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Directed Graphs

Chapter 8

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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.

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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.

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Directed Graphs

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

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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

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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.

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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

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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.

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Directed Graph Terminology

A complete digraph Has an edge between each pair of

vertices. (Each direction)

N nodes will have N * (N – 1) edges

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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

adj[i,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!

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Graph Representation

1 2 3 4 51 0 1 1 0 12 0 0 1 0 03 0 0 0 1 04 0 0 1 0 05 0 0 0 0 0

rows i

“From” Vertex

columns j

“To” Vertex

• Entry [ 1, 5 ] set to true

• Edge from vertex 1 to vertex 51

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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

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Adjacency Matrix

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

1 2 3 4 51 0 1 1 0 12 0 0 1 0 03 0 0 0 1 04 0 0 1 0 05 0 0 0 0 0

1 2 3 4 51 12345

adj adj 2

This is the number of paths of length 2 from

node 1 to node 3

01

1

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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

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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

data =

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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.

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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.)

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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

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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

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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.

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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.

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A, B, F, G, C, D, H, I, E

Depth-First Search

Start from node A. What is the sequence of nodes that

would be visited in DFS?

Click for answer

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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.

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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 algorithmwith w as the start vertex.

Note the recursion

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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.

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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 N+1.

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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

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Breadth-First Search

Notice distances from starting node.

A B D E F C H G I0 1 2 3

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Breadth-First Search

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

A B D E F C H G I0 1 2 3

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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

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Breadth-First Search Algorithm

Algorithm for BFS search of a digraph from a given starting vertex:

1. Visit the start vertex.2. Initialize queue to contain only the start

vertex.3. While queue not empty do

a. Remove a vertex v from the queue.b. For all vertices w adjacent to v do:

If w has not been visited then:i. Visit w.ii. Add w to queue.

End whileEnd of section

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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.

1. Initialize an array (vector) visited. visited[i] = false for each vertex i

2. While some element of visited is falsea. Select an unvisited vertex v.b. Set visited[v] to true.c. Use BFS or DFS to visit all vertices reachable from

vEnd while