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Tutorial 4

Tutorial 4. The Quicksort Algorithm. QuickSort. Divide: Choose a pivot, P Form a subarray with all elements ≤ P Form a subarray with all elements > P Conquer Call “ divide ” recursive call until there is all the subarraies have one element only. Combine

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Tutorial 4

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  1. Tutorial 4 The Quicksort Algorithm

  2. QuickSort • Divide: • Choose a pivot, P • Form a subarray with all elements ≤ P • Form a subarray with all elements > P • Conquer • Call “divide” recursive call until there is all the subarraies have one element only. • Combine • Nothing to do because all elements have been sorted already after the “conquer” step.

  3. Quicksort • Complexity of the Partition step: O(n). How to obtain? • Worst case partition has complexity O(n2). How to obtain? • Best case partition has complexity O(nlogn). How to obtain? • How about average case? • Why we need the randomized version Quicksort?

  4. Quicksort • Zij={zi, zi+1,zi+2,……,zj} • Each pair of elements are compared either 0 time or 1 time only. Why? • Xij=I{zi compares with zj}, it can be 1 or 0. • Total No. of comparison X

  5. Expectation of X: Average number of comparisons for randomized Quicksort Using the lemma for the Indicator function, it equals P(zi compares with zj).

  6. QuickSort • P(Xij)=Prob of {(zi is pivot) U (zj is pivot)} • P(Xij)=P(zi is pivot)+P(zj is pivot), why? • P(Xij)=1/(j-i+1)+1/(j-i+1)

  7. Quicksort • What happens if all elements have the same values? • Good news or Bad news? • What happens if all elements have been sorted? • Any ways to modify?

  8. Original Quicksort Algorithm: Hoare-Partition Hoare-Partition(A,p,r) x ← A[p]; i ← p-1; j ← r+1 While true do repeat j ← j-1 until A[j] ≤ x do repeat i ← i+1 until A[i]  x if i < j then exchange A[i], A[j] else return j

  9. True or False? • Q1: The indices i and j have values such that elements outside A[p..r] are not accessed. • Q2: When the procedure terminates, p≤ j < r. • Q3: elements of A[p..j] is less than or equal to elements of A[j+1..r].

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