# Fun With Linear Time: My Favorite Algorithm

April 30, 2019

*Boyer–Moore majority vote algorithm* — I love this algorithm because it’s amazing *and* approachable. I first saw it on a LeetCode discuss thread, and it blew everyone away. Some people were, like, irate. 169. Majority Element.

This problem, common on competitive coding sites, has a solution that was discovered in 1980 but went unpublished until 1991 because of its emphasis on Fortran and mechanical verification.

**Problem:** Given a list of votes with a majority (*n*/2 + 1), declare the leader. As in, the most frequently occurring vote. It is possible to get the result in linear time *O(n)* AND constant space *O(1)*.

Examples:

`[0, 1, 1, 0, 1, 0, 1, 1] => 1 is the majority element`

`['a', 'b', 'a', 'c', 'b', 'c', 'a', 'a'] => 'a' is the majority here`

A naive solution might look like this. We’ll use a Dictionary to keep track of all the votes as well as storing the highest number of votes we’ve seen.

```
def majority_vote(votes):
leader = None
max_votes = 0
candidates = dict()
for i in votes:
# if seen before
if i in candidates:
# count their vote
candidates[i] += 1
# and check if they're leading
if candidates[i] > max_votes:
leader = i
max_votes = candidates[i]
else:
candidates[i] = 1
return leader
```

The above accomplishes a correct solution in linear time *O(n)* using linear space *O(n)*. We can do better. One pass, **without** counting every element.

*MJRTY* or *A Fast Majority Vote Algorithm* was discovered in the Computer Science Laboratory of SRI International in 1980 by Robert S. Boyer and J Strother Moore. They were assisting a colleague who was working on fault tolerance.

In their humorous paper, they imagined a convention center filled with voters, carrying placards boasting the name of their chosen candidate. Each voter representing an index of the list.

Suppose a floor fight ensues

They opined that the voters might knock each other out simultaneously, going only for the opposing team. After the mess, the voter/s left standing would represent the majority.

Here is a bloodless way the chairman can simulate the pairing phase.

Their algorithm improves on our naive solution by removing the data structure (the Dictionary). Converted here from Fortran to Python.

```
def majority_vote_improved(votes):
# after one vote, we have a leader
leader = votes[0]
count = 1
for i in range(1, len(votes)):
# the lead may grow
if votes[i] == leader:
count += 1
else:
# or shrink
count -= 1
# and they may be replaced
if count == 0:
leader = votes[i]
count = 1
return leader
```

Thus, the majority is found in linear time *O(n)* with constant space *O(1)*.

Check out a step-by-step walkthrough on Moore’s website. More on Wikipedia too.

The entire effort of specifying MJRTY and getting the 61 verification conditions proved required about 20 man hours. […] about 55 minutes of computer time to prove the final list of 66 theorems.

MJRTY - A Fast Majority Vote Algorithm, with R.S. Boyer. In R.S. Boyer (ed.), Automated Reasoning: Essays in Honor of Woody Bledsoe, Automated Reasoning Series, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991, pp. 105-117.

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