1) Randomizing quicksort works by randomly permuting the elements of the input array before sorting or by modifying the partition procedure to randomly exchange the pivot element with another randomly chosen element. 2) This randomization means no input can elicit the worst case behavior, making the worst case less likely to occur. 3) The expected number of comparisons in the partition procedure is O(n log n), so the expected running time of randomized quicksort is O(n log n).

1. 1
Randomizing Quicksort
• Randomly permute the elements of the input
array before sorting
• OR ... modify the PARTITION procedure
– At each step of the algorithm we exchange element
A[p] with an element chosen at random from A[p…r]
– The pivot element x = A[p] is equally likely to be any
one of the r – p + 1 elements of the subarray

2. 2
Randomized Algorithms
• No input can elicit worst case behavior
– Worst case occurs only if we get “unlucky” numbers
from the random number generator
• Worst case becomes less likely
– Randomization can NOT eliminate the worst-case but
it can make it less likely!

3. 3
Randomized PARTITION
Alg.: RANDOMIZED-PARTITION(A, p, r)
i ← RANDOM(p, r)
exchange A[p] A[i]↔
return PARTITION(A, p, r)

4. 4
Randomized Quicksort
Alg. : RANDOMIZED-QUICKSORT(A, p, r)
if p < r
then q ← RANDOMIZED-PARTITION(A, p, r)
RANDOMIZED-QUICKSORT(A, p, q)
RANDOMIZED-QUICKSORT(A, q + 1, r)

5. 5
Notation
• Rename the elements of A as z1, z2, . . . , zn, with
zi being the i-th smallest element
• Define the set Zij = {zi , zi+1, . . . , zj } the set of
elements between zi and zj, inclusive
106145389 72
z1z2 z9 z8 z5z3 z4 z6 z10 z7

6. 6
Total Number of Comparisons
in PARTITION
i+1 n
=X
i n-1
∑
−
=
1
1
n
i
∑+=
n
ij 1 ijX
• Total number of comparisons X performed by
the algorithm:
• Define Xij = I {zi is compared to zj }

7. 7
Expected Number of Total
Comparisons in PARTITION
• Compute the expected value of X:
=][XE
by linearity
of expectation
the expectation of Xij is equal
to the probability of the event
“zi is compared to zj”
=





∑ ∑
−
= +=
1
1 1
n
i
n
ij
ijXE [ ]=∑ ∑
−
= +=
1
1 1
n
i
n
ij
ijXE
∑ ∑
−
= +=
=
1
1 1
}Pr{
n
i
n
ij
ji ztocomparedisz
indicator
random variable

8. 8
Comparisons in PARTITION :
Observation 1
• Each pair of elements is compared at most once
during the entire execution of the algorithm
– Elements are compared only to the pivot point!
– Pivot point is excluded from future calls to PARTITION

9. 9
Comparisons in PARTITION:
Observation 2
• Only the pivot is compared with elements in both
partitions!
z1z2 z9 z8 z5z3 z4 z6 z10 z7
106145389 72
Z1,6= {1, 2, 3, 4, 5, 6} Z8,9 = {8, 9, 10}{7}
pivot
Elements between different partitions
are never compared!

10. 10
Comparisons in PARTITION
• Case 1: pivot chosen such as: zi < x < zj
– zi and zj will never be compared
• Case 2: zi or zj is the pivot
– zi and zj will be compared
– only if one of them is chosen as pivot before any
other element in range zi to zj
106145389 72
z1z2 z9 z8 z5z3 z4 z6 z10 z7
Z1,6= {1, 2, 3, 4, 5, 6} Z8,9 = {8, 9, 10}{7}
Pr{ }?i jz is compared to z

11. 11
Probability of comparing zi with zj
= 1/( j - i + 1) + 1/( j - i + 1) = 2/( j - i + 1)
zi is compared to zj
zi is the first pivot chosen from Zij
=Pr{ }
Pr{ }
zj is the first pivot chosen from Zij
Pr{ }
OR+
•There are j – i + 1 elements between zi and zj
– Pivot is chosen randomly and independently
– The probability that any particular element is the first
one chosen is 1/( j - i + 1)

12. 12
Number of Comparisons in PARTITION
1 1 1 1
1 1 1 1 1 1 1
2 2 2
[ ] (lg )
1 1
n n n n i n n n
i j i i k i k i
E X O n
j i k k
− − − − −
= = + = = = = =
= = < =
− + +
∑ ∑ ∑∑ ∑∑ ∑
)lg( nnO=
⇒ Expected running time of Quicksort using
RANDOMIZED-PARTITION is O(nlgn)
∑ ∑
−
= +=
=
1
1 1
}Pr{][
n
i
n
ij
ji ztocomparediszXE
Expected number of comparisons in PARTITION:
(set k=j-i) (harmonic series)

13. 13
Alternative Average-Case
Analysis of Quicksort
• See Problem 7-2, page 16
• Focus on the expected running time of
each individual recursive call to Quicksort,
rather than on the number of comparisons
performed.
• Use Hoare partition in our analysis.

14. 14
Worst-Case Analysis
T(n) = max (T(q) + T(n - q - 1)) + Θ(n)
0 ≤ q ≤ n-1
1.the time to sort the left partition with q elements, plus
2.the time to sort the right partition with N-q-1 elements, plus
3.the time to build the partitions
where q ranges from 0 to n-1 since the procedure PARTITION
produces two sub-problems with total size n-1
---------- Substitution method: Guess T(n) ≤ cn2
T(n) ≤ max (cq2
+ c(n - q - 1)2
) + Θ(n)
0 ≤ q ≤ n-1
= c · max (q2
+ (n - q - 1)2
) + Θ(n)
0 ≤ q ≤ n-1
Take derivatives to get maximum at q = 0, n-1:
T(n) ≤ c(n - 1)2
+ Θ(n) ≤ cn2
- 2c(n - 1) + 1 + Θ(n) ≤ cn2
Therefore, the worst case running time is Θ(n2
)