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You can't achieve O(n) through bucket sort if you don't know if the input is random. In this problem it is not directly stated. If it was, bucket would seem to be a nice option indeed.
Radix works better for strings, sure longer than 10 digits. Constant factor for such sorting would be unnecessary big, make it slower than qsort.
And you have to be careful with count sort - its speed depends on range of numbers so,
instead of O(n log n) (for qsort n = 10^5) you get O(n) (where n = ai = 10^9).
then count works longer for big input.
Also, for input like
count sort still needs O(10^9) time.