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Randomized Quick Sort Algorithm

Randomized Quick Sort Algorithm. Lecture Note – 1 Prepared By: Muhammed Miah. Randomized Quick Sort. In traditional Quick Sort, we always pick the first element as the pivot for partitioning.

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Randomized Quick Sort Algorithm

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  1. Randomized Quick Sort Algorithm Lecture Note – 1 Prepared By: Muhammed Miah 01/16/2008

  2. Randomized Quick Sort • In traditional Quick Sort, we always pick the first element as the pivot for partitioning. • The worst case runtime is O(n2) while the expected runtime is O(nlogn) over the set of all input. • Therefore, some input are born to have long runtime, e.g., an inversely sorted list. 01/16/2008

  3. Randomized Quick Sort • In randomized Quick Sort, we pick randomly an element as the pivot for partitioning. • The expected runtime of any input is O(nlogn). • Randomized Quick Sort is a probabilistic algorithm 01/16/2008

  4. Randomized Quick Sort (S) • Input: a set of numbers S, S = {S1, S2, S3, …, Sn} • If |S| > 1 Then, • Select an element X from S at random • Partition S into 3 parts: • Random Quick Sort (L); • Random Quick Sort (R); L X R 01/16/2008

  5. Randomized Quick Sort • Example: • S = {2, 1, 3, 5, 4} 1st pivot 2 1 2 3, 5, 4 2nd pivot 3rd pivot 1 5 sorted 3, 4 5 4 4th pivot 3 4 5th pivot 3 01/16/2008

  6. Randomized Quick Sort • Properties: • Running time is unpredictable • It has randomization built into it • Nothing to do with input order • Worst unequal division – when one side has zero number and other side has all numbers • Worst case running time n2 01/16/2008

  7. Analysis of “Expected” Running Time of Randomized QS • Assume, S = {1, 2, 3, 4, …, n) Let us consider a Boolean variable, Xij 1 ≤ i, j ≤ n, j > i Xij = 1 if it is compared with j Xij = 0 otherwise # of variables we can create ≈ n2 millions of trees are generated AVG. 01/16/2008

  8. Analysis of “Expected” Running Time of Randomized QS • Xij is a random variable E[∑∑ Xij] = ∑∑ E[Xij] linearity of expectation, E[X + Y] = E[X] + E[Y] E[Xij] = how often i and j compared n-1 n n-1 n i=1 j>i i=1 j>i Horizontal definition Vertical definition 01/16/2008

  9. Analysis of “Expected” Running Time of Randomized QS • Level wise left to right (LR) order: consider set of numbers {1, 2, …, 10} assume we got the tree: level wise order: 6 4 8 3 5 9 1 10 2 If Min(LR[i], LR[j] > Min(LR of i+1, i+2, …, j-1), then i is NOT compared to j otherwise it is compared 6 4 8 5 7 9 3 1 10 2 01/16/2008

  10. Analysis of “Expected” Running Time of Randomized QS P(Xij = 1) = 2/(j-i+1) E[Xij] = 2/(j-i+1) ∑∑ E[Xij] = ∑∑ 2/(j-i+1) ≈ ∑∑ 2/(j-i) ≈ ∑∑ 2/k let, j-i = k ≤ ∑∑ 2/k n-1 n n n i=1 j>i i=1 j>i n n n n-i i=1 J=i i=1 k=1 n n i=1 k=1 01/16/2008

  11. Analysis of “Expected” Running Time of Randomized QS 2∑∑ 1/k = 2n(∑ 1/k) ∑ 1/k = 1/1 + 1/2 + 1/3 + … + 1/n (Harmonic Series) 2n(∑ 1/k) = 2n logn ≈ O(nlogn) So, “Expected” running time of randomized quick sort = O(nlogn) n n n k=1 i=1 k=1 n k=1 n k=1 01/16/2008

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