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POfair partitioning of set S into k partitions
 

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    <p>There is a set S containing N integers each with value 1&lt;=X&lt;=10^6. The problem is to partition the set S into k partitions. The value of a partition is the sum of the elements present in it. Partition is to be done in such a way the total value of the set S is fairly distributed amongst the k partitions. The mathematical meaning of <em>fair</em> also needs to be defined (e.g. the objective could be to minimize the standard deviation of the values of the partitions from the average value of the set S (which is, sum(S)/k))</p> <p>e.g. S = {10, 15, 12, 13, 30, 5}, k=3</p> <p>A good partitioning would be {30}, {10, 15}, {12, 13, 5}</p> <p>A bad partitioning would be {30, 5}, {10, 15}, {12, 13}</p> <p>First question is to mathematically express condition for one partition to be better than the other. Second question is to how to solve the problem. The problem is NP-Hard. Are there any heuristics?</p> <p>In the problem that I am trying to solve N &lt;= (k*logX)^2 and K varies from 2 to 7.</p> <p>==================================================================================</p> <p>Based on other related SO questions, there are two reasonable functions to evaluate a distribution:</p> <p>a) Minimize the value of the partition with the maximum value.</p> <p>On a second thought, this is not a good metric. Consider, a set {100, 40, 40} to be partitioned into three subsets. This metric does not distinguish between the following two distributions even though one is clearly better than the other.</p> <p>Distribution 1: {100}, {40}, {40} and Distribution 2: {100}, {40, 40}, {}</p> <p>b) Minimize the maximum of the difference of any two values in a given partition i.e minimize max|A-B| for any A, B</p>
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    1. COfor the record, `N <= (7*ln(10^6))^2 ~= 9300`; does that sound about right?
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    2. COyes thats correct
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    3. COExact same question with a really good answer here. http://stackoverflow.com/questions/6454598/stuck-with-an-interview-question-partitioning-of-an-array
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