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# Sorting In Java | Get Help In Java Data Structure The sorting problem

Given a list of n totally orderable items, rearrange the list in increasing (or decreasing) order.

• Totally orderable means that any two items can be compared: numbers, characters, strings etc.

• The items are usually called keys.

Example This is an instance of the sorting problem:

(12, 5, 8, 16, 9, 31) → (5, 8, 9, 12, 16, 31).

Often, the elements to be sorted contain keys and data:

((5, A),(2, C),(7, A),(2, B),(1, B)) → ((1, B),(2, C),(2, B),(5, A),(7, A)).

Properties of sorting algorithms

• Time complexity: worst case and average case.

• Space complexity: the amount of extra space needed by the algorithm.

Definition

An algorithm is said in-place if it does not require more than O(log n) space in addition to the input.

Stability: A sorting algorithm is stable if it does not change the order of equal elements

Example

Given the input array: {(5, A),(2, C),(7, A),(2, B),(1, B)}, where we want to sort according to the first element of the pairs (the integers), then:

{(1, B),(2, C),(2, B),(5, A),(7, A)} is a stable sorting.

{(1, B),(2, B),(2, C),(5, A),(7, A)} is not a stable sorting, since the order of (2, B) and (2, C) is reversed.

General purpose sorting algorithms

• The first type of sorting algorithms we are going to see are sorting algorithms which can be used to sort any type of keys.

• They are based on comparison only and do not assume any other property in the keys (for example, they do not require the keys to be integers or strings).

### Insertion sort

Insertion sort gradually builds the sorted array by putting each new key in its correct position.

```public static void insertionSort ( int [ ] A , int n ) {
for ( int i = 1; i < n ; i ++) {
int j = i ;
while ( j > 0 && A [ j - 1] > A [ j ]) {
int tmp = A [ j ];
A [ j ] = A [ j - 1];
A [ j - 1] = tmp ;
j - -;
}
}
}```

Worst case time complexity: O(n 2 ) (quadratic).

Average case time complexity: O(n 2 ).

Space complexity: O(1).

### Selection sort

Selection sort gradually builds the sorted array by finding the correct key for each new position

```public static void selectionSort ( int [ ] A , int n ) {
for ( int i = 0; i < n - 1; i ++) {
int min = i ;
for ( i n t j = i + 1; j < n ; j ++) {
// Search for the minimum
if ( A [ j ] < A [ min ])
min = j ;
}
// Swap A[i] with A [ min]
int tmp = A [ i ];
A [ i ] = A [ min ];
A [ min ] = tmp ;
}
}

```

Worst case time complexity: O(n 2 ) (quadratic).

Average case time complexity: O(n 2 ).

Space complexity: O(1).

Example: The array {(2, A),(2, B),(1, C)} will be sorted as {(1, C),(2, B),(2, A)}.

### Bubble sort

Bubble sort sorts the array by repeatedly swapping non-ordered adjacent keys. After each for loop iteration, the maximum is moved (or bubbled) towards the end.

```public static void bubbleSort ( int A [ ] , int n ) {
for ( int i = 0; i < n - 1; i ++) {
for ( int j = 0; j < n - 1 - i ; j ++) {
if ( A [ j ] > A [ j + 1]) {
// Swap A[j] with A[j + 1]
int tmp = A [ j ];
A [ j ] = A [ j + 1];
A [ j + 1] = tmp ;
}
}
}
}```

Worst case time complexity: O(n 2 ) (quadratic).

Average case time complexity: O(n 2 ).

Space complexity: O(1).

### Merge sort

Merge sort is a divide-and-conquer algorithms to sort an array of n elements:

• Divide the array into two equal parts.

• Sort each part apart (recursively).

• Merge the two sorted parts.

The key step in merge sort is merging two sorted arrays, which can be done in O(n).

Example

Given two arrays B = {1, 4, 6} and C = {2, 3, 7, 8}, the result of merging B and C is {1, 2, 3, 4, 6, 7, 8}.

```public static void mergeSort ( int [] A , int l , int r ) {
if ( l >= r )
return ;
int m = ( l + r ) / 2;
mergeSort (A , l , m ) ; // Sort first half
mergeSort (A , m + 1 , r ) ; // Sort second half
merge (A , l , m , r ) ; // Merge
}```

```private static void merge ( int [ ] A , int l , int m , int r ) {
int [ ] B = new i n t [ r - l + 1];
int i = l , j = m + 1 , k = 0;
while ( i <= m && j <= r )
if ( A [ i ] <= A [ j ])
B [ k ++] = A [ i ++];
else
B [ k ++] = A [ j ++];
i f ( i > m )
while ( j <= r )
B [ k ++] = A [ j ++];
else
while ( i <= m )
B [ k ++] = A [ i ++];
for ( k = 0; k < B . length ; k ++)
A [ k + l ] = B [ k ];
}```

Example:

Example Sort the array: 8, 3, 2, 9, 7, 1, 5, 4. Worst case time complexity: O(n log n) (sub-quadratic).

Average case time complexity: O(n log n).

Space complexity: O(n) (requires auxiliary memory).

### Quick sort

Quick sort is another divide-and-conquer algorithms to sort an array of n elements:

• Pick any element of the array and call it the pivot (the first element, or a randomly chosen element for example) .

• Rearrange the array so that all elements before the pivot are less or equal the pivot, and all those after the pivot are greater or equal to the pivot (partitioning).

• Recursively sort the part of the array before the pivot and the one after the pivot.

```public static void quickSort ( int [] A , int l , int r ) {
if ( l < r ) {
int s = partition (A , l , r ) ;
quickSort (A , l , s - 1) ;
quickSort (A , s + 1 , r ) ;
}
}```
```private static int partition ( int [] A , int l , int r ) {
int p = A [ l ] , i = l + 1 , j = r ;
while ( i < j ) {
while ( A [ i ] <= p && i < j )
i ++;
while ( A [ j ] > p && i < j )
j - -;
int tmp = A [ i ];
A [ i ] = A [ j ];
A [ j ] = tmp ;
}
int s ;
if ( A [ i ] <= p ) s = i ; else s = i - 1;
int tmp = A [ l ];
A [ l ] = A [ s ];
A [ s ] = tmp ;
return s ;
}```

Example

Sort the array:

5, 3, 1, 9, 8, 2, 4, 7. Worst case time complexity: O(n 2 ) (quadratic). The worst case happens when the array is already sorted for example.

Average case time complexity: O(n log n) (sub-quadratic).

Space complexity: O(1).

Example

The array {(2, A),(2, B),(1, C)} will be sorted as {(1, C),(2, B),(2, A)}.

### Specialized sorting algorithms

• We have seen that the best comparison-base algorithms are O(n log n) worst case. Can we do better?

• The answer is no. It can be proved that: no comparison-based algorithm can sort an array in less than O(n log n) in the worst case.

• This result is for general data, but when the keys are of special type (like small positive integers or strings), we can find faster sorting algorithms.

• Specialized sorting algorithms do not use comparison in general and can only be used with specific types of keys.

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