1. Bellman Ford Algorithm
Taimur khan
MS Scholar
University of Peshawar
taimurkhan803@upesh.edu.pk

2. Shortest path problem
 Shortest path network
Directed graph
Source s, Destination t
cost( v-u) cost of using edge from v to u
Shortest path problem
Find shortest directed path from s to t
Cost of path = sum of arc cost in path

3. Applications
 Netowrks (Routing )
 Robot Navigation
 Urban Traffic Planning
 Telemarketer operator scheduling
 Routing of Communication messages
 Optimal truck routing through given traffic congestion patteren
 OSPF routing protocol for IP

4. Single source shortest path
 Given graph (directed or undirected) G = (V,E) with
 weight function w: E  R and a vertex sV,
 find for all vertices vV the minimum possible weight for
 path from s to v.
 There are two algorithms
 Dijikstra’s Algorithm
 Bellman Ford algorithm

5. Relaxation
 Maintain d[v] for each v  V
 d[v] is called shortest-path weight estimate
INIT(G, s)
for each v  V do
d[v] ← ∞
π[v] ← NIL
d[s] ← 0

6. Relaxation
 RELAX(u, v)
 if d[v] > d[u]+w(u,v) then
 d[v] ← d[u]+w(u,v)
 π[v] ← u
5
2
2
9
5 7
Relax(u,v)
5
2
2
6
5 6

7. Bellman Ford
 Dijikstra Algorithm fails when there is negative edge
Solution is Bellman Ford Algorithm which can work on negative edges

8. PsuedoCode
BELMAN-FORD( G, s )
INIT( G, s )
for i ←1 to |V|-1 do
for each edge (u, v)  E do
RELAX( u, v )
for each edge ( u, v )  E do
if d[v] > d[u]+w(u,v) then
return FALSE > neg-weight cycle
return TRUE

9. s a b t
0 0 ∞ ∞ ∞
1 0
2 0
3 0
s
a
b
t
5
-2
4
6
-9

10. s a b t
0 0 ∞ ∞ ∞
1 0 5 4 ∞
2 0
3 0
s
a
b
t
5
-2
4
6
-9
5
4
∞

11. s a b t
0 0 ∞ ∞ ∞
1 0 5 4 ∞
2 0 5 3 11
3 0
s
a
b
t
5
-2
4
6
-9
4
11
5

12. s a b t
0 0 ∞ ∞ ∞
1 0 5 4 ∞
2 0 5 3 11
3 0 5 2 11
s
a
b
t
5
-2
4
6
-9
2
11

13. Any Question?
Thank you