General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

2.2. COMPUTER SCIENCE BY EXAMPLE 13
Our maze
⎧
⎫
〈a, e〉, 〈e, i〉, 〈i, j〉,
⎪⎨
〈f, j〉, 〈f, g〉, 〈g, h〉, ⎪⎬
〈d, h〉, 〈g, k〉, 〈a, b〉
⎪⎩
〈m, n〉, 〈n, o〉, 〈b, c〉
〈k, o〉, 〈o, p〉, 〈l, p〉
, a, p
⎪⎭
can be visualized as
Note that the diagram is a visualization (a representation intended for humans to process
visually) of the graph, and not the graph itself.
©: Michael Kohlhase 19
Now that we have a mathematical model for mazes, we can look at the subclass of graphs that
correspond to the mazes that we are after: unique solutions and all rooms are reachable! We will
concentrate on the first requirement now and leave the second one for later.
Unique solutions
Q: What property must the graph have for
the maze to have a solution?
A: A path from a to p.
Q: What property must it have for the maze
to have a unique solution?
A: The graph must be a tree.
Trees are special graphs, which we will now define.
Mazes as trees
Definition 7 Informally, a tree is a graph:
with a unique root node, and
each node having a unique parent.
Definition 8 A spanning tree is a tree that includes all
of the nodes.
Q: Why is it good to have a spanning tree?
A: Trees have no cycles! (needed for uniqueness)
A: Every room is reachable from the root!
So, we know what we are looking for, we can think about a program that would find spanning
trees given a set of nodes in a graph. But since we are still in the process of “thinking about the
problems” we do not want to commit to a concrete program, but think about programs in the
abstract (this gives us license to abstract away from many concrete details of the program and
concentrate on the essentials).
The computer science notion for a program in the abstract is that of an algorithm, which we
©: M
©: M