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We end up with three subarrays size 1 n 12 n 12

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Unformatted text preview: appens if we bad-split root node, then good-split the resulting What badgoodsize (n-1) node? We end up with three subarrays, size 1, (n-1)/2, (n-1)/2 Combined cost of splits = n + n-1 = 2n -1 = Θ(n) No worse than if we had good-split the root node! (n) n n-1 1 (n-1)/2 n Θ(n −1) (n-1)/2 (n-1)/2 Θ( n) (n-1)/2 29 Intuition for the Average Case Suppose, we alternate lucky and unlucky cases to get an Suppose, average behavior L(n) = 2U (n / 2) + Θ(n) lucky U (n) = L(n − 1) + Θ(n) unlucky we consequently get L(n) = 2( L(n / 2 − 1) + Θ(n / 2)) + Θ(n) = 2 L(n / 2 − 1) + Θ (n) = Θ(n log n) The combination of good and bad splits would result in T(n) = Θ(n lg n), but with slightly larger constant hidden by the Θ-notation. 30 Randomized Quicksort An algorithm is randomized if its behavior is An randomized determined not only by the input but also by values produced by a random-number generator. random Exchange A[r] with an element chosen at random Exchange from A[p…r] in Partition. Partition This ensures that the pivot...
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