The running times of the depthfirstsearch and breadthfirstsearch
algorithms are somewhat overstated by the Theorems 12.3 and
12.4. Define
as the number of vertices,
,
of , for which there exists a path from
to
. Define
as the number of edges that have these vertices as their sources.
Then the following theorem is a more precise statement of the running
times of the breadthfirstsearch and depthfirstsearch algorithms.
(This more refined statement of the running time is useful in some of
the applications of these algorithms outlined in the exercises.)
Theorem 12..5
When given as input a Graph,
, that is implemented using the
AdjacencyLists data structure, the
,
and
algorithms each run in
time.
Breadthfirst search seems to have been discovered independently by
Moore [52] and Lee [49] in the contexts of maze exploration
and circuit routing, respectively.
Adjacencylist representations of graphs were presented by
Hopcroft and Tarjan [40] as an alternative to the (then more
common) adjacencymatrix representation. This representation, as well as
depthfirstsearch, played a major part in the celebrated HopcroftTarjan
planarity testing algorithm
that can determine, in
time, if
a graph can be drawn, in the plane, and in such a way that no pair of
edges cross each other [41].
In the following exercises, an undirected graph is one in which, for
every
and
, the edge
is present if and only if the
edge
is present.
Exercise 12..1
Draw an adjacency list representation and an adjacency matrix
representation of the graph in Figure
12.7.
Figure 12.7:
An example graph.

Exercise 12..2
The
incidence matrix representation of a graph,
, is an
matrix,
, where
 Draw the incident matrix representation of the graph in
Figure 12.7.
 Design, analyze and implement an incidence matrix representation
of a graph. Be sure to analyze the space, the cost of
,
,
,
,
and
.
Exercise 12..3
Illustrate an execution of the
and
on the graph,
,
in Figure
12.7.
Exercise 12..4
Let
be an undirected graph. We say
is
connected if,
for every pair of vertices
and
in
, there is a path from
to
(since
is undirected, there is also a path from
to
). Show how to test if
is connected in
time.
Exercise 12..5
Let
be an undirected graph. A
connectedcomponent labelling
of
partitions the vertices of
into maximal sets, each of which
forms a connected subgraph. Show how to compute a connected component
labelling of
in
time.
Exercise 12..6
Let
be an undirected graph. A
spanning forest of
is a
collection of trees, one per component, whose edges are edges of
and whose vertices contain all vertices of
. Show how to compute
a spanning forest of of
in
time.
Exercise 12..7
We say that a graph
is
stronglyconnected if, for every
pair of vertices
and
in
, there is a path from
to
. Show how to test if
is stronglyconnected in
time.
Exercise 12..8
Given a graph
and some special vertex
, show how
to compute the length of the shortest path from
to
for every
vertex
.
Exercise 12..9
Give a (simple) example where the
code visits the nodes of a
graph in an order that is different from that of the
code.
Write a version of
that always visits nodes in exactly
the same order as
. (Hint: Just start tracing the execution
of each algorithm on some graph where
is the source of more than
1 edge.)
Exercise 12..10
A
universal sink in a graph
is a vertex that is the target
of
edges and the source of no edges.
^{12.1} Design and implement an algorithm that tests if a graph
, represented
as an
AdjacencyMatrix, has a universal sink. Your algorithm should
run in
time.
Footnotes
 ... edges.^{12.1}
 A universal sink,
, is also sometimes called a celebrity: Everyone in the room
recognizes
, but
doesn't recognize anyone else in the room.
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