Theoretical and Experimental DNA Computation (Natural ...

34 2 Theoretical Computer Science: A Primer
them. One solution is the adjacency matrix, which is a two-dimensional array
of integers representing the distances between towns. An example is shown in
Fig. 2.6. Here, we have five towns connected by roads of a given length. We
declare a two-dimensional array distance[5][5] to store the distance between
any two towns.
Town 0
45
15
Town 4
45
50
Town 2
50
52
Town 1
30
Town 3
Town
0
1
2
3
4
0 1
Town
2 3 4
−1 50 15 −1 45
50 −1 −1 30 −1
15 −1 −1 50 45
−1 30 50 −1 52
45 −1 45 52 −1
Fig. 2.6. Example map and its corresponding adjacency matrix
The distance is then found by reading the contents of array element
distance[firsttown][secondtown]. Notice that for each town n, distance[n][n]
has the value -1, as it makes no sense for a town to have a distance to itself.
Also, if there is no direct road connecting town x and town y, then
distance[x][y] also takes the value -1. The final thing to note about the adjacency
matrix is that as the roads are two-way, the matrix is symmetrical in
the diagonal (i.e., distance[x][y] has the same value as distance[y][x]). Obviously,
this is not the most efficient method of storing undirected graphs, but
it provides an easy illustration of the use of more complicated data structures.
Graphs
As we have seen, an example problem may be phrased thus: given a set of
towns connected by roads, what is the shortest path between town A and
town B? Another problem may ask if, given a map of mainland Europe, it
is possible to color each country red, green, or blue such that no adjacent
countries are colored the same.
The first example is referred to as an optimisation problem, as we are required
to find a path that fits some criterion (i.e., it is the shortest possible
path). The second example is known as a decision problem, as we are answering
a simple “‘yes/no” question about a structure. A large body of problems
is concerned with the mathematical structures known as graphs. Inthiscontext,
a graph is not a method of visualising data, rather a network of points
and lines. More formally, a graph, G =(V,E), is a set of points (vertices), V ,
connected by a set of lines (edges), E. Take the fragment of an English county
map depicted in Fig. 2.7.