Precise Analysis of Quicksort

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Precise Analysis of Quicksort

• Precise Analysis of Quicksort
• What are lessons learnt?
• Where can it be used?

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Quicksort Algorithm

• Given unsorted array

1
n
x
1
n
x
x
³ x
1
k
n
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Quicksort Algorithm

• Quicksort for n elements
• Partition array A1..n using pivot x (this
  takes (n1) comparisons)

• Recursively A1..k-1
• Recursively Ak1..n

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Analysis of Quicksort

• Let an the average time taken to sort an
  array of size n using Quicksort

If pivot is at position k, then an ak-1
an-k (n1)
Prob (pivot is at position k ) 1/n
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Analysis of Quicksort
Then, we have the following recurrence
?Expand the summations
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Analysis of Quicksort
Then, we have the following recurrence
?Get rid of dependence on full history
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Analysis of Quicksort
Then, we have the following recurrence
?Divide by n(n1)
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Analysis of Quicksort
Then, we have the following recurrence
? Now telescope
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Analysis of Quicksort
Then, we have the following recurrence
? Now telescope
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Analysis of Quicksort
Then, we have the following recurrence
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Analysis of Quicksort
Then, we have the following recurrence
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Analysis of Quicksort
Average running time of Quicksort
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Not that difficult, right?

• Where are the key steps?
• Get rid of full history
• Telescope
• Are there other ways?
• Simpler, but not so accurate?

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Review of the Key Steps
From here, show that
Further, show that
It follows immediately, that
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Where can it be used?

• Average height of a binary search tree
• 1.386 lg n