1. Presentation
On
Single Source Shortest Path.

2. Presenter:
Kazi Emad
ID no. 191902025
01

3. Agenda
1. Overview of Single Source Shortest
Path
2. Types of Single Source Shortest Path
Algorithm
3. Representation of Single Source
Shortest Path
4. Initialization
5. Relaxation
6. Implementation of Dijkstra's Algorithm
7. Does Dijkstra’s Algorithm Always Work?
8. Implementation of Bellman-Ford
Algorithm
9. Negative Weight Cycles in Bellman-
Ford Algorithm

4. Overview of Single Source Shortest
Path:
 In a shortest-paths problem, we are given a weighted, directed graph G = (V, E), with weight
function w : E -> R mapping edges to real-valued weights. The weight w(p) of path p = (V0,
V1….., Vk) ki is the sum of the weights of its constituent edges:
w(p)  w(vi ,vi1)
i1
 The single-source shortest path problem, in which we have to find shortest paths from a source
vertex v to all other vertices in the graph.

5. Types of Single Source Shortest Path Algorithm:
There are two types of single source shortest path algorithm. And they are:
1. Dijkstra’s Algorithm.
2. Bellman-Ford Algorithm

6. Representation of Single Source Shortest Path:
For each vertex v  V:
 d[v] = δ(s, v): a shortest-path estimate
 Initially, d[v]=∞
 Reduces as algorithms progress
 [v] = predecessor of v on a shortest path from s
 If no predecessor, [v] = NIL
  induces a tree—shortest-path tree

7. Initialization:
INITIALIZE-SINGLE-SOURCE(V, s)
1. for each v  V
2. do d[v] ← 
3. [v] ← NIL
4. d[s] ← 0

8. Relaxation:
Relaxing an edge (u, v) = testing whether we can improve the shortest path to v found so far by
going through u.
Example:
If d[v] > d[u] + w(u, v)
 update d[v] and [v]
S 8
2
4
u
v
2 6
u v
4
RELAX(u,v,w)

9. Implementation of Dijkstra's Algorithm:
Given a graph and a source vertex in the graph, find shortest paths from
source to all vertices in the given graph.
s
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t
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10. Implementation of Dijkstra's Algorithm:
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






0
distance label

11. Implementation of Dijkstra's Algorithm:
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 
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0
distance label
delmin

12. Implementation of Dijkstra's Algorithm:
s
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
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14

0
distance label
decrease key
X

X
X

13. Implementation of Dijkstra's Algorithm:
s
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
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distance label
X

X
X
delmin

14. Implementation of Dijkstra's Algorithm:
s
3
t
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2
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X

X
X

15. Implementation of Dijkstra's Algorithm:
s
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t
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2
9
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20
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
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0
X

X
X
decrease key
X 33

16. Implementation of Dijkstra's Algorithm:
s
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t
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
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X

X
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X 33
delmin

17. Implementation of Dijkstra's Algorithm:
s
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20
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0
X
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X
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X 33
44
X
X
32

18. Implementation of Dijkstra's Algorithm:
s
3
t
2
6
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2
9
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20
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14

0
X

X
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44
X
delmin
X 33X
32

19. Implementation of Dijkstra's Algorithm:
s
3
t
2
6
7
4
5
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2
9
14
15
5
30
20
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16
11
6
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0
X

X
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44
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35X
59 X
24
X 33X
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20. Implementation of Dijkstra's Algorithm:
s
3
t
2
6
7
4
5
24
18
2
9
14
15
5
30
20
44
16
11
6
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6
15
9


