## Sorting

Counting Sort

Counting sort is a sorting technique based on keys between a specific range. It works by counting the number of objects having distinct key values (kind of hashing).

```For simplicity, consider the data in the range 0 to 9.
Input data: 1, 4, 1, 2, 7, 5, 2
1) Take a count array to store the count of each unique object.
Index:     0  1  2  3  4  5  6  7  8  9
Count:     0  2  2  0   1  1  0  1  0  0

2) Modify the count array such that each element at each index
stores the sum of previous counts.
Index:     0  1  2  3  4  5  6  7  8  9
Count:     0  2  4  4  5  6  6  7  7  7

The modified count array indicates the position of each object in
the output sequence.

3) Output each object from the input sequence followed by
decreasing its count by 1.
Process the input data: 1, 4, 1, 2, 7, 5, 2. Position of 1 is 2.
Put data 1 at index 2 in output. Decrease count by 1 to place
next data 1 at an index 1 smaller than this index.```

```// Java implementation of Counting Sort
class CountingSort
{
void sort(char arr[])
{
int n = arr.length;

// The output character array that will have sorted arr
char output[] = new char[n];

// Create a count array to store count of inidividul
// characters and initialize count array as 0
int count[] = new int[256];
for (int i=0; i<256; ++i)
count[i] = 0;

// store count of each character
for (int i=0; i<n; ++i)
++count[arr[i]];

// Change count[i] so that count[i] now contains actual
// position of this character in output array
for (int i=1; i<=255; ++i)
count[i] += count[i-1];

// Build the output character array
for (int i = 0; i<n; ++i)
{
output[count[arr[i]]-1] = arr[i];
--count[arr[i]];
}

// Copy the output array to arr, so that arr now
// contains sorted characters
for (int i = 0; i<n; ++i)
arr[i] = output[i];
}

// Driver method
public static void main(String args[])
{
CountingSort ob = new CountingSort();
char arr[] = {'g', 'e', 'e', 'k', 's', 'f', 'o',
'r', 'g', 'e', 'e', 'k', 's'
};

ob.sort(arr);

System.out.print("Sorted character array is ");
for (int i=0; i<arr.length; ++i)
System.out.print(arr[i]);
}
}
```

Time Complexity: O(n+k) where n is the number of elements in input array and k is the range of input.
Auxiliary Space: O(n+k)

Points to be noted:
1. Counting sort is efficient if the range of input data is not significantly greater than the number of objects to be sorted. Consider the situation where the input sequence is between range 1 to 10K and the data is 10, 5, 10K, 5K.
2. It is not a comparison based sorting. It running time complexity is O(n) with space proportional to the range of data.
3. It is often used as a sub-routine to another sorting algorithm like radix sort.
4. Counting sort uses a partial hashing to count the occurrence of the data object in O(1).
5. Counting sort can be extended to work for negative inputs also.

Counting sort for negative numbers:   Calculate the minimum value in the array and alter the value of the array elements by subtracting the minimum value.
eg. -2 – (-2) = 0

Now all values are greater than equal to zero. Apply counting sort and in the end add the minimum value to array elements to regain the original array.

http://stackoverflow.com/questions/11001797/using-counting-sort-on-negative-values