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Another note...

Another note...

Posted Jan 19, 2012 9:23 UTC (Thu) by khim (subscriber, #9252)
In reply to: Is ignorance bliss? by cmccabe
Parent article: Denial of service via hash collisions

There are another interesting fact related to Median of Medians algorithm: k must be at least 5 only because algorithm looks for kth largest element. In this case “k == 3” just does not work (as I've noted above). However if want to find median then not only "k == 3" works, it usually works better then “k == 5”. This is true because in the "find median" task you can throw away not just “top left” xor “bottom right” elements (as pictured in Wikipedia's illustration) but you can throw away both “top left” and “bottom right” elements. This will lead to complexity of T(N) ≤ C₁*N*(1 + (⅗) + (⅗)² + …) = O(N) for “k == 5” and to complexity of T(N) ≤ C₂*N*(1 + (⅔) + (⅔)² + …) = O(N) for “k == 3”, but ⅗ (for “k == 5”) comes from “⅕ + ⅖” (for two recursive calls) while ⅔ (for “k == 3”) comes from “⅓ + ⅓” (for two recursive calls). Thus in most cases version with “k == 3” is faster (because recursion depth is smaller), but difference is small and you must correctly handle case of N=3M+1...


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