closeness and betweenness
Goal
 Task 1.
 Closeness using Djikstra's Single-Source Shortest Path
algorithm

Djikstra's algorithm is given.
 Task 2.
 Betweenness using modified Floyd-Warshall algorithms


Floyd-Warshall algorithm is given. Modify the algorithm to
determine the betweenness.
You do not have to use modified Floyd-Warshall algorithms.
You can come up with your own algorithm.
Closeness
 The closeness of a node is defined as the maximum of
all the shortest path cost between it and all the other
vertices in the graph.
Example of closeness
 l1 =Cost(shortestPath(0,1))=2
 l2=Cost (shortestPath(0,2))=7
 l3=Cost (shortestPath(0,3))=8
 C(0)=max(l1,l2,l3)=max(2,7,8)=8
2
1
5
0
2
10
1
3
Example of closeness
 l1 =Cost(shortestPath(1,0))=2
 l2=Cost (shortestPath(1,2))=5
 l3=Cost (shortestPath(1,3))=6
 C(1)=max(l1,l2,l3)=max(2,5,6)=6
2
1
5
0
2
10
1
3
Example of closeness
 l1 =Cost(shortestPath(2,0))=7
 l2=Cost (shortestPath(2,1))=5
 l3=Cost (shortestPath(2,3))=1
 C(2)=max(l1,l2,l3)=max(7,5,1)=7
2
1
5
0
2
10
1
3
Example of closeness
 l1 =Cost(shortestPath(3,0))=8
 l2=Cost (shortestPath(3,1))=6
 l3=Cost (shortestPath(3,2))=1
 C(3)=max(l1,l2,l3)=max(8,6,1)=8
2
1
5
0
2
10
1
3
 Algorithm:
Let G = (V, E) be an undirected weighted graph
Let c(i) = 0
For all vertices j in V
Let p = shortestPath(i, j)
if length(p)>c(i)
Let c(i) = length(p)
 Note: Use Djikstra's Single-Source Shortest Path
algorithm to determine the shortest path cost.
Betweenness
Let G = (V, E) be an undirected weighted graph
For all vertices a, b, in V where a != b
Let p = shortestPath(a, b)
For all vertices u in p where u != a and u != b
betweenness[u] = betweenness[u] + 1
Example of betweenness
 ShortestPath(0,1)=0,1
 ShortestPath(0,2)=0,1,2
 ShortestPath(0,3)=0,1,2,3
 ShortestPath(1,2)=1,2
 ShortestPath(1,3)=1,2,3
 ShortestPath(2,3)=2,3
 betweenness[1]=2
 betweenness[2]=2
2
1
5
0
2
10
1
3
Floyd-Warshall algorithm
 This algorithm calculates the length of the shortest path between all
nodes of a graph in O(V3) time. Note that it doesn’t actually find the
paths, only their lengths.
 G[j,i] is the shortest distance so far from i to j.
for all vertices i in V do
for all vertices j in V do
G(i,j) = weight(i,j)
for all vertices i in V do
for all vertices j in V do
if (G[j,i] > 0)
for all vertices k in V do
if (G[i,k] > 0)
if (G[j,k] == 0) or (G[j,k] > G[j,i]+G[i,k])
G[j,k] = G[j,i]+G[i,k]
Example of Floyd-Warshall
algorithm
ABCDE
A 0 10 0 5 0
B 10 0 5 5 10
C05000
D 5 5 0 0 20
E 0 10 0 20 0
Matrix G
for all vertices i in V do
for all vertices j in V do
G(i,j) = weight(i,j)
Interspace A between any two nodes in
hopes of finding a shorter path.
ABCDE
A 0 10 0 5 0
B 10 0 5 5 10
C05000
D 5 5 0 0 20
E 0 10 0 20 0
Old Matrix G
ABCDE
A 0 10 0 5 0
B 10 0 5 5 10
C05000
D 5 5 0 0 20
E 0 10 0 20 0
New Matrix G
Interspace B between any two nodes in
hopes of finding a shorter path.
ABCDE
A 0 10 0 5 0
B 10 0 5 5 10
C05000
D 5 5 0 0 20
E 0 10 0 20 0
Old Matrix G
ABC DE
A 0 10 15 5 20
B 10 0 5 5 10
C 15 5 0 10 15
D 5 5 10 0 15
E 20 10 15 15 0
New Matrix G
Interspace C between any two nodes in
hopes of finding a shorter path.
ABCDE
A 0 10 15 5 20
B 10 0 5 5 10
C 15 5 0 10 15
D 5 5 10 0 15
E 20 10 15 15 0
Old Matrix G
ABCDE
A 0 10 15 5 20
B 10 0 5 5 10
C 15 5 0 10 15
D 5 5 10 0 15
E 20 10 15 15 0
New Matrix G
Interspace D between any two nodes in
hopes of finding a shorter path.
ABCDE
A 0 10 15 5 20
B 10 0 5 5 10
C 15 5 0 10 15
D 5 5 10 0 15
E 20 10 15 15 0
Old Matrix G
ABCDE
A 0 10 15 5 20
B 10 0 5 5 10
C 15 5 0 10 15
D 5 5 10 0 15
E 20 10 15 15 0
New Matrix G
Interspace E between any two nodes in
hopes of finding a shorter path.
ABCDE
A 0 10 15 5 20
B 10 0 5 5 10
C 15 5 0 10 15
D 5 5 10 0 15
E 20 10 15 15 0
Old Matrix G
ABCDE
A 0 10 15 5 20
B 10 0 5 5 10
C 15 5 0 10 15
D 5 5 10 0 15
E 20 10 15 15 0
New Matrix G