1. CSE 373, Copyright S. Tanimoto, 2002 Graphs 2 -
Incidence and Adjacency
Representing a graph with an adjacency matrix, an incidence matrix, adjacency lists.
Graph search: depth-first, breadth-first.
Shortest paths.
CSE 373, Copyright S. Tanimoto, Graphs 2 -

2. Incidence and Adjacencyv1 v2 e
Let e =  v1, v2  be an edge of a digraph.
Then v1 is adjacent to v2.
And v2 is adjacent from v1.
e is incident to v2.
e is incident from v1.
For an undirected graph we can use the terms “adjacent to” and “incident to” for relationships in both directions.
CSE 373, Copyright S. Tanimoto, Graphs 2 -

3. Representing a Digraph with an Adjacency Matrix
G =  V, E , V = { v1, v2, , vn }, E = { e1, e2, , em } ei =  va(i), vb(i) 
An adjacency matrix for G contains n rows and n columns.
A[i,j] = 1 if  vi, vj   E;
0 otherwise.
a b c d a b c d a c b d
CSE 373, Copyright S. Tanimoto, Graphs 2 -

4. CSE 373, Copyright S. Tanimoto, 2002 Graphs 2 -
Adjacency Lists
Represent the adjacency matrix info as an array of lists, one for each row of the matrix.
The ith list contains the vertices to which vi is adjacent.
(Similar to the sparse array idea).
a b c d
a a: b, c, d
b b:
c c: d
d d: c
Efficent for getNeighbors(v)
CSE 373, Copyright S. Tanimoto, Graphs 2 -

5. CSE 373, Copyright S. Tanimoto, 2002 Graphs 2 -
Incidence Matrix
e1 e2 e3 e4 e5 a b c d
E = { a, b ,  a, d ,  a, c ,  c, d ,  d, c  }
-1: edge starts at that vertex (it’s incident from the vertex)
1: edge ends at that vertex (it’s incident to the vertex)
0: edge does not involve the vertex.
CSE 373, Copyright S. Tanimoto, Graphs 2 -

6. CSE 373, Copyright S. Tanimoto, 2002 Graphs 2 -
Using a Hash Table
If the graph is sparse,
store adjacency info in the hash table.
if  vi, vj   E, perform
put( vi, vj  , 1)
If the complementary graph is sparse (the graph is very dense), then store the complement of the edges in the hash table.
if  vi, vj   E, perform
put( vi, vj  , 0)
In either case, the keys are edges, and the values tell whether or not the edge is part of the graph.
CSE 373, Copyright S. Tanimoto, Graphs 2 -

7. CSE 373, Copyright S. Tanimoto, 2002 Graphs 2 -
Graph Search
Given a directed graph G =  V,E  and a starting vertex s, find a path from s to some goal vertex g, where g is any vertex that satisfies a goal(v) predicate. E.g., let s = a. a e g c b f d
CSE 373, Copyright S. Tanimoto, Graphs 2 -

8. CSE 373, Copyright S. Tanimoto, 2002 Graphs 2 -
Possible Solutions
There may be 0 solutions, 1 solution, or many solutions.
This particular problem seems to have at least 2 distinct solutions. P =  a,c, c,f, f,e, e,g . a e g c b f d
CSE 373, Copyright S. Tanimoto, Graphs 2 -

9. Some Additional Solutions
If cycles are permitted in the path, there may be an infinite number of solutions.
P =  a,d, d,c, c,d, ..., d,c, c,f,
f,e, e,g, g,f, ...,f,e, e,g  a e g c b d f
CSE 373, Copyright S. Tanimoto, Graphs 2 -

10. CSE 373, Copyright S. Tanimoto, 2002 Graphs 2 -
Avoiding Loops
Even if cycles do not occur in the solution, they may cause a nuisance for any search process that is not prepared for them.
Therefore, it is important to keep track of which vertices have been visited during a search.
We’ll maintain two lists: OPEN and CLOSED.
OPEN will hold the set of vertices that have been discovered so far but that have not yet been examined.
CLOSED will hold the vertices that have already been examined.
CSE 373, Copyright S. Tanimoto, Graphs 2 -

11. CSE 373, Copyright S. Tanimoto, 2002 Graphs 2 -
Depth-First Search
Set OPEN = S ; Set CLOSED =   ;
Initialize a hash table PRED. (for “predecessor”)
Put into PRED: (S, NULL);
While OPEN is not empty do {
Remove the first element of OPEN; call this V.
If Goal(V) then output the reverse of the linked list:
V, PRED(V), PRED(PRED(V)), etc., and return.
Find all W such that  V,W   E. Call this set L.
Remove from L any elements already on OPEN or CLOSED.
For each of the remaining elements V’ of L, put into PRED: (V’, V)
and put V’ onto OPEN at the front of the list.
Put V onto CLOSED. }
Output “No solution” and return.
CSE 373, Copyright S. Tanimoto, Graphs 2 -

12. CSE 373, Copyright S. Tanimoto, 2002 Graphs 2 -
Breadth-First Search
Set OPEN = S ; Set CLOSED =   ;
Initialize a hash table PRED. (for “predecessor”)
Put into PRED: (S, NULL);
While OPEN is not empty do {
Remove the first element of OPEN; call this V.
If Goal(V) then output the reverse of the linked list:
V, PRED(V), PRED(PRED(V)), etc., and return.
Find all W such that  V,W   E. Call this set L.
Remove from L any elements already on OPEN or CLOSED.
For each of the remaining elements V’ of L, put into PRED: (V’, V)
and put V’ onto OPEN at the rear of the list.
Put V onto CLOSED. }
Output “No solution” and return.
CSE 373, Copyright S. Tanimoto, Graphs 2 -

13. Comparison of DFS and B(readth)FS
In DFS, OPEN serves as a Stack
In BFS, OPEN serves as a Queue.
In both, CLOSED serves as a Set.
Both DFS and BFS find a solution if it exists.
BFS always finds a shortest solution (if one exists).
DFS finds some solution that depends upon the order in which successors (members of L) are determined and processed.
CSE 373, Copyright S. Tanimoto, Graphs 2 -

14. Vertex Visitation Order
In what order do vertices get processed (removed from OPEN and used as the value of V in the algorithm?)
DFS: a, b, c, f, e, g
BFS: a, b, c, d, f, e, g a e g c b f d
CSE 373, Copyright S. Tanimoto, Graphs 2 -