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PODetermine whether there is a subset of size n which has a standard deviation <= s
 

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    <p>Given a bunch of numbers, I am trying to determine whether there is a "clump" anywhere where numbers are very densely packed.</p> <p>To make things more precise, I thought I'd ask a more specific problem: given a set of numbers, I would like to determine whether there is a subset of size <code>n</code> which has a standard deviation &lt;= <code>s</code>. If there are many such subsets, I'd like to find the subset with the lowest standard deviation.</p> <p>So question #1 : does this formal problem definition effectively capture the intuitive concept of a "clump" of densely packed numbers?</p> <ul> <li>EDIT: I don't actually care about determining which numbers belong to this "clump", I'm much more interested in determining where the clump is centred, which is why I think that specifying <code>n</code> in advance is okay. But feel free to correct me!</li> </ul> <p>And question #2 : assuming it does, what is the best way to go about implementing something like this (in particular, I want a solution with lowest time complexity)? So far I think I have a solution that runs in <code>n log n</code>:</p> <ul> <li>First, note that the lowest-standard-deviation-possessing subset of a given size must consist of consecutive numbers. So step 1 is sort the numbers (this is <code>n log n</code>)</li> <li><p>Second, take the first <code>n</code> numbers and compute their standard deviation. If our array of numbers is 0-based, then the first <code>n</code> numbers are <code>[0, n-1]</code>. To get standard deviation, compute <code>s1</code> and <code>s2</code> as follows:</p> <ul> <li><code>s1 = sum of numbers</code></li> <li><code>s2 = sum of squares of numbers</code></li> </ul> <p>Then, <a href="http://en.wikipedia.org/wiki/Standard_deviation#Rapid_calculation_methods" rel="nofollow">wikipedia</a> says that the standard deviation is <code>sqrt(n*s2 - s1^2)/n</code>. Record this value as the highest standard deviation seen so far.</p></li> <li>Find the standard deviation of <code>[1, n]</code>, <code>[2, n+1]</code>, <code>[3, n+2]</code> ... until you hit the the last <code>n</code> numbers. To do each computation takes only constant time if you keep track of <code>s1</code> and <code>s2</code> running totals: for example, to get std dev of <code>[1, n]</code>, just subtract the 0th element from the <code>s1</code> and <code>s2</code> totals and add the nth element, then recalculate standard deviation. This means that the entire standard deviation calculating portion of the algorithm takes linear time.</li> </ul> <p>So total time complexity <code>n log n</code>.</p> <p>Is my assessment right? Is there a better way to do this? I really need this to run <em>fast</em> on fairly large sets, so the faster the better! Space is less of an issue (I <em>think</em>).</p>
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