## Sorting

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There are many sorting algorithms with just a few listed on the left. Sorting is an important problem that is easily understood and has interesting, mostly short solutions. Selection sort is one of the simpler algorithms.

## Selection Sort

Selection sort maintains a growing 'front' section of the array which is (i) sorted and (ii) less than the remainder of the array. At each step, the smallest element in the 'remainder' is selected and moved to enlarge the 'front' section.

```selection(int a[], int N)  /* in C */
/* sort a[1..N],  NB. 1 to N */
{ int i, j, smallest, aSmallest, temp;

for(i=1; i < N; i++)
{ /* invariant: a[1..i-1] sorted
and elements a[1..i-1] <= a[i..N] */

smallest = i; /* find smallest in a[i..N] */
aSmallest = a[i];

for(j=i+1; j <= N; j++)
/* a[smallest] is the least element in a[i..j-1] */
if(a[j] < aSmallest)
{ smallest=j; aSmallest=a[j]; }
/* a[smallest] is the least element in a[i..j] */

temp=a[i]; a[i]=a[smallest]; a[smallest]=temp; /*swap*/
/* a[1..i] sorted and elements a[1..i] <= a[i+1..N] */
}

/* a[1..N-1] sorted and elements a[1..N-1] <= a[N] */
/* i.e. a[1..N] sorted. */
}/*selection*/
```

At some intermediate stage, a[1..i-1] is sorted and, on an element by element basis, less than everything in a[i..N]. Find the smallest element remaining in a[i..N]:

```
select
smallest
-------
a:  1  2  3  6  5  4
-------  ^
sorted   |
& small  |
i

```

Do this by examining a[i], a[i+1], ..., a[N]:

```
a:  1  2  3  6  5  4
^     ^
|     |
i     smallest

```

Swap a[i] with a[smallest]:

```
a:  1  2  3  4  5  6
----------
sorted   ^
& small  |
|
i

```

Now a[1..i] is sorted and less than everything remaining in a[i+1..N]. (Coincidentally a[1..N] happens to be sorted in this example.) Repeat until i=N-1.

### Selection Sort Demonstration

Try other example input in the HTML FORM below, press 'go' and experiment.

input:
output:
trace:

### Complexity

#### Time

The number of comparisons of elements is

```  (N-1) + (N-2) + ... + 1
= (N-1)*N/2
```
i.e. O(N2).

#### Space

The space-complexity is O(1), just a few scalar variables. NB. We do not count the size of the array being sorted because that is given, not created specifically for this algorithm.

### Stability

Selection sort is not stable, the is the relative order of equal keys is sometimes changed. It is the swap that can do it, consider [2a,2b,1]. (Thanks to Giri Narasimhan 6/4/'05.)

### Testing

Test sort programs on a few special cases:

• the empty array!
• an array of a single element,
• an array of a small number of elements,
• some already sorted data,
• some reverse sorted data,
• some random data (several sets),
• data with duplicate keys.

### Notes

• Depending on your language, you may find yourself sorting a[1..N] or a[0..N-1]; watch those indices!
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