Graph data structures - An introduction

With most problems it is largely pretty clear which data structure is going to be appropriate to use - If you just care about storing a list and iterating over it, then you probably want an Array based structure - if you particularly care about the elements being unique you could look as a Set. If you want to capture a key-value pair dictionary data structure then you can use a Map.

Graph data structures are no different, and are very applicable to a subset of problems, and once familiar with graphs and the common algorithms it becomes quite easy to quickly identify the problem type as a graph problem.

The basics

A graph has two main elements:
  • Node - a given data point in the graph
  • Edge - a connection that joins any two Nodes. A graph can be "directed" or "un-directed" - this simply determines whether the Edge goes both ways or is purely one way. 

A graph is a data strucutre that stores a set of connected elements - the easiest way to understand the structure is with a real world example, the most famous is probably like the social-network graph. If you think about your profile on any popular social network (Facebook/LinkedIn/etc), your profile is a Node in the graph, and each of your friendship/connections is an Edge to another Node in the graph

A while ago, a Facebook intern created a visualisation of the Facebook social graph around the world - you can't really make out the individual Nodes/Edges, but you get the idea.

In Facebook's graph, it is un-directed, that is, when you become friends with another Node in the graph the relationship goes both ways - you are their friend and they are yours.

Twitter, however, is a directed graph - once you follow someone an Edge is created between your Node and theirs, but they don't automatically follow you as well, so the Edge has direction.

If you are a LinkedIn user, you may have noticed whilst browsing another user's profile a widget saying something like

"X of your connections can introduce you to someone who knows Y"

What LinkedIn is telling you, is that the shortest path between your Node and Y's Node in the graph is 3 Edges (3 "hops" - following an Edge between nodes is often called a "hop") - To be able to do this, LinkedIn is searching the social graph space to discover the shortest path (and number of unique paths that are of the shortest distance) between you and this other user.

Graph representations

There are two primary graph representations:
  • Adjacency Matrix - This is a matrix/2-D array that captures the relationship between every node. Every node is mapped against the X and Y axis, and the value in the intersecting cell determine if their is an Edge between the nodes. E.g. if we wanted to know if there was an edge between Node "4" and node "13" we would look at matrix[4][13] - the value there would tell us. Normally, value of 1 represents an edge, but other values can be used (for example, if it is a weighted graph the values could represent the weights, or if it is a directed graph it could use -ve/+ve values to represent direction).  This representation is good for "dense" graphs.
  • Adjacency List - This representation is simply a List of all Edges and a List of all Nodes. This is a simple representation and is more memory efficient for "sparse" graphs

For the code samples here, I will focus on the Adjacency List representation

Graph representation - Java

Below is a very rudimentary implementation of a Graph class in Java. It uses a Adjacency List representation and will be used in later examples I go through.

Below is a sample unit test setup that shows how a simple graph can be initialised:

In the next post I will go through basic techniques for searching and exploring graph spaces, as well as a post looking at how to solve the LinkedIn shortest path recommendation problem.