A graph is a data structure that represents a set of
entities, called nodes, connected by a set of links. A node can also
be referred to as a vertex. A link can also be referred to as an
edge or a connection. In practical applications, graphs are frequently
used to model a wide range of things: computer networks, software
program structures, project management diagrams, and so on. Graphs
are powerful models because they permit applications to benefit from
the results of graph theory research. For example, efficient methods
are available for finding the shortest path between two nodes, the
minimum cost path, and so on.
Layout of a graph
Graph layout is used in graphical user interfaces of
applications that need to display graph models. To lay out a graph
means to draw the graph so that an appropriate, readable representation
is produced. Essentially, it involves determining the location of
the nodes and the shape of the links. For some applications, the location
of the nodes is already be known (for example, based on the geographical
positions of the nodes). For other applications, the location is not
known (a pure “logical” graph) or the known location,
if used, would produce an unreadable drawing of the graph. In these
cases, the location of the nodes must be computed.
What is meant by an “appropriate” drawing
of a graph? In practical applications, it is often necessary for the
graph drawing to observe certain quality criteria. These criteria
can vary depending on the application field or on a given standard
of representation. It is often difficult to tell what a good layout
consists of. Each user can have different, subjective criteria for
qualifying a layout as “good”. However, one common goal
exists behind all the criteria and standards: the drawing must be
easy to understand and provide easy navigation through the complex
structure of the graph.
What is a good layout?
To deal with the various needs of different applications,
many classes of graph layout algorithms have been developed. A layout
algorithm addresses one or more quality criteria, depending on the
type of graph and the features of the algorithm, when laying out a
graph.
The most common criteria are:
- Minimizing the number of link crossings
- Minimizing the total area of the drawing
- Minimizing the number of bends (in orthogonal drawings)
- Maximizing the smallest angle formed by consecutive incident links
- Maximizing the display of symmetries
How can a layout algorithm meet each of these quality
criteria and standards of representation? If you look at each individual
criteria, some can be met easily, at least for some classes of graphs.
For other classes, it might be difficult to produce a drawing that
meets the criteria. For example, minimizing the number of link crossings
is relatively simple for trees (that is, graphs without cycles). For
general graphs, minimizing the number of link crossings is a mathematical NP-complete problem (that is, with all known algorithms,
the time required to perform the layout grows fast with the size of
the graph).
If you want to meet several criteria at the same time,
an optimal solution might not exist for each one individually, because
many of the criteria are mutually contradictory. Time-consuming trade-offs
might be necessary. In addition, it is not a trivial task to assign
weights to each criteria. Multicriteria optimization is, in most cases,
too complex to implement, and much too time-consuming. For these reasons,
layout algorithms are often based on heuristics and might provide
less than optimal solutions with respect to one or more of the criteria.
Fortunately, in practical terms, the layout algorithms still often
provide reasonably readable drawings.
Methods for using layout algorithms
Layout algorithms can be employed in various ways in
the various applications in which they are used. The most common ways
of using an algorithm are:
-
The layout algorithm does everything without any user intervention, except for perhaps the choice of the layout algorithm to be used. Sometimes, a set or rules can be coded to choose automatically (and dynamically) the most appropriate layout algorithm for the particular type of graph to lay out.
-
The application user is free to improve the result of the automatic layout procedure by hand. In some cases, this user can move and “pin” nodes at wanted locations and perform the layout again. In other cases, a part of the graph is automatically set as “read-only” and the user can modify the rest of the layout.
- Layout from scratchThe layout algorithm is completely redone (“from scratch”) each time the graph is changed.
- Incremental layoutWhen the layout algorithm is performed a second time on a modified graph, it tries to preserve the stability of the layout as much as possible. The layout is not performed again from scratch. The layout algorithm also tries to save CPU time by using the previous layout as an initial solution. Some layout algorithms and layout styles are incremental by nature. For others, incremental layout might be impossible.