## Wednesday, October 31, 2012

### Finding the majority element in a stream of numbers

Some time ago I came across the following question.
As input a finite stream stream of numbers is provided. Define an algorithm to find the majority element of the input. The algorithm need not provide a sensible result if no majority element exists. You may assume a transdichotomous memory model.
There are a few definitions which may not be immediately clear:
Stream
A possibly infinite set of data which may not be reused in either the forward or backward direction without explicitly storing it.
Majority element
An element in a set which occurs more than half the time.
Transdichotomous
The integer size is equal to the word size of memory. One does not need to worry about storing partial pieces of integers in separate memory units.
Unfortunately this answer isn't of my own invention, but it is interesting and succinct.

The algorithm (click to view) Using 3 registers the accumulator, the guess and the current element (next):
1. Initialize accumulator to 0
2. Accept the next element of the stream and place it into next. If there are no more elements go to step #7.
3. If accumulator is 0 place next into guess and increment accumulator.
4. Else if guess matches next increment accumulator
5. Else decrement accumulator
6. Go to step 2
7. Return the value in guess as the result

An interesting property of this algorithm is that it can be implemented in $O(n)$ time even on a single tape Turing Machine.