Algorithm to find a duplicate entry in constant space and O(n) time

Question

Given an array of N integer such that only one integer is repeated. Find the repeated integer in O(n) time and constant space. There is no range for the value of integers or

Answer 1

Jun Tarui has shown that any duplicate finder using O(log n) space requires at least Ω(log n / log log n) passes, which exceeds linear time. I.e. your question is provably unsolvable even if you allow logarithmic space.

There is an interesting algorithm by Gopalan and Radhakrishnan that finds duplicates in one pass over the input and O((log n)^3) space, which sounds like your best bet a priori.

Radix sort has time complexity O(kn) where k > log_2 n often gets viewed as a constant, albeit a large one. You cannot implement a radix sort in constant space obviously, but you could perhaps reuse your input data's space.

There are numerical tricks if you assume features about the numbers themselves. If almost all numbers between 1 and n are present, then simply add them up and subtract n(n+1)/2. If all the numbers are primes, you could cheat by ignoring the running time of division.

As an aside, there is a well-known lower bound of Ω(log_2(n!)) on comparison sorting, which suggests that google might help you find lower bounds on simple problems like finding duplicates as well.