Introduction to Programming (Java) 7/12 · The art of computer programming, volume 3 Sorting and Searching. Addison-Wesley Publishing Company, 1973. N. Wirth: Algoritmy a štruktúry

Introduction to Programming (Java) 7/12Sorting I – Associative Sorting

Introduction to Programming (Java)7/12

Michal Krátký

Department of Computer ScienceTechnical University of Ostrava

Introduction to Programming (Java)2008/2009

c©2006–2008 Michal Krátký Introduction to Programming (Java) 1/38


IntroductionInsert SortSelect SortShell Sort

Introduction

Input: order sequence (numbers, words, ...)Output: sorted sequence permutationWhy? More efficient searching

References

D. E. Knuth. The art of computer programming, volume 3Sorting and Searching. Addison-Wesley PublishingCompany, 1973.N. Wirth: Algoritmy a štruktúry údajov, Alfa, Bratislava,1989.J. Dvorský. Základy algoritmizace - sylaby. 2005. In Czech.




Sorting

we sort an order set Uan item of U is the keyeach key if often a member of a record including otherattribute values

Example:(a) A ∈ N, A = {1, 2, 3, . . .}(b) A ⊆ A1 × A2 × . . .× An, A = {(5606034215,′RottenJohnny′, . . .), (6409045321,′Vedder Eddie′, . . .), . . .}




Sorting

A sorting method is:

stable – if it preserves relative ordering (from the inputsequence point of view) items with the same key

natural – if the sorting cost depends on the orderliness ofthe input set

in situ/in place – if the method transforms the input using asmall, constant amount of extra storage space




A Classification of Sorting Algorithm

Address Sorting – there is a relation between the key andposition in the sorted sequence. No comparison is appliedin this case.

Associative/Comparisons Sorting – comparisons of twokeys is applied during the algorithm run-time

Hybrid Sorting – a combination of both methods




Associative Sorting

Cost of these algorithms is influenced by two operations:comparison of two itemsswap of items in the memory

The cost is measured by the number of comparisons andswaps.

The comparison is related to key, however whole record may beswapped. Of course it depends on the implementation.




Permutation

Definition (Permutation).Permutation of a set X with n items is a bijection f : X → X .

An Example:X = {a, b, c, d} and the permutation f : X → X is defined asfollows f (a) = c, f (b) = b, f (c) = d , f (d) = a, then

f =

(a b c dc b d a

)

The number of all permutations of the set with n items is n!




Inversion of a Permutation

Definition (Inversion of a Permutation).Let f be a permutation of X = {1, . . . , n}. We propose that acouple of different items (i , j) is the inversion of thepermutation f , if: (i − j) · (f (i)− f (j)) < 0.

An Example

f =

(1 2 3 42 4 3 1

)

An Even/Odd permutation means the even/odd number ofinversions. inv(f ) is the number of the permutation inversions.




Associative Sorting

Efficiency of the algorithm can depend on orderliness ofthe input set

We can measure the orderliness rate by the number ofinversions:

inv(f ) = 0: the input sequence is sortedinv(f ) is maximal, n(n−1)

2 : the input sequence is sorted inthe inverse orderif all permutations are uniformly possible, the average inv(f )is approximately 1

4 n(n − 1)

From this point of view, associative sorting algorithmsremove inversions by the swap operations as long as thesequence is sorted




Insert Sort

Idea: We take item by item and insert it into the sorted partof the sequence.Array is broken into two parts: sorted part (from 0 to i − 1)and unsorted part (from i to n − 1).We select an item from the unsorted part and insert it inthe sorted part in order to this part was still sorted.This step continues while the unsorted part is empty.In the start of the algorithm the sorted part includes onlyone item.




Insert Sort, An Example

Input: 44 | 55 12 42 94 18 6 6744 55 | 12 42 94 18 6 6712 44 55 | 42 94 18 6 6712 42 44 55 | 94 18 6 6712 42 44 55 94 | 18 6 6712 18 42 44 55 94 | 6 676 12 18 42 44 55 94 | 67

Output: 6 12 18 42 44 55 67 94 |

Sorted Part →




Example 7.1, setRandom()

/ / Set ar ray ’ s i tems at random values ,/ / the maximal i tem value i s maxValues t a t i c void setRandom ( i n t [ ] data , i n t maxValue ){

for ( i n t i = 0 ; i < data . leng th ; i ++){

data [ i ] = ( i n t ) ( Math . random ( ) ∗ maxValue ) ;}

}




Example 7.1, print()

s t a t i c void p r i n t ( i n t [ ] data ){


System . out . p r i n t ( data [ i ] ) ;i f ( i ! = data . length −1){

System . out . p r i n t ( " , " ) ;}

}System . out . p r i n t l n ( " " ) ;

