Graphs in Data Structure

Lecture: Graphs Data Structures
Graphs

Graph Definitions
• A graph G is denoted by G = (V, E) where
– V is the set of vertices or nodes of the graph
– E is the set of edges or arcs connecting the vertices in V
• Each edge E is denoted as a pair (v,w) where v,w ε V
For example in the graph below
1
2
5
6 4
3
V = {1, 2, 3, 4, 5, 6}
E = {(1, 2) (2, 5) (3, 6) (4, 6) (5, 6)}
• This is an example of an unordered
or undirected graph

Graph Definitions (contd.)
• If the pair of vertices is ordered then the
graph is a directed graph or a di-graph
1
2
5
6 4
3 Here,
V = {1, 2, 3, 4, 5, 6}
E = {(1, 2) (2, 5) (5, 6) (6, 3) (6, 4)}
• Vertex v is adjacent to w iff (v,w) ε E
• Sometimes an edge has another component called a weight or
cost. If the weight is absent it is assumed to be 1

Graph Definitions: Path
• A path is a sequence of vertices w1, w2,
w3, ....wn such that (wi, wi+1) ε E
• Length of a path = # edges in the path
• A loop is an edge from a vertex onto itself. It
is denoted by (v, v)
• A simple path is a path where no vertices
are repeated along the path
• A cycle is a path with at least one edge such
that the first and last vertices are the same,
i.e. w1 = wn

Graph Definitions: Connectedness
• A graph is said to be connected if there is a path from every vertex to
every other vertex
• A connected graph is strongly connected if it is a connected graph as
well as a directed graph
• A connected graph is weakly connected if it is
– a directed graph that is not strongly connected, but,
– the underlying undirected graph is connected
1
2
5
6 4
3
1
2
5
6 4
3
Strong or Weak?

Applications of Graphs
• Driving Map
– Edge = Road
– Vertex = Intersection
– Edge weight = Time required to cover the road
• Airline Traffic
– Vertex = Cities serviced by the airline
– Edge = Flight exists between two cities
– Edge weight = Flight time or flight cost or both
• Computer networks
– Vertex = Server nodes
– Edge = Data link
– Edge weight = Connection speed
• CAD/VLSI

Representing Graphs: Adjacency Matrix
• Adjacency Matrix
– Two dimensional matrix of size n x n where n is
the number of vertices in the graph
– a[i, j] = 0 if there is no edge between vertices i
and j
– a[i, j] = 1 if there is an edge between vertices i
and j
– Undirected graphs have both a[i, j] and a[j, i] = 1
if there is an edge between vertices i and j
– a[i, j] = weight for weighted graphs
• Space requirement is Θ(N2
)
• Problem: The array is very sparsely populated. For example if a
directed graph has 4 vertices and 3 edges, the adjacency matrix has 16
cells only 3 of which are 1

Representing Graphs: Adjacency List
• Adjacency List
– Array of lists
– Each vertex has an array entry
– A vertex w is inserted in the list for vertex v if
there is an outgoing edge from v to w
– Space requirement = Θ(E+V)
– Sometimes, a hash-table of lists is used to
implement the adjacency list when the vertices
are identified by a name (string) instead of an
integer

Adjacency List Example
1
2
5
6 4
3
1
2
3
4
5
6
2
5
6
43
Graph Adjacency List
• An adjacency list for a weighted graph should contain two elements in
the list nodes – one element for the vertex and the second element for
the weight of that edge

Negative Cost Cycle
• A negative cost cycle is a cycle such that the sum of the costs of the
edges is negative
• The more we cycle through a negative cost cycle, the lower the cost
of the cycle becomes
2
3
4
4 2
-12
Cost of the path v1-v5
• No traversal of cycle: 6
• One traversal of cycle: 0
• Two traversals of cycle: -6
• Three traversals of cycle: -12
...
5
1
5
1
• Negative cost cycles are not allowed when we traverse a graph to
find the weighted shortest path

Application of Graphs
Using a model to solve a complicated traffic lightUsing a model to solve a complicated traffic light
problemproblem
• GIVEN: A complex intersection.
• OBJECTIVE: Traffic light with minimum phases.
• SOLUTION:
• Identify permitted turns, going straight is a “turn”.
• Make group of permitted turns.
• Make the smallest possible number of groups.
• Assign each phase of the traffic light to a group.

Using a model to solve a complicated trafficUsing a model to solve a complicated traffic
light problemlight problem
A
B
C
D
E
An intersection

Roads C and E are one way, others are two way.
There are 13 permitted turns.
Some turns such as AB (from A to B) and EC can be carried
out simultaneously.
Other like AD and EB cross each other and can not be carried
out simultaneously.
The traffic light should permit AB and EC simultaneously, but
should not allow AD and EB.

A
B
C
D
E
An intersection
AB & AC
AD & EB

SOLUTION:
• We model the problem using a structure called graph G(V,E).
• A graph consists of a set of points called vertices, and lines
connecting the points, called edges.
•Drawing a graph such that the vertices represent turns.
• Edges between those turns that can NOT be performed
simultaneously.

A
B
C
D
E
An intersection
AB AC AD
BA BC BD
DA DB DC
EA EB EC ED
Partial graph of incompatible turns

Partial table of incompatible turns

SOLUTION:
The graph can aid in solving our problem.
A coloring of a graph is an assignment of a color to
each vertex of the graph, so that no two vertices
connected by an edge have the same color.
Our problem is of coloring the graph of incompatible
turns using as few colors as possible.

More on SOLUTION (Graph coloring):
The problem has been studied for decades.
The theory of algorithms tells us a lot about it.
Unfortunately this belongs to a class of problems called
as NP-Complete problems.
For such problems, all known solutions are basically “try
all possibilities”
In case of coloring, try all assignments of colors.

Approaches to attempting NP-Complete problems:
1.If the problem is small, might attempt to find an
optimal solution exhaustively.
2.Look for additional information about the problem.
3.Change the problem a little, and look for a good, but
not necessarily optimal solution.
An algorithm that quickly produces good but not
necessarily optimal solutions is called a heuristic.

A reasonable heuristic for graph coloring is the greedy
algorithm.
Try to color as many vertices as possible with the first
color, and then as many uncolored vertices with the
second color, and so on.
The approach would be:
1. Select some uncolored vertex, and color with new color.
2. Scan the list of uncolored vertices. For each uncolored
vertex, determine whether it has an edge to any vertex
already colored with the new color. If there is no such
edge, color the present vertices with the new color.

This is called “greedy”, because it colors a vertex,
whenever it can, without considering potential
drawbacks.
1 5
3
4
21 2
3
4
5
1 5
3
1
3
5 2
4

Graphs in Data Structure

More Related Content

What's hot (20)

Viewers also liked (20)

Similar to Graphs in Data Structure (20)

Recently uploaded (20)

Graphs in Data Structure