I haven't done an algorithms post in awhile, so the usual disclaimer first: If you don't find programming algorithms interesting, stop reading. This post is not for you.

On the other hand, if you do find algorithms interesting, in addition to this post, you might also want to read my other posts with the algorithms tag.

One common reason for this problem could be fault-tolerant computing. You perform multiple redundant computations and then verify that a majority of the results agree.

Their solution boils down to an observation (with proof) that the first phase of Boyer-Moore can be solved by combining the results for sub-sequences of the original input as long as both the candidate

[1, 1, 1, 2, 1, 2, 1, 2, 2] (Majority = 1)

If you were to run Boyer-Moore's first pass on this, you'd end up with:

candidate = 1

count = 1

If you were to split the array up into two parts and run Boyer-Moore on each of them, you'd get something like:

split 1:

[1, 1, 1, 2, 1]

candidate = 1

count = 3

split 2:

[2, 1, 2, 2]

candidate = 2

count = 2

You can then basically run Boyer-Moore over the resulting candidate, count pairs the same as you would if it were a list containing only the value candidate repeated count times. So for instance, Part 1's result could be considered the same as [1, 1, 1] and Part 2's as [2, 2]. However knowing that these are the same value repeated means you can generate the result for each part in constant time using something like the following python:

candidate = 0

count = 0

for candidate_i, count_i in parallel_output:

if candidate_i = candidate

count += count_i

else if count_i > count:

count = count_i - count

candidate = candidate_i

else

count = count - count_i

This algorithm can be run multiple times as well to combine parallel outputs in a tree-like fashion if necessary for additional performance.

As a final step, a distributed count needs to be performed to verify the final candidate.

###

On the other hand, if you do find algorithms interesting, in addition to this post, you might also want to read my other posts with the algorithms tag.

### Problem Statement

Imagine that you have a non-sorted list of values. You want to know if there is a value that is present in the list for more than half of the elements in that list. If so what is that value? If not, you need to know that there is no majority element. You want to accomplish this as efficiently as possible.One common reason for this problem could be fault-tolerant computing. You perform multiple redundant computations and then verify that a majority of the results agree.

### Simple Solution

Sort the list, if there is a majority value it must now be the middle value. To confirm it's the majority, run another pass through the list and count it's frequency.

The simple solution is O(n lg n) due to the sort though. We can do better!

### Boyer-Moore Algorithm

The Boyer-Moore algorithm is presented in this paper: Boyer-Moore Majority Vote Algorithm. The algorithm uses O(1) extra space and O(N) time. It requires exactly 2 passes over the input list. It's also quite simple to implement, though a little trickier to understand how it works.

In the first pass, we generate a single candidate value which is the majority value if there is a majority. The second pass simply counts the frequency of that value to confirm. The first pass is the interesting part.

In the first pass, we need 2 values:

- A candidate value, initially set to any value
- A count, initially set to zero.

For each element in our input list, we first examine the count value. If the count is equal to 0, we set the candidate to the value at the current element. Next, first compare the element's value to the current candidate value. If they are the same, we increment count by 1. If they are different, we decrement count by 1.

In python:

candidate = 0

count = 0

for value in input:

if count == 0:

candidate = value

if candidate == value:

count +=1

else:

count -= 1

At the end of all of the inputs, the candidate will be the majority value if a majority value exists. A second O(N) pass can verify that the candidate is the majority element (an exercise left for the reader).

#### Explanation

To see how this works, we only need to consider cases that contain a majority value. If the list does not contain a majority value, the second pass will trivially reject the candidate.

First, consider a list where the first element is not the majority value, for example this list with majority value 0:

[5, 5,

When processing the first element, we assign the value of 5 to candidate and 1 to count. Since 5 is not the majority value, at some point in the list our algorithm

List Values:

[5, 5, 0, 0, ...

Count value:

[1, 2, 1, 0, ...

At the point that count returns to zero, we have consumed exactly the same number of 5's as other elements. If all of the other elements were the majority element as in this case, we've consumed 2 majority elements and 2 non-majority elements. This is the largest number of majority elements we could have consumed, but even still the majority element must still be a majority of the

We can see similarly that if the first element was a majority element and count at some point drops to zero, then we can also see that the majority element is still the majority of the remainder of the input list since again we have consumed an equal number of majority and non-majority elements.

This in turn demonstrates that the range of elements from the time candidate is first assigned to when count drops to zero can be discarded from the input without affecting the final result of the first pass of the algorithm. We can repeat this over and over again discarding ranges that prefix our input until we find a range that is a suffix of our input where count never drops to zero.

Given an input list suffix where count never drops to zero, we must have more values that equal the first element than values that do not. Hence, the first element (candidate) must be the majority of that list and is the only possible candidate for the majority of the full input list, though it is still possible there is no majority at all.

First, consider a list where the first element is not the majority value, for example this list with majority value 0:

[5, 5,

**0**,**0**,**0**, 5,**0**,**0**, 5]When processing the first element, we assign the value of 5 to candidate and 1 to count. Since 5 is not the majority value, at some point in the list our algorithm

__must__find another value to pair with every 5 we've seen so far, thus count will drop to zero at some point before the last element in the list. In the above example, this occurs at the 4th' element:List Values:

[5, 5, 0, 0, ...