}




Example 7.1, insertSort()

s t a t i c void i n s e r t S o r t ( i n t [ ] data ){

i n t i , j , v ;for ( i = 0 ; i < data . leng th ; i ++){

v = data [ i ] ;j = i ;while ( ( j > 0 ) & & ( data [ j −1] > v ) ){

data [ j ] = data [ j −1];j−−;

}data [ j ] = v ;

}}




Example 7.1, file Example0701

public class Example0701{

public s t a t i c void main ( S t r i n g [ ] args ){

f i n a l i n t SIZE = 3 0 ;f i n a l i n t MAX_VALUE = 1000 ;i n t [ ] data = new i n t [ SIZE ] ;

setRandom ( data , MAX_VALUE) ;p r i n t ( data ) ;i n s e r t S o r t ( data ) ;p r i n t ( data ) ;

}

/ / methods . . .}




Insert Sort, Const Analysis 1/2

The number of comparisons in the i-th step Ci : 1 ≤ Ci ≤ i − 1.If all permutations of n keys are uniformly possible: Ci = i/2.

The number of swaps Si = Ci .

The overall number of comparisons and swaps are as follows:

Cmin = n − 1 Smin = 2(n − 1)

Cavg = 14(n2 + n + 2) Savg = 1

4(n2 + 9n − 10)

Cmax = 12(n2 + n)− 1 Smax = 1

2(n2 + 3n − 4)




Insert Sort, Const Analysis 2/2

If the input sequence is sorted, Cmin and Smin appear. Wecall this case as the best case.

On the other hand, if the input sequence is sorted in theinverse order, Cmax and Smax appear. We call this case asthe worst case.

Consequently, this algorithm is natural and stable




Binary Insert Sort

Obviously, the right position of the current item is possibleto find by the binary searching

In this way, we can write the binary insert sort algorithm




Example 7.2, binaryInsertSort() 1/3

s t a t i c void b i n a r y I n s e r t S o r t ( i n t [ ] data ){

i n t i , j , value ;i n t low , hi , m;

for ( i = 1 ; i < data . leng th ; i ++){

value = data [ i ] ;





/ / f i n d the r i g h t p o s i t i o n/ / by the b inary searching a lgo r i t hmlow =0;h i = i ;while ( low < h i ){

m = ( low + h i ) / 2 ;i f ( data [m] < = value ){

low = m + 1 ;}else{

h i = m;}

} / / wh i le





for ( j = i ; j > h i ; j −−) / / swap i tems{

data [ j ] = data [ j −1];}data [ h i ] = value ;

} / / f o r} / / b i n a r y I n s e r t S o r t




Example 7.2, file Example0702




setRandom ( data , MAX_VALUE) ;p r i n t ( data ) ;b i n a r y I n s e r t S o r t ( data ) ;p r i n t ( data ) ;

}

/ / methods . . .}




Binary Insert Sort, Cost Analysis

The number of comparisons is as follows:

C =n∑

i=1

dlog ie

However, the number of swaps is the same: O(n2).




Select Sort

Idea: We select the minimal item from the unsorted part ofthe array.

This item is inserted into the sorted part as the last item.

Consequently, the size of the sorted part in increased by 1.




Select Sort, Example

Input: 44 55 12 42 94 18 6 676 | 55 12 42 94 18 44 676 12 | 55 42 94 18 44 676 12 18 | 42 94 55 44 676 12 18 42 | 94 55 44 676 12 18 42 44 | 55 94 676 12 18 42 44 55 | 94 67

Output: 6 12 18 42 44 55 67 | 94

Sorted Part →




Example 7.3, setRandom()

/∗ ∗∗ Set random values i n the array ,∗ maximal value < maxValue .∗ ∗ /

s t a t i c void setRandom ( double [ ] data ,double maxValue )

{for ( i n t i = 0 ; i < data . leng th ; i ++){

data [ i ] = Math . random ( ) ∗ maxValue ;}

}




Example 7.3, print()

/ / p r i n t the ar rays t a t i c void p r i n t ( double [ ] data ){


System . out . p r i n t ( data [ i ] ) ;i f ( i ! = data . length −1){

System . out . p r i n t ( " , " ) ;}

}System . out . p r i n t l n ( " " ) ;

