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8.Sorting in linear time
Computer Theory Lab.
8.1 Lower bound for sorting
The decision tree model
a1:a2


a2:a3


<1,2,3>

a1:a3

<1,3,2>
Chapter 8
a1:a3

<2,1,3>

<3,1,2>
a2:a3

<2,3,1>

<3.2,1>
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Computer Theory Lab.
Theorem 9.1. Any decision tree that sorts n elements has
height ( n log n ) .
Proof:
n! l  2h ,
h  log( n!)  (n lg n).
Corollary 9.2 Heapsort and merge sort are asymptotically
optimal comparisons.
Chapter 8
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Computer Theory Lab.
8.2 Counting sort

Assume that each of the n input elements is an
integer in the range 1 to k for some integer k.
COUNTING_SORT(A,B,k)
1 for i 1 to k
2
do c [ i ]  0
3 for j 1 to length[A]
4
6 for i  2 to k
7
8 ► c[i] now contains the number of
do c [ A[ j ]]  c [ A[ j ]] 1
5 ► c[i] now contains the number
of elements equal to i
elements less than or equal to i
9 for j  length [ A ] downto 1
10
11
Chapter 8
do c [ i ]  c [ i ]  c [ i 1 ]
do B[ c [ A[ j ]]]  A[ j ]
c [ A[ j ]]  c [ A[ j ]] 1
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Computer Theory Lab.

Analysis: O( k  n )

Special case: O( n ) when k  O( n ) .

Property: stable (number with the same value appear in the
output array in the same order as they do in the input array.)
Chapter 8
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Computer Theory Lab.
The operation of Counting-sort on an input array A[1..8]
Chapter 8
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Computer Theory Lab.
8.3 Radix sort
Used by the card-sorting machines you can
now find only in computer museum.
RADIX_SORT(A,d)
1 for i  1 to d
2 do use a stable sort to sort array A on digit i
Chapter 8
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Computer Theory Lab.
Analysis:
O( d( n  k ))  O( dn  dk )
NOTE: not in-place (in-place: only constant number of elements
of the input array are ever stored outside the array.)
Lemma 8.3
Given n d-digit numbers in which each digit can take on up to k
possible values, RADIX-SORT correctly sorts these number in
(d(n + k)) time.
Chapter 8
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Computer Theory Lab.
Lemma 8.4
Given n b-bit numbers and any positive integer r ≤ b, RADIX- SORT
correctly sorts these numbers in ((b/r)(n+2r)) time.
Proof : Choose d = b/r.
IF b < lg n, choose r = b.
  ((b/r)(n+2r)) = (bn/logn)
Chapter 8
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Computer Theory Lab.
8.4 Bucket sort
BUCKET_SORT(A)
1 n  length [ A ]
2 for i  1 to n
3
do insert A[ i ] into
list B  nA[ i ] 
4 for i  1 to n-1
do sort list B [ i ] with insertion sort
6 concatenate B [ 0 ], B [1 ],..., B [ n  1 ] together in
5
order
Chapter 8
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Computer Theory Lab.
Analysis
The running time of bucket sort is
n1
T (n)  (n)   O(ni2 ).
i 0
taking expectations of both sides and using linearity of
expectation, we have
n 1

2 
E[T (n)]  E (n)   O(ni )
i 0


n 1
 (n)   E[O(ni2 )]
i 0
n 1
 (n)   O[ E (ni2 )]
Chapter 8
i 0
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Computer Theory Lab.
We claim that
E[ni2 ]  2  1 / n
We define indicator random variables
Xij = I {A[j] falls in bucket i}
for i = 0, 1, …, n-1 and j = 1, 2,…,n. thus,
ni 
Chapter 8
n
 X ij .
j 1
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Computer Theory Lab.
2
 n

2


E[ni ]  E  X ij  
 j 1  


n n

 E    X ij X ik 
 j 1k 1



n


 E   X ij2    X ij X ik 
1 j  n 1 k  n
 j 1

k j


n
  E[ X ij2 ] 
j 1
Chapter 8
  E[ X ij X ik ],
1 j  n 1 k  n
k j
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Computer Theory Lab.
Indicator random variable Xij is 1 with probability 1/n and 0
otherwise, and therefore
1
 1
E[ X ij2 ]  1   0  1  
n
 n
1

n
When k  j, the variables Xij and Xik are independent, and hence
E[ X ij X ik ]  E[ X ij ]E[ X ik ]
Chapter 8
1 1
 
n n
1
 2.
n
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Computer Theory Lab.
E[ni2 ] 
n
1
1
n    2
j 1
1 j  n 1 k  n n
k j
1
1
 n   n(n  1)  2
n
n
n 1
 1 
n
1
 2 
n
We can conclude that the expected time for bucket sort is
(n)+n·O(2-1/n)= (n).
Chapter 8
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Computer Theory Lab.
Another analysis:
n 1
n 1
n 1
2
2
 O( E [ ni ])  O(  E [ ni ])  O(  (1 ))  O( n )
i 0
i 0
i 0
Because
E( ni2 )  Var [ ni ]  E 2 [ ni ]
E [ ni ]  np  1
where p=
Var [ ni ]  np(1  p )  1 
E( ni2 )  1 
1
n
1
n
1 2
1
 1  2   (1 )
n
n
(Basic Probability Theory)
Chapter 8
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