Count value:

[1, 2, 1, 0, ...

At the point that count returns to zero, we have consumed exactly the same number of 5's as other elements. If all of the other elements were the majority element as in this case, we've consumed 2 majority elements and 2 non-majority elements. This is the largest number of majority elements we could have consumed, but even still the majority element must still be a majority of the

*remainder*of the input list (in our example, the remainder is ...**0**, 5,**0**,**0**, 5]). If some of the other elements were not majority elements (for example, if the value was 4 instead), this would be even more true.We can see similarly that if the first element was a majority element and count at some point drops to zero, then we can also see that the majority element is still the majority of the remainder of the input list since again we have consumed an equal number of majority and non-majority elements.

This in turn demonstrates that the range of elements from the time candidate is first assigned to when count drops to zero can be discarded from the input without affecting the final result of the first pass of the algorithm. We can repeat this over and over again discarding ranges that prefix our input until we find a range that is a suffix of our input where count never drops to zero.

Given an input list suffix where count never drops to zero, we must have more values that equal the first element than values that do not. Hence, the first element (candidate) must be the majority of that list and is the only possible candidate for the majority of the full input list, though it is still possible there is no majority at all.

### Fewer Comparisons

The above algorithm makes 2 passes through our list, and so requires 2N comparisons in the worst case. It requires another N more if you consider the comparisons of count to 0. There is another, more complicated, algorithm that operates using only 3N/2 - 2 comparisons, but requires N additional storage. The paper (Finding a majority among N votes) also proves that 3N/2 - 2 is optimal.

Their approach is to rearrange all of the elements so that no two adjacent elements have the same value and keep track of the leftovers in a "bucket".

In the first pass, you start with an empty rearranged list and an empty "bucket". You take elements from your input and compare with the last element on the rearranged list. If they are equal you place the element in the "bucket". If they are not equal, you add the element to the end of the list and then move one element from the bucket to the end of the list as well. The last value on your list at the end of this phase is your majority candidate.

In the second pass, you repeatedly compare the candidate to the last value on the list. If they are the same, you discard two values from the end of the list. If they are different, you discard the last value from the end of the list and a value from the bucket. In this way you always pass over two values with one comparison. If the bucket ever empties, you are done and have no majority element. If you remove all elements from the rearranged list without emptying the bucket your candidate is the majority element.

Given the extra complexity and storage, I doubt this algorithm would have better real performance than Boyer-Moore in all but some contrived cases where equality comparison is especially expensive.

### Distributed Boyer-Moore

Of course, Gregable readers probably know that I like to see if these things can be solved in parallel on multiple processors. It turns out that someone has done all of the fun mathematical proof to show how to solve this in parallel: Finding the Majority Element in Parallel.Their solution boils down to an observation (with proof) that the first phase of Boyer-Moore can be solved by combining the results for sub-sequences of the original input as long as both the candidate

**and**count values are preserved. So for instance, if you consider the following array:[1, 1, 1, 2, 1, 2, 1, 2, 2] (Majority = 1)

If you were to run Boyer-Moore's first pass on this, you'd end up with:

candidate = 1

count = 1

If you were to split the array up into two parts and run Boyer-Moore on each of them, you'd get something like:

split 1:

[1, 1, 1, 2, 1]

candidate = 1

count = 3

[2, 1, 2, 2]

candidate = 2

count = 2

You can then basically run Boyer-Moore over the resulting candidate, count pairs the same as you would if it were a list containing only the value candidate repeated count times. So for instance, Part 1's result could be considered the same as [1, 1, 1] and Part 2's as [2, 2]. However knowing that these are the same value repeated means you can generate the result for each part in constant time using something like the following python:

candidate = 0

count = 0

for candidate_i, count_i in parallel_output:

if candidate_i = candidate

count += count_i

else if count_i > count:

count = count_i - count

candidate = candidate_i

else

count = count - count_i

This algorithm can be run multiple times as well to combine parallel outputs in a tree-like fashion if necessary for additional performance.

As a final step, a distributed count needs to be performed to verify the final candidate.

### Birds of a Feather

If you are one of the handful of people interested in voting algorithms and advanced software algorithms like this, you are the type of person I'd like to see working with me at Google. If you send me your resume (ggrothau@gmail.com), I can make sure it gets in front of the right recruiters and watch to make sure that it doesn't get lost in the pile that we get every day.

## 3 comments:

This is a nice post, it explains the points in a nice simple way. Algorithms can be tricky for many programmers to learn, however when you break things down in an ABC format, the understanding becomes fun to learn as well as challenging.

I would like to share a link with you which you may find useful as well as intuitive. http://norvig.com/sudoku.html

This was the only page where I found a proof. Just 1 addition if I may. If there is a majority the suffix after reaching 0 for the last time must start with the majority element.(since we would reach 0 otherwise) Your proof that in the 0 ending blocks the majority element can't be majority is the key for proving that this algorithm works. Thanks again.

Really nice proof of this. I was missing the key that in blocks of elements which end in 0 count the majority element can't be in majority. Also if there is a majority element the block of elements after the last time count went to 0 must start with the majority element.

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