An array of length N can contain values 1,2,3 … N^2. Is it possible to sort in O(n) time?

Question

Given an array of length N. It can contain values from ranging from 1 to N^2 (N squared) both inclusive, values are integral. Is it possible to sort this array in O(N) time?

Answer 1

It is possible to sort any array of integers with a well defined maximum value in O(n) time using a radix sort. This is likely the case for any list of integers you encounter. For example if you were sorting a list of arbitrary precision integers it wouldn't be true. But all the C integral types have well-defined fixed ranges.

Answer 2

Write each integer in base N, that is each x can be represented as (x1, x2) with x = 1 + x1 + x2*N. Now you can sort it twice with counting sort, once on x1 and once on x2, resulting in the sorted array.

Answer 3

Yes, you can, using radix sort with N buckets and two passes. Basically, you treat the numbers as having 2 digits in base N.