1. Thinking about Algorithms Abstractly
Searching in a Graph

2. Representations of Graphs
Adjacency Matrices
Adjacency Lists

3. Adjacency Matrices
Graphs G = (V, E) can be represented by adjacency matrices
G[v1..v|V |, v1..v|V |], where the rows and columns are indexed by the nodes, and the entries G[vi, vj] represent the edges. In the case of unlabeled graphs, the entries are just boolean values.
            A B C D 1

4. Adjacency Matrices
In case of labeled graphs, the labels themselves may be introduced into the entries.
                   A B C D 10 4 1 15 9
Adjacency matrices require O(|V |2) space, and so they are space-efficient only when they are dense (that is, when the graphs have many edges). Time-wise, the adjacency matrices allow easy addition and deletion of edges.

5. Adjacency Lists
A representation of the graph consisting of a list of nodes, with each node containing a list of its neighboring nodes.
                  
                                  
This representation takes O(|V | + |E|) space.

6. Graph Traversal
Depth-First Traversal
Breadth-First Traversal

7. Depth-First Traversal
algorithm  dft(x)     
visit(x)     
FOR each y such that (x,y) is an edge DO       
IF <y was not visited yet >
THEN          dft(y) 

8. Depth-First Traversal
A recursive algorithm implicitly recording a “backtracking” path from the root to the node currently under consideration

9. Depth-First Traversal

10. Depth-First Traversal

11. Depth-First Traversal

12. Depth-First Traversal

13. Depth-First Traversal
Depth first search is another way of traversing graphs, which is closely related to preorder traversal of a tree.
the Breath-first search tree is typically "short and bushy", the DFS tree is typically "long and stringy".

14. Breadth-First Traversal
Visit the nodes at level i before the nodes of level i+1.
                                      

15. Breadth-First Traversal
visit(start node) 
queue <- start node 
WHILE queue is not empty DO    
x <- queue    
FOR each y such that (x,y) is an edge                    and y has not been visited yet DO      
visit(y)      
queue <- y    
END 

16. Breadth-First Traversal

17. Breadth-First Traversal

18. Breadth-First Traversal

19. Breadth-First Traversal
Each vertex is clearly
marked at most once,
added to the list at most once (since that happens only when it's marked), and
removed from the list at most once.
Since the time to process a vertex is proportional to the length of its adjacency list, the total time for the whole algorithm is O(m).
A tree T constructed by the algorithm is called a breadth first search tree.
The traversal goes a level at a time, left to right within a level (where a level is defined simply in terms of distance from the root of the tree).

20. Breadth-First Traversal
Every edge of G can be classified into one of three groups.
Some edges are in T themselves.
Some connect two vertices at the same level of T.
The remaining ones connect two vertices on two adjacent levels. It is not possible for an edge to skip a level.
Breadth-first search tree really is a shortest path tree starting from its root.

21. Relation between BFS and DFS
dfs(G) {
list L = empty
tree T = empty
choose a starting vertex x
search(x)
while(L nonempty)
remove edge (v,w) from end of L
if w not yet visited
add (v,w) to T
search(w) }
bfs(G) {
list L = empty tree
T = empty
choose a starting vertex x
search(x)
while(L nonempty)
remove edge (v,w) from start of L
if w not yet visited
add (v,w) to T
search(w) }
search(vertex v) {
visit(v);
for each edge (v,w)
add edge (v,w) to end of L }

22. Relation between BFS and DFS
Both of these search algorithms now keep a list of edges to explore; the only difference between the two is
while both algorithms adds items to the end of L,
BFS removes them from the beginning, which results in maintaining the list as a queue
DFS removes them from the end, maintaining the list as a stack.

23. BFS and DFS in directed graphs
The same search routines work essentially unmodified for directed graphs.
The only difference is that when exploring a vertex v, we only want to look at edges (v,w) going out of v; we ignore the other edges coming into v.
For BFS in directed graphs each edge of the graph either
connects two vertices at the same level
goes down exactly one level
goes up any number of levels.
For DFS, each edge either connects an ancestor to
a descendant
a descendant to an ancestor
one node to a node in a previously visited subtree.

24. Searching

26. Searching

27. Searching

28. Searching

30. Searching

31. Searching

33. Searching

34. Searching

35. Searching

48. End

49. End