8.sorting in linear time hsu, lih-hsing. computer theory lab. chapter 8p.2 8.1 lower bound for...
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8.Sorting in linear time
Hsu, Lih-Hsing
Chapter 8 P.2
Computer Theory Lab.
8.1 Lower bound for sortingThe decision tree model
a1:a2
a2:a3 a1:a3
<1,2,3> a1:a3
<3,1,2><1,3,2>
a2:a3<2,1,3>
<3.2,1><2,3,1>
Chapter 8 P.3
Computer Theory Lab.
Theorem 9.1. Any decision tree that sorts n elements has
height ( log )n n .
Proof:
).lg()!log(
,2!
nnnh
ln h
Corollary 9.2 Heapsort and merge sort are asymptotically
optimal comparisons.
Chapter 8 P.4
Computer Theory Lab.
8.2 Counting sortAAAssssssuuummmeee ttthhhaaattt eeeaaaccchhh ooofff ttthhheee nnn iiinnnpppuuuttt eeellleeemmmeeennntttsss iiisss aaannn
iiinnnttteeegggeeerrr iiinnn ttthhheee rrraaannngggeee 111 tttooo kkk fffooorrr sssooommmeee iiinnnttteeegggeeerrr kkk...
COUNTING_SORT(A,B,k)
1 for i 1 to k
2 do c i[ ] 0
3 for j 1 to length[A]
4 do c A j c A j[ [ ]] [ [ ]] 1
5 ► c[i] now contains the number
of elements equal to i
6 for i 2 to k
7 do c i c i c i[ ] [ ] [ ] 1
8 ► c[i] now contains the number of
elements less than or equal to i
9 for j length A [ ] downto 1
10 do B c A j A j[ [ [ ]]] [ ]
11 c A j c A j[ [ ]] [ [ ]] 1
Chapter 8 P.5
Computer Theory Lab.
A n a l y s i s : O k n( )
S p e c i a l c a s e : O n( ) w h e n k O n ( ) .
P r o p e r t y : sss ttt aaa bbb lll eee ( n u m b e r w i t h t h e s a m e v a l u e a p p e a r i n t h e
o u t p u t a r r a y i n t h e s a m e o r d e r a s t h e y d o i n t h e i n p u t a r r a y . )
Chapter 8 P.6
Computer Theory Lab.
The operation of Counting-sort on an input array A[1..8]
Chapter 8 P.7
Computer Theory Lab.
8.3 Radix sort Used by the card-sorting machines you can now fi
nd 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 P.8
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 P.9
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 P.10
Computer Theory Lab.
8.4 Bucket sort
B U C K E T _ S O R T ( A )
1 n l e n g t h A [ ]
2 f o r i 1 t o n
3 d o i n s e r t A i[ ] i n t o
l i s t B n A i[ ]
4 f o r i 1 t o n - 1
5 d o s o r t l i s t B i[ ] w i t h i n s e r t i o n s o r t
6 c o n c a t e n a t e B B B n[ ] , [ ] , . . . , [ ]0 1 1 t o g e t h e r i n
o r d e r
Chapter 8 P.11
Computer Theory Lab.
AnalysisThe running time of bucket sort is
.)()()(1
0
2
n
iinOnnT
taking expectations of both sides and using linearity of expectation, we have
1
0
2
1
0
2
1
0
2
)]([)(
)]([)(
)()()]([
n
ii
n
ii
n
ii
nEOn
nOEn
nOnEnTE
Chapter 8 P.12
Computer Theory Lab.
We claim that
nnE i /12][ 2
We define indicator random variablesXij = I {A[j] falls in bucket i}
for i = 0, 1, …, n-1 and j = 1, 2,…,n. thus,
.1
n
jiji Xn
Chapter 8 P.13
Computer Theory Lab.
,][][
][
1 11
2
1 11
2
1 1
2
1
2
njjk
nkikij
n
jij
njjk
nkikij
n
jij
n
jik
n
kij
n
jiji
XXEXE
XXXE
XXE
XEnE
Chapter 8 P.14
Computer Theory Lab.
Indicator random variable Xij is 1 with probability 1/n and 0
otherwise, and therefore
n
nnXE ij
1
110
11][ 2
When k j, the variables Xij and Xik are independent, and hence
.1
11
][][][
2n
nn
XEXEXXE ikijikij
Chapter 8 P.15
Computer Theory Lab.
n
nn
nnn
nn
nnnE
njjk
nk
n
ji
12
11
1)1(
1
11][
2
1 12
1
2
We can conclude that the expected time for bucket sort is (n)+n·O(2-1/n)= (n).
Chapter 8 P.16
Computer Theory Lab.
Another analysis:
O E n O E n O O nii
ni
i
n
i
n( [ ]) ( [ ]) ( ( )) ( )2
0
1 2
0
1
0
11
Because
E n Var n E n
E n np p=n
Var n np pn
E nn n
i i i
i
i
i
( ) [ ] [ ]
[ ]
[ ] ( )
( ) ( )
2 2
2 2
11
1 11
111 2
11
where
(Basic Probability Theory)