Graphs

1
Graphs

• CS-240/341

2
Uses for Graphs

• computer networks and routing
• airline flights
• geographic maps
• course prerequisite structures
• tasks for completing a job
• plumbing/hydraulic systems
• electrical circuits
• automatons/finite state machines
• grammar mappings

3
Graphs

• Used for representing many-to-many relationships
• can take two forms
• directed (digraph) - a finite set of elements
  called vertices (nodes) connected by a set of
  directed edges (arcs)
• G V E where
• V V1,V2,V3Vn E (Vm1,Vn1),(Vm2,Vn2)(Vmx,Vn
  y)
• undirected (graph) a finite set of vertices
  connected by a set of undirected (bidirectional)
  edges

1
2
3
4
5
4
Edges
B
A
25

• Represent a relationship between two vertices
• to in a digraph represents that A is adjacent
  to B
• from in a digraph B is adjacent from A
• in an undirected graph an edge merely represents
  that the vertices are adjacent
• Edges may also have an associated weight or value
• if the vertices represent cities then the edge
  may represent a connection (road) between them
  and the weight may represent the distance and the
  direction may represent a one way street
• if the graph represents a network of active
  components (p-n junctions, diodes/transistors)
  then the direction represents the direction of
  the pn junctions and the edges represent the way
  the components are connected and the weights
  represent the forward resistance of the junction
  (backward resistance is infinity)

A
5
Digraph Abstract Data Type

• Data Elements
• collection of vertices and a collection of edges
  and weights
• Basic Operations
• Create an empty digraph
• Check if the digraph is empty
• Destroy a digraph
• Insert a new vertex
• Insert a new edge between two existing vertices
• Delete a vertex and all directed edges to or from
  it
• Delete a directed edge between to vertices
• search for an edge value

6
Digraph representation

• Adjacency Matrix Representation

to
from
0 1 2 3 4 0 0 5 2 4
0 1 0 0 5 0 0 2 0 1 0 0
6 3 0 0 0 0 0 4 0 0 0
0 0
4
0
5
3
3
2
1
2
4
5
6
7
Digraph Representation

• Adjacency List
• use an array of pointers to be the heads of a
  list of adjacencies for each vertex

4
0 1 2 3 4
1 5
2 3
3 4
0
2 5
5
3
1 2
4 6
3
2
1
2
4
5
6
8
Graph Traversal
0 1 2 3 4
1 5
2 3
3 4
2 5
1 2
4 6

• Depth First

1. Visit the start vertx V mark it as visited 2.
For each vertex adjacent to V do the following
if w has not been visited, apply the
depth-first search algorithm starting with
w connecting the vertices along the way will
produce a spanning tree
4
0
visited
Spanning tree
5
V
3
3
4
0
2
5
1
3
3
2
4
5
6
2
1
2
4
5
6
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Graph Traversal

• Breadth First

1. Visit the start vertex V 2 . Initialize a
queue to contain only the start vertex 3. While
the queue is not empty do the following
Rempve vertex V from the queue For all
vertices w adjacent to v do the following
If w has not been visited then
Visit W Add W to the
queue
10
Spanning Trees

• The set of vertices and edges produced by a DFS
  traversal produces what is known as a spanning
  tree
• if the DFS doesnt include all of the vertices,
  then do another DFS starting at a vertex not yet
  in the tree. This may have to be done a number of
  times but will eventually produce a set disjoint
  (not connected) spanning trees called a spanning
  forest.

11
Degree of a vertex

• Digraph
• In degree is the number of edges coming into a
  vertex
• Out degree is the number of edges going out of a
  vertex
• Undirected Graph
• degree is the number of edges touching a vertex
• the highest degree of any vertex is the degree of
  the graph

In degree 3 Out degree 2
3
12
Complete Graphs

• A graph is complete if there is an edge
  connecting every pair of verticies

13
Trees

• A tree is a special case of a graph
• a binary tree is an order 2 graph
• If a tree is represented by its adjacencies
  (matrix or list) it can be traversed BFS or DFS
• A BFS done starting at the root should yield the
  same traversal path as a preorder tree traversal

14
Connectedness

• A graph is said to be connected if there is a
  path from every node to every other node
• do a DFS starting at each vertex
• that vertex is complete is the visited array is
  full
• vertices not visited are not connected

15
NP-Complete Problems

• NP - nondeterministic polynomial
• cannot be solved in polynomial time
• use a heuristic solution
• guess an answer (nondeterministic) , try to prove
  it wrong, if you cant prove it wrong it must be
  part of the solution set
• P deterministic polynomial