Graph Theory

1
Graph Theory

• Arnold Mesa

2
Basic Concepts

• A graph G (V,E) is defined by a set of vertices
  and edges

Edge (e1)
v2
Vertex (v1)
v1
v3
A Graph with 3 vertices and 3 edges
3
Basic Concepts (cont.)

• A vertex (v1) is said to be adjacent to (v2) if
  and only if they share a common edge (e1).

e1
e2
v1
v3
v2
4
Basic Concepts (cont.)

• A directed graph or digraph is a graph where
  direction matters.
• Movement in the graph is determined by the
  direction of the edge.

v3
v2
v1
v4
v5
v6
5
Basic Concepts (cont.)

• A path in a graph is a sequence of vertices (v1,
  v2, v3,,vn).
• The path length is determined by how many edges
  are on the path.

2
v3
v2
1
3
v1
v4
v5
v6
6
Adjacency List Representation
v3
v2
v1
v4
v5
v6
v1 v2 v2 v3 v6 v3 v4
v6 v4 v3 v5 v5 v6 v6 v1
Bi-Directional paths are allowed
7
More terms

• An edge may have a component known as a weight or
  cost associated to it.

4
6
v1
v3
v2
8
Adjacency Matrix Representation
1
v3
v2
6
6
3
4
6
v1
v4
4
7
v5
v6
4
v1
v3
v4
v2
v5
v6
v1
0
0
0
0
0
6
v2
0
0
0
0
1
4
v3
0
0
0
3
5
6
v4
0
0
0
0
0
7
v5
0
0
0
0
0
6
v6
0
0
0
0
0
4
9
Graph Algorithms

• Shortest-Path Algorithms
• Unweighted Shortest Paths
• Floyds Algorithm
• Dijkstras Algorithm
• All-Pairs Shortest Path

• Network Flow Problems
• Maximum-Flow Algorithm

• Minimum Spanning Tree
• Prims Algorithm
• Kruskals Algorithm

10
Shortest Path Algorithms

• Unweighted Shortest Paths
• Starting with a vertex s, we find the shortest
  path from s to all the other vertices

v2
v3
2
1
0
s
v1
v5
v4
2
1
v7
v6
11
Shortest Path Algorithms

• Floyds Algorithm
• Suppose we have the following adjaceny matrix

v1
v3
v4
v2
v1
0
2
13
9
v1
v2
0
1
7
10
9
v3
0
13
2
14
1
v4
0
7
4
11
v2
7
v4
13
v3
12
Matrix Addition
v1
v3
v4
v2
v1
0
2
13
9
v2
0
1
7
10
v3
0
13
2
14
v4
0
7
4
11

• We first look at row and column v1
• Then we look at each square v not in row or
  column v1 (for
  example look at (v2,v3) 10)
• Now look at the corresponding row and column in
  v1. (1, 13)
• Add the two numbers together.
  
  (113 14)
• If the sum is less than v, replace v with the
  sum. (Is 14 lt 13?
  No, so dont replace it.)

13
Matrix Addition
v1
v3
v4
v2
v1
0
2
13
9
v2
0
1
7
10
v3
0
13
2
14
v4
0
7
4
11

• How about (v2, v4)? (1 2) 3
• Since this is less than 7, we replace the 7
  with a 3.

v1
v3
v4
v2
v1
0
2
13
9
v2
0
1
3
10
v3
0
13
2
14
v4
0
7
4
11
14
What is the purpose of doing this?
15
v1
v1
v3
v4
v2
1
2
v1
0
2
13
9
7
v2
v2
0
1
7 3
10
v4
v3
0
13
2
14
v4
0
7
4
11
v3
If we take the path from v2 to v4 it would cost
us a total of 7 units However, if we move from v2
to v1. Then move from v1 to v4. The total cost
would be 3 units. We have essentially found the
shortest path at this moment from v2 to v4.
16
Now do the same algorithm with row and column v2.

