1. Single Source Shortest Paths Bellman-Ford Algorithm2 4
1 1 b -3 -5 3
source a b c e f  2 -3 1 3 4 -5
What is a shortest path from a to b ?
a -> b = 2
a -> e -> c -> b = 0

2. Bellman-Ford Algorithmc f 2 4
1 1 b -3 -5 3
source
Initially:
The source is assigned d-attribute 0, every other vertex is assigned ∞. No vertex has a parent vertex.
current attributes d and π
a.d, a.π
b.d, b.π
c.d, c.π
e.d, e.π
f.d, f. π
0, NIL
, NIL
We implement the algorithm to visit the edges in each phase in this order:
(a,b), (a,e), (f,a), (b,c), (c,b), (e,c), (c,e), (f,e), (f,c)

3. Bellman-Ford Algorithmc f 2 4
1 1 b -3 -5 3
source
Phase 1:
Each edge is visited and relaxed once, starting with edge (a,b).
current attributes d and π
a.d, a.π
b.d, b.π
c.d, c.π
e.d, e.π
f.d, f. π
0, NIL
->2,NIL->a
, NIL
We implement the algorithm to visit the edges in each phase in this order:
(a,b), (a,e), (f,a), (b,c), (c,b), (e,c), (c,e), (f,e), (f,c)

4. Bellman-Ford Algorithmc f 2 4
1 1 b -3 -5 3
source
Phase 1:
We are relaxing the next edge, (a,e).
current attributes d and π
a.d, a.π
b.d, b.π
c.d, c.π
e.d, e.π
f.d, f. π
0, NIL
2, a
, NIL
->-3,NIL->a
We implement the algorithm to visit the edges in each phase in this order:
(a,b), (a,e), (f,a), (b,c), (c,b), (e,c), (c,e), (f,e), (f,c)

5. Bellman-Ford Algorithmc f 2 4
1 1 b -3 -5 3
source
Phase 1:
We are relaxing the next edge, (f,a).
current attributes d and π
a.d, a.π
b.d, b.π
c.d, c.π
e.d, e.π
f.d, f. π
still 0, NIL
2, a
, NIL
-3, a
We implement the algorithm to visit the edges in each phase in this order:
(a,b), (a,e), (f,a), (b,c), (c,b), (e,c), (c,e), (f,e), (f,c)

6. Bellman-Ford Algorithmc f 2 4
1 1 b -3 -5 3
source
Phase 1:
We are relaxing the next edge, (b,c).
current attributes d and π
a.d, a.π
b.d, b.π
c.d, c.π
e.d, e.π
f.d, f. π
0, NIL
2, a
->3,NIL->b
-3, a
, NIL
We implement the algorithm to visit the edges in each phase in this order:
(a,b), (a,e), (f,a), (b,c), (c,b), (e,c), (c,e), (f,e), (f,c)

7. Bellman-Ford Algorithmc f 2 4
1 1 b -3 -5 3
source
Phase 1:
We are relaxing the next edge, (c,b).
current attributes d and π
a.d, a.π
b.d, b.π
c.d, c.π
e.d, e.π
f.d, f. π
0, NIL
still 2, a
3, b
-3, a
, NIL
We implement the algorithm to visit the edges in each phase in this order:
(a,b), (a,e), (f,a), (b,c), (c,b), (e,c), (c,e), (f,e), (f,c)

8. Bellman-Ford Algorithmc f 2 4
1 1 b -3 -5 3
source
Phase 1:
We are relaxing the next edge, (e,c). This results in a second update of the attributes of c within the same phase.
current attributes d and π
a.d, a.π
b.d, b.π
c.d, c.π
e.d, e.π
f.d, f. π
0, NIL
2, a
3->-1,b->e
-3, a
, NIL
We implement the algorithm to visit the edges in each phase in this order:
(a,b), (a,e), (f,a), (b,c), (c,b), (e,c), (c,e), (f,e), (f,c)

9. Bellman-Ford Algorithmc f 2 4
1 1 b -3 -5 3
source
Phase 1:
We are relaxing the next edge, (c,e).
current attributes d and π
a.d, a.π
b.d, b.π
c.d, c.π
e.d, e.π
f.d, f. π
0, NIL
2, a
-1, e
still -3, a
, NIL
We implement the algorithm to visit the edges in each phase in this order:
(a,b), (a,e), (f,a), (b,c), (c,b), (e,c), (c,e), (f,e), (f,c)

10. Bellman-Ford Algorithmc f 2 4
1 1 b -3 -5 3
source
Phase 1:
We are relaxing the next edge, (f,e).
current attributes d and π
a.d, a.π
b.d, b.π
c.d, c.π
e.d, e.π
f.d, f. π
0, NIL
2, a
-1, e
still -3, a
, NIL
We implement the algorithm to visit the edges in each phase in this order:
(a,b), (a,e), (f,a), (b,c), (c,b), (e,c), (c,e), (f,e), (f,c)

11. Bellman-Ford Algorithmc f 2 4
1 1 b -3 -5 3
source
Phase 1:
We are relaxing the next edge, (f,c).
This completes Phase 1.
current attributes d and π
a.d, a.π
b.d, b.π
c.d, c.π
e.d, e.π
f.d, f. π
0, NIL
2, a
still -1, e
-3, a
, NIL
We implement the algorithm to visit the edges in each phase in this order:
(a,b), (a,e), (f,a), (b,c), (c,b), (e,c), (c,e), (f,e), (f,c)