14

0
X

X
X
44
X
35X
59 X
delmin
X 33X
32

21. Implementation of Dijkstra's Algorithm:
s
3
t
2
6
7
4
5
24
18
2
9
14
15
5
30
20
44
16
11
6
19
6
15
9


14

0
X

X
X
44
X
35X
59 XX51
X 34
X 33X
32

22. Implementation of Dijkstra's Algorithm:
s
3
t
2
6
7
4
5
18
2
9
14
15
5
30
20
44
16
11
6
19
6
15
9


14

0
X

X
X
44
X
35X
59 XX51
X 34
delmin
X 33X
32
24

23. Implementation of Dijkstra's Algorithm:
s
3
t
2
6
7
4
5
18
2
9
14
15
5
30
20
44
16
11
6
19
6
15
9


14

0
X

X
X
44
X
35X
59 XX51
X 34
24
X50
X45
X 33X
32

24. Implementation of Dijkstra's Algorithm:
s
3
t
2
6
7
4
5
18
2
9
14
15
5
30
20
44
16
11
6
19
6
15
9


14

0
X

X
X
44
X
35X
59 XX51
X 34
24
X50
X45
delmin
X 33X
32

25. Implementation of Dijkstra's Algorithm:
s
3
t
2
6
7
4
5
18
2
9
14
15
5
30
20
44
16
11
6
19
6
15
9


14

0
X

X
X
44
X
35X
59 XX51
X 34
24
X50
X45
X 33X
32

26. Implementation of Dijkstra's Algorithm:
s
3
t
2
6
7
4
5
18
2
9
14
15
5
30
20
44
16
11
6
19
6
15
9


14

0
X

X
X
44
X
35X
59 XX51
X 34
X50
X45
delmin
X 33X
32
24

27. Implementation of Dijkstra's Algorithm:
s
3
t
2
6
7
4
5
24
18
2
9
14
15
5
30
20
44
16
11
6
19
6
15
9


14

0
X

X
X
44
X
35X
59 XX51
X 34
X50
X45
X 33X
32

28. Implementation of Dijkstra's Algorithm:
s
3
t
2
6
7
4
5
24
18
2
9
14
15
5
30
20
44
16
11
6
19
6
15
9


14

0
X

X
X
44
X
35X
59 XX51
X 34
X50
X45
X 33X
32

29. Implementation of Dijkstra's Algorithm:
S 2 3 4 5 6 7 t
S 0 9 inf inf inf 14 15 inf
2 0 9 33 inf inf 14 15 inf
6 0 9 32 inf 44 14 15 inf
7 0 9 32 inf 35 14 15 59
3 0 9 32 inf 34 14 15 51
5 0 9 32 45 34 14 15 50
4 0 9 32 45 34 14 15 50
t 0 9 32 45 34 14 15 50
Tabular Solution of the Graph for Shortest Path

30. Does Dijkstra’s Algorithm Always Work?
No. Dijkstra’s Algorithm doesn’t always work in case of negative weight
edges.
So, we have a new algorithm named “Bellman-Ford Algorithm” which can
also perform in negative weight edges.

31. Implementation of Bellman-Ford Algorithm:

32. Initialize all the
distances
Implementation of Bellman-Ford Algorithm:

33. iterate over all
edges/vertices
and apply update
rule
Implementation of Bellman-Ford Algorithm:

34. check for negative
cycles
Implementation of Bellman-Ford Algorithm:

35. G
S
F
E
A
D
B
C
10
8
1
-1
-1
3
1
1
2
-2
-4
How many edges is
the shortest path
from s to:
A:
Implementation of Bellman-Ford Algorithm:

36. G
S
F
E
A
D
B
C
10
8
1
-1
-1
3
1
1
2
-2
-4
How many edges is
the shortest path
from s to:
A: 3
Implementation of Bellman-Ford Algorithm:

37. G
S
F
E
A
D
B
C
10
8
1
-1
-1
3
1
1
2
-2
-4
How many edges is
the shortest path
from s to:
A: 3
B:
Implementation of Bellman-Ford Algorithm:

38. G
S
F
E
A
D
B
C
10
8
1
-1
-1
3
1
1
2
-2
-4
How many edges is
the shortest path
from s to:
A: 3
B: 5
Implementation of Bellman-Ford Algorithm:

39. G
S
F
E
A
D
B
C
10
8
1
-1
-1
3
1
1
2
-2
-4
How many edges is
the shortest path
from s to:
A: 3
B: 5
D:
Implementation of Bellman-Ford Algorithm:

40. G
S
F
E
A
D
B
C
10
8
1
-1
-1
3
1
1
2
-2
-4
How many edges is
the shortest path
from s to:
A: 3
B: 5
D: 7
Implementation of Bellman-Ford Algorithm:

41. G
S
F
E
A
D
B
C
10
8
1
-1
-1
3
1
1
2
-2
-4
0 





Iteration: 0
Implementation of Bellman-Ford Algorithm:

42. G
S
F
E
A
D
B
C
10
8
1
-1
-1
3
1
1
2
-2
-4
0 10




8
Iteration: 1
Implementation of Bellman-Ford Algorithm:

43. G
S
F
E
A
D
B
C
10
8
1
-1
-1
3
1
1
2
-2
-4
0 10


12
9
8
Iteration: 2
Implementation of Bellman-Ford Algorithm:

44. G
S
F
E
A
D
B
C
10
8
1
-1
-1
3
1
1
2
-2
-4
0 5
10

8
9
8
Iteration: 3
A has the correct
distance and path
Implementation of Bellman-Ford Algorithm:

45. G
S
F
E
A
D
B
C
10
8
1
-1
-1
3
1
1
2
-2
-4
0 5
6
11
7
9
8
Iteration: 4
Implementation of Bellman-Ford Algorithm:

46. G
S
F
E
A
D
B
C
10
8
1
-1
-1
3
1
1
2
-2
-4
0 5
5
7
147
9
8
Iteration: 5
B has the correct
distance and path
Implementation of Bellman-Ford Algorithm:

47. G
S
F
E
A
D
B
C
10
8
1
-1
-1
3
1
1
2
-2
-4
0 5
5
6
107
9
8
Iteration: 6
Implementation of Bellman-Ford Algorithm:

48. G
S
F
E
A
D
B
C
10
8
1
-1
-1
3
1
1
2
-2
-4
0 5
5
6
97
9
8
Iteration: 7
D (and all other
nodes) have the
correct distance
and path
Implementation of Bellman-Ford Algorithm:

49. Negative Weight Cycles in Bellman-Ford
Algorithm:
• Negative edges are OK, as long as there are no negative weight cycles (otherwise paths with
arbitrary small “lengths” would be possible)
• Shortest-paths can have no cycles (otherwise we could improve them by removing cycles)
– Any shortest-path in graph G can be no longer than n – 1 edges, where n is the number of
vertices.

50. Thank You