# Determining the particular number in $O(n)$ time and space (worst case)

$\newcommand\ldotd{\mathinner{..}}$Given that $A[1\ldotd n]$ are integers such that $0\le A[k]\le m$ for all $1\le k\le n$, and the occurrence of each number except a particular number in $A[1\ldotd n]$ is an odd number. Try to find the number whose occurrence is an even number.

There is an $\Theta(n\log n)$ algorithm: we sort $A[1\ldotd n]$ into $B[1\ldotd n]$, and break $B[1\ldotd n]$ into many pieces, whose elements' value are the same, therefore we can count the occurrence of each element.

I want to find a worst-case-$O(n)$-time-and-$O(n)$-space algorithm.

Supposing that $m=\Omega(n^{1+\epsilon})$ and $\epsilon>0$, therefore radix sort is not acceptable. $\DeclareMathOperator{\xor}{xor}$ Binary bitwise operations are acceptable, for example, $A\xor A$.

• Aryabhata's answer below shows that the general case is not good, but perhaps you have further restrictions available? A simple (but big) restriction would be to enforce that all the entries in the array are $O(n)$ in size. This would give a pretty trivial linear algorithm. Jul 23 '12 at 4:02
• @LukeMathieson: I deleted that answer, as I am not yet convinced that the paper I cited will work without any modification, and besides, OP seems to be interested only in the uniform cost RAM model. Jul 23 '12 at 4:28
• @Aryabhata: hehe, well the answer that's not there then! Out of interesting, and perhaps useful for Frank, what did you think was the problem with adapting the result in the paper? A quick skim suggested it applied, but I obviously didn't read into it. Jul 23 '12 at 4:51
• @LukeMathieson: The fact that the other elements need to appear an odd number of times in the current problem. Since, I skimmed over the proof too... Jul 23 '12 at 5:36
• It would be interesting if you are interested in theoretic results or in practical solutions. From the theory point of view, my first quick response is, that you can sort a list of integers faster than $O(n\log n)$. There is a deterministic algorithm by Han that runs in $O(\log\log n)$ time. For randomized algorithms, even better results are known, e.g. Han and Thorup have found a $O(n \sqrt{\log\log n})$ expected time algorithm. However, I think that your problem shouldn't require sorting. Jul 23 '12 at 7:01

Here is an idea for a simple algorithm; just count all occurrences!

1. Find $m = \max A$. -- time $\Theta(n)$
2. "Allocate" array $C[0..m]$. -- time $O(1)$¹
3. Iterate over $A$ and increase $C[x]$ by one whenever you find $A[\_]=x$. If $C[x]$ was $0$, add $x$ to a linear list $L$. -- time $\Theta(n)$
4. Iterate over $L$ and find the element $x_e$ with $C[x_e]$ even. -- time $O(n)$.
5. Return $x_e$.

All in all, this gives you a linear-time algorithm which may use (in the sense of allocating) lots of memory. Note that being able to random-access $C$ in constant time independently of $m$ is crucial here.

An additional $O(n)$ bound on space is more difficult with this approach; I don't know of any dictionary data-structure that offers $O(1)$ time lookup. You can use hash-tables for which here are implementations with $O(1 + k/n)$ expected lookup time ($n$ the table's size, $k$ the number of stored elements) so you can get arbitrarily good with linear space -- in expectation. If all values in $A$ map to the same hash value, you are screwed.

1. On a RAM, this is implicitly done; all we need is the start position and maybe the end position.

An almost trivial solution - which uses however $\Theta(n)$ space - is to use a hash map. Recall that a hash map has amortized runtime $\mathcal{O}(1)$ for adding and looking up elements.

Hence, we can use the following algorithm:

1. Allocate a hash map $H$. Iterate over $A$. For each element $i \in A$, increase the number of occurences seen, i.e. $H(i)++$.

2. Iterate through the key set of the hash map, and check which of the keys has an even count of occurences.

Now this is a simple algorithm which doesn't really use any large trick, but sometimes even this suffices. If not, you might want to specifiy what space restrictions you impose.

• I would still like to know, if there is a non-randomized $O(n)$ time algorithm using polynomial space. In particular, is there any theoretical evidence that finding the only even-occurring item is harder than finding the only odd-occurring item? Jul 23 '12 at 11:01
• @A.Schulz I think that it is the $O(n)$-expected-time algorithm by using hash table. I remember that somebody told me an $O(n)$-algorithm (or for some special case, say, odd=1 and even=2) maybe with stack, but I cannot recall it. Jul 23 '12 at 12:11
• Not every hashtable implementation has this property; usually, lookup is not $O(1)$, not even amortized (afaik). In fact, a prior discussion has not yielded any implementation that has constant time lookup. Can you be more specific?
– Raphael
Jul 23 '12 at 15:21