Randomized%20Quick%20Sort%20Algorithm

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Randomized Quick Sort Algorithm

• Lecture Note 1
• Prepared By Muhammed Miah

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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.

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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

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Randomized Quick Sort (S)

• Input a set of numbers S,
• S S1, S2, S3, , Sn
• If S gt 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
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Randomized Quick Sort

• Example
• S 2, 1, 3, 5, 4

1st pivot
2
1
2
3, 5, 4
2nd pivot
3rd pivot
5
1
sorted
3, 4
5
4
4th pivot
3
4
5th pivot
3
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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

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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 gt i
• Xij 1 if it is compared with j
• Xij 0 otherwise
• of variables we can create n2

X12 X13 Xn-1 n SUM
1 if X1 and X2 are compared, 0 otherwise SUM(X12, X13, , Xn-1 n)


AVG(X12) AVG(X13)
millions of trees are generated
AVG.
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Analysis of Expected Running Time of Randomized
QS

• Xij is a random variable
• E? ? Xij ? ? EXij
• linearity of expectation,
• EX Y EX EY
• EXij how often i and j compared

n-1
n
n-1
n
i1
jgti
i1
jgti
Horizontal definition
Vertical definition
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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(LRi, LRj gt Min(LR of i1, i2, ,
  j-1), then
• i is NOT compared to j
• otherwise it is compared

6
8
4
7
9
5
3
1
10
2
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Analysis of Expected Running Time of Randomized
QS

• P(Xij 1) 2/(j-i1)
• EXij 2/(j-i1)
• ? ? EXij ? ? 2/(j-i1)
• ? ? 2/(j-i) ? ? 2/k let, j-i k
• ? ? 2/k

n-1
n
n
n
i1
jgti
i1
jgti
n
n
n
n-i
i1
Ji
i1
k1
n
n
i1
k1
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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
k1
i1
k1
n
k1
n
k1