What are good examples of problems that graphs can

Question

After reading Stevey Yegge's Get That Job At Google article, I found this little quote interesting:

Whenever someone gives you a problem, think graphs. They are the most fundamental and flexible way of representing any kind of a relationship, so it's about a 50–50 shot that any interesting design problem has a graph involved in it. Make absolutely sure you can't think of a way to solve it using graphs before moving on to other solution types. This tip is important!

What are some examples of problems that are best represented and/or solved by graph data structures/algorithms?

One example I can think of: navigation units (ala Garmin, TomTom), that supply road directions from your current location to another, utilize graphs and advanced pathing algorithms.

What are some others?

Answer 1

I am very very interested in graph theory and ive used it solved so many different kinds of problem. You can solve a lot of Path related problem, matching problem, structure problems using graph.

• Path problems have a lot of applications.

  This was in a career cup's interview question. Say you want to find the longest sum of a sub array. For example, [1, 2, 3, -1] has the longest sum of 6. Model it as a Directed Acyclic Graph (DAG), add a dummy source, dummy destination. Connect each node with an edge which has a weight corresponding to the number. Now use the Longest Path algorithm in the DAG to solve this problem.

  Similarly, Arbitrage problems in financial world or even geometry problems of finding the longest overlapping structure is a similar path problem.

• Some obvious ones would be the network problems (where your network could have computers people, organisation charts, etc).
  You can glean a lot of structural information like

  • which point breaks the graph into two pieces
  • what is the best way to connect them
  • what is the best way to reach one place to another
  • is there a way to reach one place from another, etc.

• I've solved a lot of project management related problems using graphs. A sequence of events can be pictured as a directed graph (if you don't have cycles then thats even better). So, now you can

  • sort the events according to their priority
  • you can find the event that is the most crucial (that is would free a lot of other projects)
  • you can find the duration needed to solve the total project (path problem), etc.

• A lot of matching problems can be solved by graph. For example, if you need to match processors to the work load or match workers to their jobs. In my final exam, I had to match people to tables in restaurants. It follows the same principle (bipartite matching -> network flow algorithms). Its simple yet powerful.

• A special graph, a tree, has numerous applications in the computer science world. For example, in the syntax of a programming language, or in a database indexing structure.

• Most recently, I also used graphs in compiler optimization problems. I am using Morgan's Book, which is teaching me fascinating techniques.

The list really goes on and on and on. Graphs are a beautiful math abstraction for relation. You really can do wonders, if you can model it correctly. And since the graph theory has found so many applications, there are many active researches in the field. And because of numerous researches, we are seeing even more applications which is fuelling back researches.

If you want to get started on graph theory, get a good beginner discrete math book (Rosen comes to my mind), and you can buy books from authors like Fould or Even. CLRS also has good graph algorithms.

Answer 2

An example most people are familiar: build systems. Make is the typical example, but almost any good build system relies on a Directed Acyclic Graph. The basic idea is that the direction models the dependency between a source and a target, and you should "walk" the graph in a certain order to build things correctly -> this is an example of topological sort.

Another example is source control system: again based on a DAG. It is used for merging, for example, to find common parent.

Answer 3

The following are based on graph theory:

• Binary trees and other trees such as Red-black trees, splay trees, etc.
• Linked lists
• Anything that's modelled as a state machine (GUIs, network stacks, CPUs, etc)
• Decision trees (used in AI and other applications)
• Complex class inheritance

Answer 4

OCR. Picture a page of text scanned at an angle, with some noise in the image, where you must find the space between lines of text. One way is to make a graph of pixels, and find the shortest path from one side of the page to the other, where the difference in brightness is the distance between pixels.

This example is from the Algorithm Design Manual, which has lots of other real world examples of graph problems.

Answer 5

Anything that can be modelled as a foreign key in a relational database is essentially an edge and nodes in a graph.

Maybe that will help you think of examples, since most things are readily modelled in a RDBMS.

Answer 6

Your source code is tree structured, and a tree is a type of graph. Whenever you hear people talking about an AST (Abstract Syntax Tree), they're talking about a kind of graph.

Pointers form graph structures. Anything that walks pointers is doing some kind of graph manipulation.

The web is a huge directed graph. Google's key insight, that led them to dominate in search, is that the graph structure of the web is of comparable or greater importance than the textual content of the pages.

State machines are graphs. State machines are used in network protocols, regular expressions, games, and all kinds of other fields.

It's rather hard to think of anything you do that does not involve some sort of graph structure.