Negative edge weights are found in various applications of graphs, hence the usefulness of this algorithm

1. BELLMAN
FORD
ALGORITHM
MD.SAJJADUL ISLAM
ID: 172-35-2088
SOFTWARE ENGNIEERING
DAFFODIL INTERNATIONAL UNIVERSITY

2. B E L L M A N - F O R D
Richard E. Bellman Lester R.Ford ,Jr.

3. • The algorithm was first proposed by Alfonso Shimbel
in 1955
• Named after Richard Bellman and Lester Ford Jr who
published it in 1956 and 1958
• Edward F. Moore also published the same algorithm
in 1957, and for this reason it is also sometimes
called the Bellman–Ford–Moore algorithm

4. B E L L M A N – F O R D
A L G O R I T H M
 Negative edge weights are found in various applications of graphs,
hence the usefulness of this algorithm. If a graph contains a "negative
cycle" ( a cycle whose edges sum to a negative value) that is reachable
from the source, then there is no cheapest path: any path that has a
point on the negative cycle can be made cheaper by one more walk
around the negative cycle. In such a case, the Bellman–Ford algorithm
can detect negative cycles and report their existence.

5. S H O R T E S T PAT H
P R O B L E M
Shortest path network
Driected 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
δ (S , Vi-1) + W(Vi-1,Vi)
Vi
V1
V2
V3
δ (S , Vi)=

6. Dijkstra’s Algorithm fails when theme is negative edge
Ex: Selects vertex v immediately after s . But short path s to v is
s x
v y
4
6
-9
2
S-X-Y-V
Solution is Bellman Ford Algorithm which can work on negative edge.

7. BELLMAN-FOR D ( G ,s )
INITIALIZE SINGLE – SOURCE ( G , s )
for i 1 to |V|-1 do computaion
for each edge ( u, v ) € G.E do
RELAX ( U , V )
for each edge ( u, v ) € E do
if d[V] > d[U] + W( U , V ) then check
return FALSE
return TRUE

8. B E L L M A N – F O R D
E X A M P L E
A
B D
E
C
Vertices A B C D E
Cost 0 α α α α
Prevertices ─ ─ ─ ─ ─1
5
2
-3
-2
2
1
2
0 α
α α
α
1 s t :

9. B E L L M A N – F O R D
E X A M P L E
A
B D
E
C
1
5
2
-3
-2
2
1
2
0
5α
α α
α
2
Vertices A B C D E
Cost 0 α 5 α 2 α α
Pre ─ ─ A A ─

10. B E L L M A N – F O R D
E X A M P L E
A
B D
E
C
1
5
2
-3
-2
2
1
2
0
5α
α α
α
20
Vertices A B C D E
Cost 0 α 5 α 0 2 α α
Pre ─ ─ A A B ─

11. B E L L M A N – F O R D
E X A M P L E
A
B D
E
C
1
5
2
-3
-2
2
1
2
0
5α
α α
202
Vertices A B C D E
Cost 0 α 2 5 α 0 2 α α 6
Pre ─ C A A B C

12. B E L L M A N – F O R D
E X A M P L E
A
B D
E
C
1
5
2
-3
-2
2
1
2
0
5α
α α
202
Vertices A B C D E
Cost 0 α 2 5 α 0 2 α α 6 2
Pre ─ C A A B C D

13. B E L L M A N – F O R D
E X A M P L E
A
B D
E
C
1
5
2
-3
-2
2
1
2
0
5α
α α
202
Vertices A B C D E
Cost 0 α 2 5 α 0 2 α α 6
2
Pre ─ C A A B C D

14. B E L L M A N – F O R D
E X A M P L E
A
B D
E
C
Vertices A B C D E
Cost 0 2 5 0 2
Pre ─ C A B D
1
5
2
-3
-2
2
1
2
0 5
2 0
2
2nd:

15. B E L L M A N – F O R D
E X A M P L E
A
B D
E
C
Vertices A B C D E
Cost 0 2 5 0 2
Pre ─ C A B D
1
5
2
-3
-2
2
1
2
0
2 0
2
5

16. B E L L M A N – F O R D
E X A M P L E
A
B D
E
C
Vertices A B C D E
Cost 0 2 5 0 2
Pre ─ C A B D
1
5
2
-3
-2
2
1
2
0
2 0
2

17. B E L L M A N – F O R D
E X A M P L E
A
B D
E
C
Vertices A B C D E
Cost 0 2 5 0 2
Pre ─ C A B D
1
5
2
-3
-2
2
1
2
0
2 0
2

18. B E L L M A N – F O R D
E X A M P L E
A
B D
E
C
Vertices A B C D E
Cost 0 2 5 0 2
Pre ─ C A B D
1
5
2
-3
-2
2
1
2
0
2 0
2

19. B E L L M A N – F O R D
E X A M P L E
A
B D
E
C
Vertices A B C D E
Cost 0 2 5 0 2
Pre ─ C A B D
1
5
2
-3
-2
2
1
2
0 5
2 0
2

20. B E L L M A N – F O R D
E X A M P L E
Vertices A B C D E
Cost 0 2 5 0 2
Pre ─ C A B D
A
B D
E
C
1
5
2
-3
-2
2
1
2
0 5
2 0
2
Shortest path are –
E = ACBDE = {5+(-3)+(-2)+2 }= 2