/*
Counting sort is a linear tine sorting algorithm that sort in O(n+k) time when elements are in range from 1 to k.

What if the elements are in range from 1 to n2? 
We can’t use counting sort because counting sort will take O(n2) which is worse than comparison based sorting algorithms.
Can we sort such an array in linear time?
Radix Sort is the answer. The idea of Radix Sort is to do digit by digit sort starting from least significant digit to 
most significant digit. Radix sort uses counting sort as a subroutine to sort.
*/

/*
The Radix Sort Algorithm
1) Do following for each digit i where i varies from least significant digit to the most significant digit.
………….a) Sort input array using counting sort (or any stable sort) according to the i’th digit.

Example:
Original, unsorted list:

170, 45, 75, 90, 802, 24, 2, 66
Sorting by least significant digit (1s place) gives: [*Notice that we keep 802 before 2, because 802 occurred before 2 
in the original list, and similarly for pairs 170 & 90 and 45 & 75.]

170, 90, 802, 2, 24, 45, 75, 66
Sorting by next digit (10s place) gives: [*Notice that 802 again comes before 2 as 802 comes before 2 in the previous 
list.]

802, 2, 24, 45, 66, 170, 75, 90
Sorting by most significant digit (100s place) gives:

2, 24, 45, 66, 75, 90, 170, 802

*/


/*
Let there be d digits in input integers. Radix Sort takes O(d*(n+b)) time where b is the base for representing numbers, 
for example, for decimal system, b is 10. What is the value of d? If k is the maximum possible value, then d would be
O(logb(n)).

So overall time complexity is O((n+b)*\log_b(k)). Which looks more than the time complexity of comparison based sorting 
algorithms for a large k. Let us first limit k. Let k <= nc where c is a constant. In that case, the complexity becomes 
O(n\log_b(n)). But it still doesn’t beat comparison based sorting algorithms.
What if we make value of b larger?. What should be the value of b to make the time complexity linear? If we set b as n, 
we get the time complexity as O(n). In other words, we can sort an array of integers with range from 1 to nc if the 
numbers are represented in base n (or every digit takes \log_2(n) bits).

Is Radix Sort preferable to Comparison based sorting algorithms like Quick-Sort?
If we have \log_2(n) bits for every digit, the running time of Radix appears to be better than Quick Sort for a wide 
range of input numbers. The constant factors hidden in asymptotic notation are higher for Radix Sort and Quick-Sort uses 
hardware caches more effectively. Also, Radix sort uses counting sort as a subroutine and counting sort takes extra space
to sort numbers.


*/

Create RadixSort.java