CMPS 101
Algorithms and Abstract Data Types
Spring 2006
Programming Assignment 4
Due Wednesday May 31, 10:00 pm
In this assignment you will build a Graph module in C, implement Depth First Search (DFS), and use
your Graph module to find the strongly connected components of a directed graph. Begin by reading
sections 22.3-22.5 in the text.
A directed graph (or digraph)
),( EVG
is said to be strongly connected if for every pair of vertices
Vvu ,
, u is reachable from v and v is reachable from u. Most directed graphs are not strongly
connected. In general we say a subset
VX
is strongly connected if every vertex in X is reachable from
every other vertex in X. A strongly connected subset which is maximal with respect to this property is
called a strongly connected component of G. In other words,
VX
is a strongly connected component
of G if and only if (i) X is strongly connected, and (ii) the addition of one more vertex to X would create a
subset which is not strongly connected.
Example
1 2 3 4
G
5 6 7 8
It’s easy to see that there are 4 strong components of G:
}5 ,2 ,1{
1
C
,
}4 ,3{
2
C
,
}7 ,6{
3
C
, and
}8{
4
C
.
To find the strong components of a digraph G first call
)(DFS G
, and as vertices are finished, place them
on a stack. When this is complete the stack stores the vertices ordered by decreasing finish times. Next
compute the transpose
T
G
of G, which is obtained by reversing the directions on all edges of G. Finally,
run
)(DFS
T
G
, but in the main loop (lines 5-7) process vertices by decreasing finish times. (Here we
mean finish times from the first call to DFS.) This is accomplished by simply popping vertices off the
stack in loop 5-7 of DFS. When this process is complete the trees in the resulting DFS forest constitute
the strong components of G. (Note the strong components of G are identical to the strong components of
T
G
.) See the algorithm (Strongly-Connected-Components) and proof of correctness in section 22.5 of
the text.
In this assignment you will create a graph module in C which represents a directed graph by an array of
adjacency lists. Your graph module will, among other things, provide the capability of running DFS, and
computing the transpose of a directed graph. DFS requires that vertices posses attributes for color (white,
black, grey), discover time, finish time, and parent. Here is a catalog of required functions and their
prototypes:
/* Constructors-Destructors */
GraphRef newGraph(int n);
void freeGraph(GraphRef *G);
/* Access functions */
int getOrder(GraphRef G);
