People who like this sort of thing will find this the sort of thing they like'

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• People who like this sort of thing will find this
  the sort of thing they like.
• Abraham Lincoln (1809 - 1865), in a book
  review

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CSE 502NFundamentals of Computer Science

• Fall 2004
• Lecture 8
• Quicksort, randomized quicksort,
• running time analysis
• (CLRS 7.1-7.4)

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Quicksort

• Sorts array A of n numbers in-place
• Divide partition rearrange Apr into
  subarrays around a pivot point q such that each
  element in Apq-1 Aq each element in
  Aq1r
• Conquer sort subarrays and via recursive calls
• Combine since subarrays are sorted in place,
  entire array is Apr sorted

Quicksort(A,p,r) q Partition(A,p,r)
Quicksort(A,p,q-1) Quicksort(A,q1,r)
Partition(A,p,r) i p-1 x Ar for j
p to r-1 if Aj x then i
exchange Ai Aj exchange
Ai1 Ar return i1
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Correctness of Quicksort

• Loop invariant (for loop in Partition)
• If p k i, then Ak x
• If i1 k j-1, then Ak gt x
• If k r, then Ak x
• Initialization prior to first iteration, i p-1
  and j p
• There are no values between p and i (condition 1)
• There are no values between i1 and j-1
  (condition 2)
• Third condition satisfied by initialization of x
• Maintenance two cases depending on outcome of if
  compare
• If Aj gt x, j is incremented, condition 2 holds,
  no other entries affected
• If Aj x, i is incremented, Ai and Aj are
  swapped, j is incremented
• Ai x, (condition 1)
• Aj-1 gt x, (condition 2, by loop invariant)

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Correctness of Quicksort

• Termination at termination j r, and all
  entries have been partitioned into one of the
  three sets described by loop invariant
• If p k i, then Ak x
• If i1 k j-1, then Ak gt x
• If k r, then Ak x
• Recursive calls operate on regions of at most n-1
  elements
• Api where all elements are Ai1
• Ai2r where all elements are gt Ai1
• Running time of Partition is Q(n) where n r p
  1

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Quicksort Performance

• Performance depends on whether or not the
  partitioning is balanced or unbalanced
• If unbalanced, worst-case running time is no
  better than insertion sort
• T(n) T(n 1) T(0) Q(n) T(n 1) Q(n)
  Q(n2)
• Occurs when array is already sorted when
  insertion sort is O(n)
• If evenly balanced, running time is equal to
  merge sort
• T(n) 2T(n/2) Q(n) O(n lg n)
• How balanced does the partitioning need to be to
  achieve O(n lg n)?
• How probable is the balanced or unbalanced case?

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Randomized Quicksort

• Sorts array A of n numbers in-place
• Randomly chooses pivot point to yield reasonably
  balanced partitions on average

Randomized-Quicksort(A,p,r) q
Randomized-Partition(A,p,r) Randomized-Quicksor
t(A,p,q-1) Randomized-Quicksort(A,q1,r)
Randomized-Partition(A,p,r) i Random(p,r)
exchange Ar Ai return Partition(A,p,r)