Randomizing quicksort algorith with example
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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
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](https://image.slidesharecdn.com/randomizingquicksort-170224145012/85/Randomizing-quicksort-algorith-with-example-1-320.jpg)
![3
Randomized PARTITION
Alg.: RANDOMIZED-PARTITION(A, p, r)
i ← RANDOM(p, r)
exchange A[p] A[i]↔
return PARTITION(A, p, r)](https://image.slidesharecdn.com/randomizingquicksort-170224145012/85/Randomizing-quicksort-algorith-with-example-3-320.jpg)
![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](https://image.slidesharecdn.com/randomizingquicksort-170224145012/85/Randomizing-quicksort-algorith-with-example-7-320.jpg)
![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)](https://image.slidesharecdn.com/randomizingquicksort-170224145012/85/Randomizing-quicksort-algorith-with-example-12-320.jpg)
Randomizing quicksort algorith with example
- 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 )