Dulmage–Mendelsohn decomposition
Okay kiddo, so let's say you have a big group of friends and you want to play a game where everyone can play together. But some friends don't really get along with each other very well, so you want to split everyone up into smaller groups where they all like each other and can have fun playing the game.
That's kind of what the Dulmage-Mendelsohn decomposition is all about. It's a way to take a big group of things (like people, or numbers, or whatever) and split them up into smaller groups where they all work well together.
But instead of just randomly splitting everyone up, we use something called a "bipartite graph" to help us figure out which things should go together.
Okay, now you're probably wondering what a bipartite graph is. Basically, imagine you have two groups of things - let's say some red balls and some blue balls. Now imagine you want to connect each red ball to some of the blue balls. But you don't want any two red balls to be connected to the same blue ball, and you don't want any two blue balls to be connected to the same red ball.
So you draw a bunch of lines between the red and blue balls, like this:
```
(red ball 1)---(blue ball 1)
(red ball 1)---(blue ball 2)
(red ball 2)---(blue ball 2)
(red ball 3)---(blue ball 3)
(red ball 4)---(blue ball 3)
(red ball 5)---(blue ball 4)
```
This kind of "two-part" graph is called a "bipartite graph."
Now, imagine that each red ball represents one person, and each blue ball represents one hobby that person likes. We can use the bipartite graph to figure out which people like the same hobbies. For example, we can see that:
- Person 1 likes hobbies 1 and 2.
- Person 2 likes hobby 2.
- Person 3 likes hobby 3.
- Person 4 likes hobby 3.
- Person 5 likes hobby 4.
Now, here comes the cool part. The Dulmage-Mendelsohn decomposition is a way to split up these people and hobbies into smaller groups, where everyone in each group likes each other.
To do this, we start by "covering" as many hobbies as possible with just one person. In other words, we try to find a group of hobbies that only one person likes, and we put that person in a group by themselves.
In this case, we can see that Person 1 is the only one who likes hobbies 1 and 2, so we put them in a group together:
Group 1: {Person 1}
Next, we look for any "uncovered" hobbies - hobbies that nobody in Group 1 likes. In this case, there's only one uncovered hobby: hobby 2. And Person 2 is the only one who likes hobby 2, so we put them in a group together:
Group 2: {Person 2}
Now we look for uncovered hobbies again. This time, there are two uncovered hobbies: hobbies 3 and 4. But Person 3 and Person 4 both like hobby 3, so we put them in a group together:
Group 3: {Person 3, Person 4}
That leaves just hobby 4, which only Person 5 likes, so they go in a group by themselves:
Group 4: {Person 5}
And that's it - we've "decomposed" the big group of people and hobbies into smaller groups where everyone likes each other! In this case, we ended up with four groups:
- Group 1: {Person 1}
- Group 2: {Person 2}
- Group 3: {Person 3, Person 4}
- Group 4: {Person 5}
And that's basically what the Dulmage-Mendelsohn decomposition is all about. Using the bipartite graph to help us split up a big group into smaller groups where everyone gets along.
Does that make sense, kiddo?
That's kind of what the Dulmage-Mendelsohn decomposition is all about. It's a way to take a big group of things (like people, or numbers, or whatever) and split them up into smaller groups where they all work well together.
But instead of just randomly splitting everyone up, we use something called a "bipartite graph" to help us figure out which things should go together.
Okay, now you're probably wondering what a bipartite graph is. Basically, imagine you have two groups of things - let's say some red balls and some blue balls. Now imagine you want to connect each red ball to some of the blue balls. But you don't want any two red balls to be connected to the same blue ball, and you don't want any two blue balls to be connected to the same red ball.
So you draw a bunch of lines between the red and blue balls, like this:
```
(red ball 1)---(blue ball 1)
(red ball 1)---(blue ball 2)
(red ball 2)---(blue ball 2)
(red ball 3)---(blue ball 3)
(red ball 4)---(blue ball 3)
(red ball 5)---(blue ball 4)
```
This kind of "two-part" graph is called a "bipartite graph."
Now, imagine that each red ball represents one person, and each blue ball represents one hobby that person likes. We can use the bipartite graph to figure out which people like the same hobbies. For example, we can see that:
- Person 1 likes hobbies 1 and 2.
- Person 2 likes hobby 2.
- Person 3 likes hobby 3.
- Person 4 likes hobby 3.
- Person 5 likes hobby 4.
Now, here comes the cool part. The Dulmage-Mendelsohn decomposition is a way to split up these people and hobbies into smaller groups, where everyone in each group likes each other.
To do this, we start by "covering" as many hobbies as possible with just one person. In other words, we try to find a group of hobbies that only one person likes, and we put that person in a group by themselves.
In this case, we can see that Person 1 is the only one who likes hobbies 1 and 2, so we put them in a group together:
Group 1: {Person 1}
Next, we look for any "uncovered" hobbies - hobbies that nobody in Group 1 likes. In this case, there's only one uncovered hobby: hobby 2. And Person 2 is the only one who likes hobby 2, so we put them in a group together:
Group 2: {Person 2}
Now we look for uncovered hobbies again. This time, there are two uncovered hobbies: hobbies 3 and 4. But Person 3 and Person 4 both like hobby 3, so we put them in a group together:
Group 3: {Person 3, Person 4}
That leaves just hobby 4, which only Person 5 likes, so they go in a group by themselves:
Group 4: {Person 5}
And that's it - we've "decomposed" the big group of people and hobbies into smaller groups where everyone likes each other! In this case, we ended up with four groups:
- Group 1: {Person 1}
- Group 2: {Person 2}
- Group 3: {Person 3, Person 4}
- Group 4: {Person 5}
And that's basically what the Dulmage-Mendelsohn decomposition is all about. Using the bipartite graph to help us split up a big group into smaller groups where everyone gets along.
Does that make sense, kiddo?