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POApproximated closest pair algorithm
 

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    <p>I have been thinking about a variation of the <a href="http://en.wikipedia.org/wiki/Closest_pair_of_points_problem" rel="nofollow">closest pair problem</a> in which the only available information is the set of distances already calculated (we are not allowed to sort points according to their x-coordinates).</p> <p>Consider 4 points (A, B, C, D), and the following distances:</p> <pre><code>dist(A,B) = 0.5 dist(A,C) = 5 dist(C,D) = 2 </code></pre> <p>In this example, I don't need to evaluate <code>dist(B,C)</code> or <code>dist(A,D)</code>, because it is guaranteed that these distances are greater than the current known minimum distance. </p> <ol> <li><p>Is it possible to use this kind of information to reduce the O(n²) to something like O(nlogn)?</p></li> <li><p>Is it possible to reduce the cost to something close to O(nlogn) if I accept a kind of approximated solution? In this case, I am thinking about some technique based on reinforcement learning that only converges to the real solution when the number of reinforcements go to infinite, but provides a great approximation for small n.</p></li> <li><p>Processing time (measured by the big O notation) is not the only issue. To keep a very large amount of previous calculated distances can also be an issue.</p></li> <li><p>Imagine this problem for a set with 10⁸ points.</p></li> </ol> <p>What kind of solution should I look for? Was this kind of problem solved before?</p> <p>This is not a classroom problem or something related. I have been just thinking about this problem. </p>
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    1. COWhy "the distance between dist(A,B)=0.5, dist(A,C)=5, dist(C,D)=2, for instance, I don't need to evaluate dist(B,C) and dist(A,D)" ? I think this is similar to shortest path problem,when the relationship of these distances is not transitive, thus your statement is not correct
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    2. CO@PhamTrung Given `dist(A,B) = 0.5; dist(A,C) = 5`, the closest B and C can possibly be is if A, B, and C are colinear, and in that order. This leads to `dist(B,C) >= 4.5`, which is greater than the current shortest path, so there's no need to evaluate.
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    3. CO@trojansdestroy, Yes!
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