,0] </code></pre> <p>Now, given a point <em>dest</em>, I need to find a starting point <em>p</em> and a set of vectors <em>moves</em> that will bring me to <em>dest</em> in the most efficient manner. Efficiency is defined as "fewest number of moves", not necessarily "least linear distance": it's permissible to select a <em>p</em> that's further from <em>dest</em> than other candidates if the move set is such that you can get there in fewer moves. The vectors in <em>moves</em> 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.</p> <p>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 <em>dest</em> points. I want to optimize for speed, not memory. It's acceptable for the algorithm to fail (to find no possible paths to <em>dest</em>).</p> <h3>Update w/ Accepted Solution</h3> <p>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 &lt;<em>dest</em>, <em>p+vectors</em>>, 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 <em>dest</em> lookups happen in constant time.</p>
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    1. COActually, given your constraints, you've got about 1024 possible positions you can get to from 100 starting points, so about a hundred thousand possible destinations (which may not be distinct). Precalculation into a hash table might work here.
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    2. COI don't think that you mean "orthogonal" in your introductory parenthesis.
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    3. CO@Svante, you're right. Corrected to "isomorphic".
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