Facebook has a list of friends (note that friends are a bi-directional thing on Facebook. If I'm your friend, you're mine). They also have lots of disk space and they serve hundreds of millions of requests everyday. They've decided to pre-compute calculations when they can to reduce the processing time of requests. One common processing request is the "You and Joe have 230 friends in common" feature. When you visit someone's profile, you see a list of friends that you have in common. This list doesn't change frequently so it'd be wasteful to recalculate it every time you visited the profile (sure you could use a decent caching strategy, but then I wouldn't be able to continue writing about mapreduce for this problem). We're going to use mapreduce so that we can calculate everyone's common friends once a day and store those results. Later on it's just a quick lookup. We've got lots of disk, it's cheap.

Assume the friends are stored as Person->[List of Friends], our friends list is then:

A ->B C D
B ->A C D E
C ->A B D E
D ->A B C E
E ->B C D

Each line will be an argument to a mapper. For every friend in the list of friends, the mapper will output a key-value pair. The key will be a friend along with the person. The value will be the list of friends. The key will be sorted so that the friends are in order, causing all pairs of friends to go to the same reducer. This is hard to explain with text, so let's just do it and see if you can see the pattern. After all the mappers are done running, you'll have a list like this:

For map(A -> B C D) :

(A B) -> B C D
(A C) -> B C D
(A D) -> B C D

For map(B -> A C D E) : (Note that A comes before B in the key)

(A B) -> A C D E
(B C) -> A C D E
(B D) -> A C D E
(B E) -> A C D E
For map(C -> A B D E) :

(A C) -> A B D E
(B C) -> A B D E
(C D) -> A B D E
(C E) -> A B D E
For map(D -> A B C E) :

(A D) -> A B C E
(B D) -> A B C E
(C D) -> A B C E
(D E) -> A B C E
And finally for map(E -> B C D):

(B E) -> B C D
(C E) -> B C D
(D E) -> B C D
Before we send these key-value pairs to the reducers, we group them by their keys and get:

(A B) -> (A C D E) (B C D)
(A C) -> (A B D E) (B C D)
(A D) -> (A B C E) (B C D)
(B C) -> (A B D E) (A C D E)

Each line will be passed as an argument to a reducer. The reduce function will simply intersect the lists of values and output the same key with the result of the intersection. For example, reduce((A B) -> (A C D E) (B C D)) will output (A B) : (C D) and means that friends A and B have C and D as common friends.

The result after reduction is:

(A B) -> (C D)
(A C) -> (B D)
(A D) -> (B C)
(B C) -> (A D E)
(B D) -> (A C E)
(B E) -> (C D)
(C D) -> (A B E)
(C E) -> (B D)
(D E) -> (B C)

Now when D visits B's profile, we can quickly look up(B D)and see that they have three friends in common,(A C E).