Chapter 7 Quicksort Ack: This presentation is based on the lecture slides from Hsu, Lih- Hsing, as...

Chapter 7

QuicksortAck: This presentation is based on the lecture slides from Hsu, Lih-Hsi

ng, as well as various materials from the web.

20071026 chap07 Hsiu-Hui Lee 2

7.1 Description of quicksort

Quicksort an n-element array:• Divide: Partition the array into two subarrays aro

und a pivot x such that elements in lower subarray ≤ x ≤ elements in upper subarray.

≤ x x ≥ x

• 2.Conquer: Recursively sort the two subarrays.• 3. Combine: Trivial.

Key: Linear-time partitioning subroutine.


QUICKSORT(A,p,r)

1 if p r

2 then q PARTITION A p r ( , , )

3 QUICKSORT(A,p,q-1)

4 QUICKSORT(A,q+1,r)

Initial call: QUICKSORT(A, 1, n)


Partition(A, p, r)1 x A[r]2 i p – 13 for j p to r -1 4 do if A[j] ≤ x5 then i i + 16 exchange A[i] A[j]7 exchange A[i+1] A[r] 8 return i+1


Loop Invariant

1. All entries in A[p..i] are ≤ pivot.

2. All entries in A[i+1..j-1] are > pivot.

3. A[r] = pivot.


The operation of Partition on a sample array1 x A[r]2 i p – 13 for j p to r -1 4 do if A[j] ≤ x5 then i i + 16 exchange A[i] A[j]7 exchange A[i+1] A[r] 8 return i+1


Two cases for one iteration of procedure Partition

Complexity: Partition on A[p…r] is (n) where n = r –p +1

1 x A[r]2 i p – 13 for j p to r -1 4 do if A[j] ≤ x5 then i i + 16 exchange A[i] A[j]7 exchange A[i+1] A[r] 8 return i+1

Why???(Exercise 7.1-3)


Another Partitioning subroutine

Invariant: x ≤x ≥x ?

p i j r

Partition(A, p, r) 1 x A[p]2 i p3 for j p+1 to r 4 do if A[j] ≤ x5 then i i + 16 exchange A[i] A[j]7 exchange A[i] A[p] 8 return i

Select first element as pivot


Another partitioning example

6 10 13 5 8 3 2 11

i j


7.2 Performance of quicksort

Worst-case partition:

)(

)()1(

)()0()1()(

2n

nnT

nTnTnT

Best-case partition:

)lg()(

)()2/(2)(

nnnT

nnTnT

Why???(Exercise 7.2-1)

Why???

The input array is already completely sorted.No better than insertion sort.


• Quicksort’s average running time is much closer to the best case than to the worst case.

• Imagine that PARTITION always produces 9-to-1 split. • Any split of constant proportionality will yield a recur

sion tree of depth , where the cost at each level is O(n).

Balanced partition: )lg()( nnnT

)lg()(

)()10/()10/9()(

nnnT

nnTnTnT

)(lg n


Intuition for the average case T(n) = (n lg n)

• Splits will not always be constant.• Mix of good and bad splits throughout

the recursion tree


7.3 Randomized quicksort• We have assumed that

all input permutations are equally likely. (not always true)

• To correct this, add randomization to randomly permute the input array

• Instead, we use random samplingpicking one element at random.

• Don’t always use A[r] as the pivot, and randomly pick an element from the subarray that is being sorted.


RANDOMIZED_PARTITION(A, p, r)

1 ),( rpRANDOMi

2 exchange ][][ iArA

3 return PARTITION(A, p, r)

RANDOMIZED_QUICKSORT(A, p, r)

1 if p r

2 then ),,(_ rpAPARTITIONRANDOMIZEDq

3 RANDOMIZED_QUICKSORT(A, p, q-1)

4 RANDOMIZED_QUICKSORT(A, q+1, r)


7.4 Analysis of quicksort7.4.1 Worst-case analysis

)())1()(()( max10

nqnTqTnTnq

guess T n cn( ) 2

2

2

22

10

22

10

)()1(2

)())1((

)())1(()(

max

max

cn

nnccn

nqnqc

nqnccqnT

nq

nq

pick the constant c large enough so that the 2 1c n( ) term dominates

the ( )n term.

T n n( ) ( ) 2


7.4.2 Expected running time

• Running time and comparisons• Lemma 7.1

• Let X be the number of comparisons performed in line 4 of partition over the entire execution of Quicksort on an n-element array. Then the running rime of Quicksort is O(n+X)


we define

{zi is compared to zj},

{zi is compared to zj}

IX ij

.1

1 1

n

i

n

ijijXX

1

1 1

1

1 1

1

1 1

Pr

][

n

i

n

ij

n

i

n

ijij

n

i

n

ijij

XE

XEXE


Pr{zi is compared to zj} = Pr{zi or zj is first pivot chosen from Zij}

= Pr{zi is first pivot chosen from Zij}

+ Pr{zj is first pivot chosen from Zij}

1

1

1

1

ijij 1

2

ij

1

1 1

.1

2][

n

i

n

ij ijXE


)lg(

)(lg

2

1

2

1

2][

1

1

1

1 1

1

1 1

1

1 1

nnO

nO

k

k

ijXE

n

i

n

i

n

k

n

i

in

k

n

i

n

ij

by pp. 1060 A.7

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