quick sort pp t 2114

7/26/2019 Quick Sort Pp t 2114

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Quick Sort

Divide: Partition the array into two sub-arrays

A[p . . q-1] and A[q+1 . . r] such that each element o

A[p . . q-1] is less than or equal to A[q]! which in turn

less than or equal to each element o A[q+1 . . r]


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"onquer: #ort the two sub-arrays A[p . . q-1] and

A[q+1 . . r] by recursive calls to quic$ sort.


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"ombine: #ince the sub-arrays are sorted in place! no

wor$ is needed to combine them.


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%&'"(#)*,A! p! r

i p r

then qPA*'')/,A! p! r

%&'"(#)*,A! p! q-1

%&'"(#)*,A! q+1! r


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PA*'')/,A! p! r

0A[r]

ip-1


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or

p to r-1

do i A[] 2 0

then i

i+1

e0chan3e A[i]A[]

e0chan3e A[i+1]

A[r]

return i+1


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i p, j r

2 8 7 1 3 5 6 4

(a)


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p, i j r

2 8 7 1 3 5 6 4

(b)


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p, i j r

2 8 7 1 3 5 6 4

(c)


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p, i j r

2 8 7 1 3 5 6 4

(d)


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p i j r

2 1 7 8 3 5 6 4

(e)


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p i j r

2 1 3 8 7 5 6 4

(f)


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p i j r

2 1 3 8 7 5 6 4

(g)


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p i r

2 1 3 8 7 5 6 4

(h)


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p i r

2 1 3 4 7 5 6 8

(i)


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4orst-case partitionin3:

he partitionin3 routine produces one sub-problem

with n-1 elements and another sub-problem with 5

elements. #o the partitionin3 costs 6,n time.


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he recurrence or the runnin3 time

,n2 ,n-1 + ,5 + 6,n

2,n-1 + 6,n

2----------------- 6,n7


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he 6,n7 runnin3 time occurs when the input

array is already completely sorted 8 a common

situation in which insertion sort runs in O,n time


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9est-case partitionin3:

he partitionin3 procedure produces two

sub-problems! each o sie not more than

n;7.


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,n 2 7,n;7 + 6,n

2 ----- O,n l3 n


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he equal balancin3 o the two sides o the

partition at every level o the recursion

produces aster al3orithm.


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9alanced partitionin3:

#uppose! the partitionin3 al3orithm always

produces


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,n 2 ,


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9alanced partitionin3: he recursion tree


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'n act! a


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'ntuition or the avera3e case:

't is unli$ely that the partitionin3 always happens

in the same way at every level.


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'ntuition or the avera3e case:

'n the avera3e case! PA*')/ produces a mi0 o

=3ood> and =bad> splits.


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Quick Sort'ntuition or the avera3e case:

he combination o the bad split ollowed by the 3ood split

produces three arrays o sies 5! ,n-1;7-1! and ,n-1;7 at a

combined partitionin3 cost o 6,n + 6,n-12 6,n

n

n1

(n1)!21 (n1)!2

"

#(n)


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Quick Sort'ntuition or the avera3e case:

A sin3le level o partitionin3 produces two sub-arrays o sie

,n-1;7 at a cost o 6,n.

n

(n1)!2 (n1)!2

#(n)


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$ %ando&i'ed erion of Quick Sort

'nstead o always usin3 A[r] as the pivot! we will

use a randomly chosen element rom the sub-array

A[p..r].


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9ecause the pivot element is randomly chosen!

we e0pect the split o the input array to be

reasonably well balanced on avera3e.


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*A/D)?'@D-PA*'')/,A! p! r

i*A/D)?,p! r

e0chan3e A[r]A[i]

return PA*'')/,A! p! r


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*A/D)?'@D-%&'"(#)*,A! p! r

i pr then

q*A/D)?'@D-PA*'')/,A! p! r

*A/D)?'@D-%&'"(#)*,A! p! q-1

*A/D)?'@D-%&'"(#)*,A! q+1! r

quick sort pp t 2114

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