08 - Order Statistics Data Structures and Algorithms Andrei...

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Order Statistics Data Structures and Algorithms Andrei Bulatov
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Algorithms – Order Statistics 8-2 Medians and Statistics We consider the following problems: - Find a minimal and maximal elements in a sequence - Find i-th smallest element in a sequence The i-th smallest element of a sequence is called its i-th order statistic The median of a sequence of length n is its (n+1)/2-th order statistic, if n is odd n/2 and n/2 + 1-th order statistic In either case, the medians are (n + 1)/2 and (n + 1)/2 -th order statistics
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Algorithms – Order Statistics 8-3 The Problem The Selection Problem Instance : A sequence A of n numbers, and i, 1 i n Objective : Find the i-th order statistics of A Finding maximum / minimum, median are particular case of the Selection problem Can be solved in O(n log n) by first sorting and then taking the i-th element Our goal is to do this in O(n)
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Algorithms – Order Statistics 8-4 Maximum and Minimum Finding maximum (or minimum) is easy Minimum(A) set min:=A[1] for i=2 to length(A) do if min>A[i] then set min:=A[i] endfor return min This algorithm runs in O(n) time
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Algorithms – Order Statistics 8-5 Maximum and Minimum (cntd) If we need to find both maximum and minimum we can do slightly better that running the algorithm twice: we need 3/2 n comparisons instead of 2n Minimum-maximum(A)
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