Quicksort

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Quicksort
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Properties of QuickSort

• Runs in ?(n2) worst case, but ?(n lg n) on
  average
• Constants in the ?(n lg n) are small, so its
  often the best practical sorting algorithm
• Sorts in place
• Several variations
• randomized partitioning
• random sampling

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Overview of QuickSort

• Divide The array p..r is partitioned into two
  non-empty subarrays Ap..q and Aq1 .. r such
  that each element in the first array is less than
  or equal to each item in the second
• Conquer Sort recursively
• Combine Since they are already sorted

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QuickSort

• QUICKSORT (A, p, r)
• if p lt r
• then q ? PARTITION (A, p, r)
• QUICKSORT (A, p, q)
• QUICKSORT (A, q1, r)
• Note obviously, PARTITION is the key here!

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Partitioning

• PARTITION selects a pivot element for
  partitioning
• We have a range p to r in A with two variables i
  and j, and a pivot point x, initialized to r.
• At the beginning of each loop
• Everything between p and i is ? x
• Everything between i and j -1 gt x

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This is in the correct spot!
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PARTITION

• PARTITION (A, p, r)
• 1 x ? Ar
• 2 i ? p - 1
• 3 for j ? p to r 1
• 4 do if Aj ? x
• 5 then i ? i 1
• 6 exchange Ai ? Aj
• 7 exchange Ai 1 ? Ar
• 8 return i 1

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Performance

• Depends heavily on whether the partitioning is
  balanced
• Unbalanced can run as slow as insertion sort!
• Worst-case two subproblems with
• n 1 elements
• 0 elements
• T (n) T (n 1) T (0) ?(n)
• Decomposes to ?(n2)

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Whats really going on...

• Average time not close to that...
• Suppose that we have a 9-to-1 split
• T (n) T (9n/10) T (n/10) cn
• Then, a subproblem at depth i is 9in/10i
• Solving for i yields a tree of depth log10/9 n
• Thus, the height of the tree is still
  logarithmic!
• Even a 99-to-1 split is logarithmic
• Any split of constant proportionality yields a
  tree of depth ?(lg n)
• Note 80 of the time, PARTITION produces a split
  more balanced than 9-to-1

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Randomized Quicksort

• Uses random sampling
• Use a randomly chosen element from the array
• Swap Ar with the chosen element
• RANDOMIZED-PARTITION (A, p, r)
• 1 i ? RANDOM (p, r)
• 2 exchange Ar ? Ai
• 3 return PARTITION (A, p, r)

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Summary

• Runs in O(n log n) on average
• Runs in O(n2) in worst-case
• It uses partitioning to sort
• Randomized partitioning helps
• Quicksort is preferred, because of its low
  constant factors