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Manipulating C++ Graph Data Structures with the Boost Graph Library

Quickly becoming a de facto standard C++ library, the Boost library includes a powerful graph data structure that's also easy to use. Jeff Cogswell discusses some interesting theory behind graphs, and explains the Boost library's graph structures.
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In advanced mathematics, a graph is a special data structure made up of lines (called edges) connected together at endpoints (called vertices or nodes). This definition of the term graph is in contrast to a popular definition used in early math classes, where a graph is simply a plot of a function.

Many computer science and mathematical algorithms make use of graphs. Because graphs are so important in many algorithms, a data structure for a graph is equally important. Most languages don’t have graph data structures built in; indeed, C++ doesn’t. Therefore, when you need a graph data structure, you either have to code one yourself or make use of a third-party library offering a graph data structure.

The Boost library is quickly becoming a de facto standard C++ library, and it includes a graph data structure that’s easy to use yet powerful. In this article, I discuss the theory behind graphs, and then we explore the Boost library’s graph structures.

Grasping Graph Theory

As I already mentioned, a graph is simply a set of vertices (the points) connected by edges (the lines). When two vertices are connected by an edge, those two vertices are said to be adjacent. If you enjoy mathematics, you might like to know that this definition can be described a bit more technically by stating that a graph consists of a finite set of vertices along with a finite set of unordered pairs of distinct vertices; you can see that the pairs of vertices represent the edges of the graph.

Graphs provide a visual way of displaying underlying information. For example, a large airline serves many cities. The airline’s web site might provide a map showing cities as dots, and lines connecting the cities representing flights between those cities. This map could be considered a graph. Similar graphs might include those showing a trucking company’s delivery routes between cities, or bus routes between cities.

Although, technically speaking, a graph consists simply of vertices connected by edges, this limited definition isn’t particularly useful. Often, graphs are more useful when they have more associated information. For example, a flight map might also show the flight numbers associated with each flight. In this representation, a graph consists of vertices, edges, and numbers associated with the edges. When numbers are associated with edges, such graphs are called weighted graphs.

Certainly, some graphs might need more than one piece of data associated with each edge. For example, in addition to the flight numbers, the airline map might show the length of each flight. In terms of computer science, however, such information would still be considered individual pieces of data in the form of individual structures or objects associated with each edge, with each structure containing a flight number and a flight time

Similarly, a graph might need to have information attached to the vertices, such as city names in the case of the airline map. In mathematics, you can attach numbers to vertices and use such numbers in calculations.

Another somewhat limiting aspect of a traditional definition of a graph is that the edges connecting the vertices don’t have directions. In real life, a system that can be modeled by a graph will likely have directions. For example, if you travel by train somewhere, you definitely want to feel comfortable that the tracks you’re riding on from one city to the next are carrying the train in a single direction! When you add directions to graphs, you have what mathematicians call a directed graph, or digraph for short.

Important to graph theory is another type of graph, limited by the restriction that the graph can be drawn on a sheet of paper without any edges crossing. Such a graph is called a planar graph. The term planar is used because the graph can be drawn on a single plane. Of course, this plane might not always be flat in the geometrical sense; the plane could exist on a curved surface such as the surface of the Earth, in which case it still maps to two dimensions. That’s getting into another interesting area of mathematics called topology, however, which we won’t cover here.

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