12. Bellman-Ford Algorithmc f 2 4
1 1 b -3 -5 3
source
Phase 2:
Each edge is visited and relaxed once, starting with edge (a,b).
current attributes d and π
a.d, a.π
b.d, b.π
c.d, c.π
e.d, e.π
f.d, f. π
0, NIL
still 2, a
-1, e
-3, a
, NIL
We implement the algorithm to visit the edges in each phase in this order:
(a,b), (a,e), (f,a), (b,c), (c,b), (e,c), (c,e), (f,e), (f,c)

13. Bellman-Ford Algorithmc f 2 4
1 1 b -3 -5 3
source
Phase 2:
We are relaxing the next edge, (a,e).
current attributes d and π
a.d, a.π
b.d, b.π
c.d, c.π
e.d, e.π
f.d, f. π
0, NIL
2, a
-1, e
still -3, a
, NIL
We implement the algorithm to visit the edges in each phase in this order:
(a,b), (a,e), (f,a), (b,c), (c,b), (e,c), (c,e), (f,e), (f,c)

14. Bellman-Ford Algorithmc f 2 4
1 1 b -3 -5 3
source
Phase 2:
We are relaxing the next edge, (f,a).
current attributes d and π
a.d, a.π
b.d, b.π
c.d, c.π
e.d, e.π
f.d, f. π
still 0, NIL
2, a
-1, e
-3, a
, NIL
We implement the algorithm to visit the edges in each phase in this order:
(a,b), (a,e), (f,a), (b,c), (c,b), (e,c), (c,e), (f,e), (f,c)

15. Bellman-Ford Algorithmc f 2 4
1 1 b -3 -5 3
source
Phase 2:
We are relaxing the next edge, (b,c).
current attributes d and π
a.d, a.π
b.d, b.π
c.d, c.π
e.d, e.π
f.d, f. π
0, NIL
2, a
still -1, e
-3, a
, NIL
We implement the algorithm to visit the edges in each phase in this order:
(a,b), (a,e), (f,a), (b,c), (c,b), (e,c), (c,e), (f,e), (f,c)

16. Bellman-Ford Algorithmc f 2 4
1 1 b -3 -5 3
source
Phase 2:
We are relaxing the next edge, (c,b).
current attributes d and π
a.d, a.π
b.d, b.π
c.d, c.π
e.d, e.π
f.d, f. π
0, NIL
2->0,a->c
-1, e
-3, a
, NIL
We implement the algorithm to visit the edges in each phase in this order:
(a,b), (a,e), (f,a), (b,c), (c,b), (e,c), (c,e), (f,e), (f,c)

17. Bellman-Ford Algorithmc f 2 4
1 1 b -3 -5 3
source
Phase 2:
We are relaxing the next edge, (e,c).
current attributes d and π
a.d, a.π
b.d, b.π
c.d, c.π
e.d, e.π
f.d, f. π
0, NIL
0, c
still -1, e
-3, a
, NIL
We implement the algorithm to visit the edges in each phase in this order:
(a,b), (a,e), (f,a), (b,c), (c,b), (e,c), (c,e), (f,e), (f,c)

18. Bellman-Ford Algorithmc f 2 4
1 1 b -3 -5 3
source
Phase 2:
We are relaxing the next edge, (c,e).
current attributes d and π
a.d, a.π
b.d, b.π
c.d, c.π
e.d, e.π
f.d, f. π
0, NIL
0, c
-1, e
still -3, a
, NIL
We implement the algorithm to visit the edges in each phase in this order:
(a,b), (a,e), (f,a), (b,c), (c,b), (e,c), (c,e), (f,e), (f,c)

19. Bellman-Ford Algorithmc f 2 4
1 1 b -3 -5 3
source
Phase 2:
We are relaxing the next edge, (f,e).
current attributes d and π
a.d, a.π
b.d, b.π
c.d, c.π
e.d, e.π
f.d, f. π
0, NIL
0, c
-1, e
still -3, a
, NIL
We implement the algorithm to visit the edges in each phase in this order:
(a,b), (a,e), (f,a), (b,c), (c,b), (e,c), (c,e), (f,e), (f,c)

20. Bellman-Ford Algorithmc f 2 4
1 1 b -3 -5 3
source
Phase 2:
We are relaxing the next edge, (f,c).
This completes Phase 2.
current attributes d and π
a.d, a.π
b.d, b.π
c.d, c.π
e.d, e.π
f.d, f. π
0, NIL
0, c
still -1, e
-3, a
, NIL
We implement the algorithm to visit the edges in each phase in this order:
(a,b), (a,e), (f,a), (b,c), (c,b), (e,c), (c,e), (f,e), (f,c)

21. Bellman-Ford Algorithmc f 2 4
1 1 b -3 -5 3
source
Phase 3:
As in every phase, each edge is relaxed once. In this phase, no successful relaxation occurs, thus, we have found the final result.
final attributes d and π
a.d, a.π
b.d, b.π
c.d, c.π
e.d, e.π
f.d, f. π
0, NIL
0, c
-1, e
-3, a
, NIL
We implement the algorithm to visit the edges in each phase in this order:
(a,b), (a,e), (f,a), (b,c), (c,b), (e,c), (c,e), (f,e), (f,c)

22. The Shortest Path Tree rooted at a found by Bellman-Fordc f 2 4
1 1 b -3 -5 3
source
Vertex f is not part of the shortest path tree as it is
not reachable from the source.