}




Example 7.3, selectSort() 1/2

s t a t i c void s e l e c t S o r t ( double [ ] data ){

i n t i , j , min ;double t ;

for ( i = 0 ; i < data . leng th ; i ++){

min = i ;




Example 7.3, selectSort() 2/2

for ( j = i + 1 ; j < data . leng th ; j ++){

i f ( data [ j ] < data [ min ] ){

min = j ;}

}t = data [ min ] ;data [ min ] = data [ i ] ;data [ i ] = t ;

}}




Example 7.3, file Example0703.java



f i n a l i n t SIZE = 2 0 ;f i n a l double MAX_VALUE = 1 0 0 . 0 ;

double [ ] data = new double [ SIZE ] ;setRandom ( data , MAX_VALUE) ;p r i n t ( data ) ;s e l e c t S o r t ( data ) ;p r i n t ( data ) ;

}

/ / methods . . .}




Select Sort, Cost Analysis

The number of comparisons C does not depend on the initialordering. In this way, this algorithm is less natural then InsertSort.

C =12(n2 − n)

The number of swaps S:

Smin = 3(n − 1)

Smax =

⌈(n2

4

)⌉+ 3(n − 1)

Savg.= n(ln n + γ)

Select Sort is commonly supposed as more efficient than InsertSort, exception a sorted or almost sorted sequence.




Shell Sort

Obviously, the number of inversions is decreased by one ifwe change the adjacent items.

Therefore, the complexity is quadratic in the average andworst cases.

Obviously, if more faraway items are changed, the numberof inversions falls quickly to zero.




Shell Sort

Shell Sort, D. L. Shell, 1959

We sort items in the hi distance for the i-th iteration,i = t , t − 1, . . . , 0, hi+1 > hi , h1 = 0, t > 0

The hi value is the i-th step of the method

Result: This algorithm sorts shorted sequences for the firstiterations, it sorts longer sequences to be more sorted forthe next iterations




Shell Sort

Obviously, the cost of Shell Sort depends on a sequence ofsteps, there are the following step sequences:

A: h1 = 1, hi+1 = 2 ∗ hi + 1 – Hibbard

B: h1 = 1, h2 = 3, hi+1 = 2 ∗ hi − 1 (for i > 2)

C: h1 = 1, hi+1 = 3 ∗ hi + 1 – D. Knuth, 19691, 4, 13, 40, 121, 364,1093, 3280, 9841, ...

D: h1 = 1, hi+1 = 4i+1 + 3 ∗ 2i + 1R. Sedgewick (20–30% faster than C)1, 8, 23, 77, 281, 1073, 4193, 16577, ...

The complete cost analysis is still an open issue.




Shell Sort, Example

Array Index: 0 1 2 3 4 5 6 7

Input: 44 18 12 42 94 55 6 67 h = 4, i = 644 18 6 42 94 55 12 67 h = 1, i = 118 44 6 42 94 55 12 67 h = 1, i = 26 18 44 42 94 55 12 67 h = 1, i = 36 18 42 44 94 55 12 67 h = 1, i = 56 18 42 44 55 94 12 67 h = 1, i = 66 12 18 42 44 55 94 67 h = 1, i = 7

Output: 6 12 18 42 44 55 67 94




Example 7.4, shellSort() 1/2

s t a t i c void s h e l l S o r t ( i n t data [ ] ){

i n t i , j , h , value ;

for ( h = 1 ; h <= data . leng th ; h = 3∗h +1) ;h / = 3 ;

do{




Example 7.4, shellSort() 2/2

for ( i = h ; i < data . leng th ; i ++){

value = data [ i ] ;j = i ;while ( ( j >= h ) & & ( data [ j−h ] > value ) ){

data [ j ] = data [ j−h ] ;j −= h ;

} / / wh i ledata [ j ] = value ;

p r i n t ( data ) ;} / / f o rh / = 3 ;

} while ( h ! = 0 ) ;} / / s h e l l S o r t




Example 7.4, file Example0704.java




setRandom ( data , MAX_VALUE) ;p r i n t ( data ) ;s h e l l S o r t ( data ) ;p r i n t ( data ) ;

}

/ / methods . . .}


Documents

Introduction to Programming (Java) 7/12 · The art of computer programming, volume 3 Sorting and Searching. Addison-Wesley Publishing Company, 1973. N. Wirth: Algoritmy a štruktúry