v1
v3
v4
v2
v1
v3
v4
v2
v1
v1
0
2
13
9
0
2
13
9
v2
v2
0
1
3
10
0
1
3
10
v3
v3
0
13
2
14
0
3
2
5
v4
v4
0
7
4
11
0
7
4
11
After row and column v3.
v1
v3
v4
v2
v1
v3
v4
v2
v1
v1
0
2
13
9
0
2
13
9
v2
v2
0
1
3
10
0
1
3
10
v3
v3
0
3
2
0
3
2
5
5
v4
v4
0
7
4
11
0
6
4
7
17
Finally, after row and column v4.
v1
v3
v4
v2
v1
v3
v4
v2
v1
v1
0
2
13
9
0
2
6
8
v2
v2
0
1
3
10
0
1
3
7
v3
v3
0
3
2
0
3
2
5
5
v4
v4
0
6
4
7
0
6
4
7
Although we did not see it in this example. It
is possible a square gets updated more than once.
After Floyds Algorithm is complete, we have
the shortest path/cost path from any node to any
other node in the graph. Time Complexity
O(V3)
18
Shortest Path Algorithms

• Dijkstras Algorithm
• Example of a greedy algorithm.
• A greedy algorithm is one that takes the shortest
  path at any given moment without looking ahead.
  Many times a greedy algorithm is not the best
  choice.

Greedy path
1
9
v2
s
v1
v4
2
2
v3
19
Dijkstras Algorithm
Pick a starting point s and declare it known
(True). We sill start with v1.
known is a true/false value. Whether we have
delclared a vertex known (true) or not
(false) dv is the cost from the previous vertex
to the current vertex pv is the previous vertex
in the path to the current vertex
known dv pv
V v1 v2 v3 v4
T 0 0
F 0 0
F 0 0
v1
s
F 0 0
v2
v4
v3
20
v1
9
known dv pv
V v1 v2 v3 v4
11
F 0 0
2
1
F 0 0
v2
v4
7
2
F 0 0
13
4
F 0 0
10
14
v3
v1 goes to v2 and v4. What is the cost of each
edge? v1 to v2 9, v1 to v4 2, and v1 to v3
13 Now we declare 1 as known (True) and update
dv and pv We have a choice now. Do we pick
v2 or v4 to explore?
known dv pv
V v1 v2 v3 v4
T 0 0
F 9 v1
F 13 v1
F 2 v1
21
v1
9
11
We pick v4. Why? Greedy Algorithm We want the
shortest path. v4 goes into v3 and v2 v4 to v3
4 v4 to v2 7 Now v4 is declared known. We
update the table like before.
2
1
v2
v4
7
2
13
4
10
14
v3
known dv pv
V v1 v2 v3 v4
T 0 0
Next we explore the next vertex that led from v1.
Namely v2
F 7 v4
F 4 v4
T 2 v1
22
v1
9
11
v2 goes to v1 and v3 v2 to v1 1 v2 to v3
10 Now we declare v2 as known and update the
table again.
2
1
v2
v4
7
2
13
4
10
14
v3
known dv pv
V v1 v2 v3 v4

• But wait! There is a problem.
• When updating v3 we see that the cost 10 is
  higher than the cost 4.
• We like the cost 4 so we will keep v3 the same.

T 1 v2
T 7 v4
F 4 v4
T 2 v1
23
v1
9
11
The last vertex to look at is v3. v3 goes to v2,
v1 and v4. v3 to v2 2 v3 to v1 13 v3 to v4
14 v3 is now declared known. Like before we
update as before, keeping in mind to keep lower
cost paths.
2
1
v2
v4
7
2
13
4
10
14
v3
known dv pv
V v1 v2 v3 v4
T 1 v2
T 2 v3
T 4 v4
T 2 v1
24
So How Do We Use This?
known dv pv
V v1 v2 v3 v4
T 1 v2
T 2 v3
Easy!
T 4 v4

• Take any vertex you want as a starting point.
• Now locate that vertex in the pv column.
• Work backwards to the vertex you want to end up
  in.

T 2 v1
Lets look at an example Lets say we want the
shortest path from v1 to v2. We see that v1
goes into v4 at a cost of 2. Now v4 goes into v3
at a cost of 4. Lastly v3 goes into v2 at a cost
of 2. 2 4 2 8
25
v1
9
11
If we took a direct path from v1 to v2 it would
cost us 9. But we found a way to get from v1 to
v2 at a cost of 8.
2
1
v2
v4
7
2
13
4
10
14
v3
known dv pv
V v1 v2 v3 v4
T 1 v2
Time complexity of Dijkstras Algorithm is O(V2)
T 2 v3
T 4 v4
T 2 v1
26
The End