What are good examples of problems that graphs can solve better than the alternative?

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?

Computer Networks: Graphs model intuitively model computer networks and the Internet. Often nodes will represent end-systems or routers, while edges represent connections between these systems.

Data Structures: Any data structure that makes use of pointers to link data together is making use of a graph of some kind. This includes tree structures and linked lists which are used all the time.

Pathing and Maps: Trying to find shortest or longest paths from some location to a destination makes use of graphs. This can include pathing like you see in an application like Google maps, or calculating paths for AI characters to take in a video game, and many other similar problems.

Constraint Satisfaction: A common problem in AI is to find some goal that satisfies a list of constraints. For example, for a University to set it's course schedules, it needs to make sure that certain courses don't conflict, that a professor isn't teaching two courses at the same time, that the lectures occur during certain timeslots, and so on. Constraint satisfaction problems like this are often modeled and solved using graphs.

Molecules: Graphs can be used to model atoms and molecules for studying their interaction and structure among other things.

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.

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.

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.

Well, many program optimization algorithms that compilers use are based on graphs (e.g., figure out call graph, flow control, lots of static analysis).

Many optimization problems are based on graph. Since many problems are reducable to graph colouring and similar problems, then many other problems are also graph based.

I'm not sure I agree that graphs are the best way to represent every relation and I certainly try to avoid these "got a nail, let's find a hammer" approaches. Graphs often have poor memory representations and many algorithms are actually more efficient (in practice) when implemented with matrices, bitsets, and other things.

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.

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

One popular example is garbage collection.

The collector starts with a set of references, then traverses all the objects they reference, then all the objects referenced there and so on. Everything it finds is added into a graph of reachable objects. All other objects are unreachable and collected.

To find out if two molecules can fit together. When developing drugs one is often interested in seeing if the drug molecules can fit into larger molecules in the body. The problem with determining whether this is possible is that molecules are not static. Different parts of the molecule can rotate around their chemical bindings so that a molecule can change into quite a lot of different shapes.

Each shape can be said to represent a point in a space consisting of shapes. Solving this problem involves finding a path through this space. You can do that by creating a roadmap through space, which is essentially a graph consisting of legal shapes and saying which shape a shape can turn into. By using a A* graph search algorithm through this roadmap you can find a solution.

Okay that was a lot of babble that perhaps wasn't very understandable or clear. But my point was that graphs pop up in all kinds of problems.

Graphs are not data structures. They are mathematical representation of relations. Yes, you can think and theoretize about problems using graphs, and there is a large body of theory about it. But when you need to implement an algorithm, you are choosing data structures to best represent the problem, not graphs. There are many data structures that represent general graphs, and even more for special kinds of graphs.

In your question, you mix these two things. The same theoretical solution may be in terms of graph, but practical solutions may use different data structures to represent the graph.

Graphs are great for managing dependencies.

I recently started to use the Castle Windsor Container, after inspecting the Kernel I found a GraphNodes property. Castle Windsor uses a graph to represent the dependencies between objects so that injection will work correctly. Check out this article.

I have also used simple graph theory to develop a plugin framework, each graph node represent a plugin, once the dependencies have been defined I can traverse the graph to create a plugin load order.

I am planning on changing the algorithm to implement Dijkstra's algorithm so that each plugin is weighted with a specific version, thus a simple change will only load the latest version of the plugin.

I with I had discovered this sooner. I like that quote "Whenever someone gives you a problem, think graphs." I definitely think that's true.

IMHO most of the domain models we use in normal applications are in some respect graphs. Already if you look at the UML diagrams you would notice that with a directed, labeled graph you could easily translate them directly into a persistence model. There are some examples of that over at [Neo4j] 1

Cheers

/peter

Social connections between people make an interesting graph example. I've tried to model these connections at the database level using a traditional RDMS but found it way too hard. I ended up choosing a graph database and it was a great choice because it makes it easy to follow connections (edges) between people (nodes).

Profiling and/or Benchmarking algorithms and implementations in code.

You could take a look at some of the examples in the Neo4j wiki, > http://wiki.neo4j.org/content/Domain_Modeling_Gallery

and the projects that Neo4j is used in (the known ones)

> http://wiki.neo4j.org/content/Neo4j_In_The_Wild .

Otherwise, Recommender Algorithms are a good use for Graphs, see for instance PageRank, and other stuff at

> https://github.com/tinkerpop/gremlin/wiki/pagerank

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.

You can utilise graphs anywhere you can define the problem domain objects into nodes and the solution as the flow of control and/or data amongst the nodes.

Considering the fact that trees are indeed connected-acyclic graphs, there are even more areas you can use the graph theory.

Analysing transaction serialisability in database theory.

Basically nearlly all common data structures like trees, lists, queues, etc, can be thought as type of graph, some with different type of constraint.

To my experiences, I have used graph intensively in network flow problems, which is used in lots of areas like telecommunication network routing and optimisation, workload assignment, matching, supply chain optimisation and public transport planning.

Another interesting area is social network modelling as previous answer mentioned.

There are far more, like integrated circuit optimisation, etc.