Directed Graphs

1
Directed Graphs

• Chapter 8

2
Objectives

• You will be able to
• Say what a directed graph is.
• Describe two ways to represent a directed graph
• Adjacency matrix.
• Adjacency lists.
• Describe two ways to traverse a directed graph
• Depth first search
• Breadth first search
• Manually perform each kind of traversal on a
  directed graph on paper.

3
Directed Graphs

• A directed graph
• A finite set of elements
• Called vertices or nodes
• Can hold values
• A finite set of directed
• Arcs or edges
• Connect pairs of vertices
• Often called a digraph.

4
Directed Graphs

• Applications of directed graphs
• Analyze electrical circuits
• Find shortest routes
• Develop project schedules
• State diagrams

5
Graph Terminology

• Multigraph
• A digraph in which there can be more than one arc
  between a given pair of nodes.
• Pseudograph
• A multigraph in which there can be an arc from a
  node back to itself.
• Graph
• Arcs are bidirectional

6
Directed Graphs

• Trees are a special kind of directed graph.
• One node (the root) has no incoming edge.
• Every other node can be reached from the root by
  a unique path.
• Graphs differ from trees as ADTs
• Insertion of a node does not require an incoming
  edge or may have multiple edges.

7
Directed Graph as an ADT

• A directed graph is defined as a collection of
  data elements
• Called nodes or vertices
• And a finite set of direct arcs or edges
• Ordered pairs of nodes.
• Operations include
• Constructors
• Insert node, edge
• Delete node, edge
• Search for a value in a node, starting from a
  given node

8
Directed Graph Terminology

• Weighted digraph
• Each arc has a "cost" or "weight"
• Example Distance between cities connected by
  highways.
• Classic problem
• Find the shortest route from one city to another.

9
Directed Graph Terminology

• A complete digraph
• Has an edge between each pair of vertices.
• (Each direction)
• N nodes will have N (N 1) edges

10
Directed Graph Representation

• Adjacency matrix representation
• Identify nodes with consecutive integers 1,
  2, n
• The adjacency matrix is an n by n matrix.
• Call it adj
• adji,j is
• 1 (true) if vertex j is adjacent to vertex i
• There is a directed arc from i to j
• 0 (false) otherwise

Direction matters!
11
Graph Representation
columns j To Vertex
1 2 3 4 5
1 0 1 1 0 1
2 0 0 1 0 0
3 0 0 0 1 0
4 0 0 1 0 0
5 0 0 0 0 0

• Entry 1, 5 set to true
• Edge from vertex 1 to vertex 5

rows i From Vertex
1
1
0
12
Adjacency Matrix Terminology

• Out-degree of ith vertex (node)
• Number of arc emanating from that node
• Sum of 1's in row i
• In-degree of jth vertex (node)
• Number of arcs coming into that node
• Sum of the 1's in column j

13
Adjacency Matrix

• Consider the sum of the products of the pairs of
  elements from row i and column j

adj 2
adj
1 2 3 4 5
1 1
2
3
4
5
1 2 3 4 5
1 0 1 1 0 1
2 0 0 1 0 0
3 0 0 0 1 0
4 0 0 1 0 0
5 0 0 0 0 0
This is the number of paths of length 2 from node
1 to node 3
1
1
0
14
Adjacency Matrix

• This is matrix multiplication
• What is adj 3?
• The value in each entry would represent
• The number of paths of length 3
• From node i to node j
• Consider the meaning of the generalization of adj
  n

15
Adjacency Matrix

• Deficiencies in adjacency matrix representation
• Data must be stored in separate matrix
• When there are few edges the matrix is sparse.
• Wasted space

16
Adjacency List Representation

• Solving problem of wasted space
• Better to use an array of pointers to linked
  row-lists.
• This is called an Adjacency List representation.

17
Searching a Graph

• Recall that with a tree we search from the root.
• But with a digraph
• There is no distinguished vertex.
• There may not be a vertex from which every other
  vertex can be reached.
• May not be possible to traverse entire
  digraph(regardless of starting vertex.)

18
Searching a Graph

• We must determine which nodes are reachable from
  a given node
• Two standard methods of searching
• Depth first search
• Breadth first search

19
Depth-First Search

• Start from a given vertex v
• Visit first neighbor w, of v
• Then visit first neighbor of w which has not
  already been visited.
• Continue descent until we reach a node with no
  unvisited neighbors.
• When no unvisited neighbors
• Back up to last visited node
• Visit next unvisited neighbor of that node

20
Depth-First Search

• Same as pre-order traversal of a tree except we
  have to keep track of nodes visited and avoid
  going back to them.

21
Depth-First Search

• Start from node A.
• What is the sequence of nodes that would be
  visited in depth first search?

Click for answer
A, B, E, F, H, C, D, G
Same as pre-order traversal of the tree.
22
Depth-First Search

• Start from node A.
• What is the sequence of nodes that would be
  visited in DFS?

Click for answer
A, B, F, G, C, D, H, I, E
23
Depth-First Search

• DFS uses backtracking to return to vertices that
  were seen earlier and
• already processed or
• skipped over on an earlier pass
• Recursion is a natural technique for this task.

24
Depth-First Search

• Algorithm to perform DFS search of digraph from a
  specified starting vertex

1.
Visit the start vertex, v
2.
For each vertex w adjacent to v do
If
w
has not been visited,
apply the depth-first search algorithm
with
w
as the start vertex.
Note the recursion
25
Breadth-First Search

• A different search technique
• At each point in the search, visit all previously
  unvisited neighbors of current node before
  advancing to their neighbors.

26
Breadth-First Search

• Start from a given vertex v
• Visit all neighbors of v
• Then visit all previously unvisited neighbors of
  first neighbor w of v.
• Then visit all previously unvisited neighbors of
  second neighbor x of v etc.
• Continue, visiting all vertices at distance N
  from starting vertex before moving on to vertices
  at distance N1.

27
Breadth-First Search

• Start from node containing A
• What is a sequence of nodes which would be
  visited in BFS?

Click for answer
A, B, D, E, F, C, H, G, I
28
Breadth-First Search

• Notice distances from starting node.

A B D E F C H G I
3
1
2
0
29
Breadth-First Search

• Breadth-First Search defines a tree consisting of
  nodes reachable from the starting node.

30
Breadth-First Search Algorithm

• While visiting each node on a given level
• store its ID so that we can return to it after
  completing this level.
• So that nodes adjacent to it can be visited.
• First node visited on given level should befirst
  node to which we return upon completion of
  that level.What data structure does this imply?

A queue
31
Breadth-First Search Algorithm

• Algorithm for BFS search of a digraph from a
  given starting vertex
• Visit the start vertex.
• Initialize queue to contain only the start
  vertex.
• While queue not empty do
• Remove a vertex v from the queue.
• For all vertices w adjacent to v do If w
  has not been visited then
• Visit w.
• Add w to queue.
• End while

End of section
32
Directed Graph Traversal

• Algorithm to traverse digraph must
• Visit each vertex exactly once.
• BFS or DFS forms basis of traversal.
• Mark vertices when they have been visited.
• Initialize an array (vector) visited. visitedi
  false for each vertex i
• While some element of visited is false
• Select an unvisited vertex v.
• Set visitedv to true.
• Use BFS or DFS to visit all vertices reachable
  from v
• End while