Data Structure-shortest path dijkstra’s

1. .Shortest Path Problems
- Greedy Approach

2. INTRODUCTION
Dijkstra’s algorithm calculates the least distance
from a starting vertex to a destination vertex from
the given weighted graph
Input:
• A weighted graph
• Starting vertex
• Destination vertex
Output:
A graph which connects all the vertices with least
distance

3. Dijkstra's Algorithm
Input:
a weighted digraph G=(V,E) with positive edge weights
a source node s V∈
Initialization:
d[s]=0
for each vertex x V-s∈
d[x]=infinity
Mark all the vertices as unprocessed
Iteration:
for i=1 to |V|
Choose an unprocessed vertex x from V with minimum d[x]
Mark x as processed
for all y adj(x)∈
if d[y] > d[x]+w(x,y)
d[y] = d[x]+w(x,y)

4. Dijkstra's Algorithm With PATH
Input:
a weighted digraph G=(V,E) with positive edge weights
a source node s V∈
Initialization:
d[s]=0 predecessor [ s ] = undefined
for each vertex x V-s∈
d[x]=infinity
Mark all the vertices as unprocessed
Iteration:
for i=1 to |V|
Choose an unprocessed vertex x from V with minimum d[x]
Mark x as processed
for all y adj(x)∈
if d[y] > d[x]+w(x,y)
d[y] = d[x]+w(x,y)
predecessor [ y ] = x

5. Printing the Distance and path
for each vertex v
{
print Distance d[v]
print_path(v);
}
print_path(v)
{
if (v is undefined)
return;
else
print_path ( prdecessor[v] ); // note recursion
print v
}

6. C Code
for(i=1; i<=numofvertices; i++)
{
printf("nDistance : %d ::: Path: ",distance[i]);
print_path(i);
}
void print_path(int i)
{
if (i==-1)
return;
else
print_path(prdecessor[i]);
Assumes that vertices are
numbered from 1.
distance array and
predecessor array are
computed by dijkstra
algorithm
-1 is undefined vertex

7. Dijkstra's Shortest Path Algorithm
Find shortest path from s to t.
7
s
3
t
2
6
7
4
5
24
18
2
9
14
15
5
30
20
44
16
11
6
19
6

8. Dijkstra's Shortest Path Algorithm 8
s
3
t
2
6
7
4
5
24
18
2
9
14
15
5
30
20
44
16
11
6
19
6
∞
∞
∞
∞
∞
∞
∞
0
distance label
S = { }
PQ = { s, 2, 3, 4, 5, 6, 7, t }

9. Dijkstra's Shortest Path Algorithm 9
s
3
t
2
6
7
4
5
24
18
2
9
14
15
5
30
20
44
16
11
6
19
6
∞
∞
∞
∞
∞
∞
∞
0
distance label
S = { }
PQ = { s, 2, 3, 4, 5, 6, 7, t }
delmin

10. Dijkstra's Shortest Path Algorithm
10
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
distance label
S = { s }
PQ = { 2, 3, 4, 5, 6, 7, t }
decrease key
∞X
∞
∞X
X

11. Dijkstra's Shortest Path Algorithm 11
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
distance label
S = { s }
PQ = { 2, 3, 4, 5, 6, 7, t }
∞X
∞
∞X
X
delmin

12. Dijkstra's Shortest Path Algorithm
12
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
S = { s, 2 }
PQ = { 3, 4, 5, 6, 7, t }
∞X
∞
∞X
X

13. Dijkstra's Shortest Path Algorithm
13
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
S = { s, 2 }
PQ = { 3, 4, 5, 6, 7, t }
∞X
∞
∞X
X
decrease key
X 33

14. Dijkstra's Shortest Path Algorithm
14
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
S = { s, 2 }
PQ = { 3, 4, 5, 6, 7, t }
∞X
∞
∞X
X
X 33
delmin

15. Dijkstra's Shortest Path Algorithm15
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
S = { s, 2, 6 }
PQ = { 3, 4, 5, 7, t }
∞X
∞
∞X
X
X 33
44
X
X
32

16. Dijkstra's Shortest Path Algorithm16
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
S = { s, 2, 6 }
PQ = { 3, 4, 5, 7, t }
∞X
∞
∞X
X
44
X
delmin
∞X 33X
32

17. Dijkstra's Shortest Path Algorithm
17
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
S = { s, 2, 6, 7 }
PQ = { 3, 4, 5, t }
∞X
∞
∞X
X
44
X
35X
59 X
24
∞X 33X
32

18. Dijkstra's Shortest Path Algorithm 18
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
S = { s, 2, 6, 7 }
PQ = { 3, 4, 5, t }
∞X
∞
∞X
X
44
X
35X
59 X
delmin
∞X 33X
32

19. Dijkstra's Shortest Path Algorithm
19
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
S = { s, 2, 3, 6, 7 }
PQ = { 4, 5, t }
∞X
∞
∞X
X
44
X
35X
59 XX51
X 34
∞X 33X
32

20. Dijkstra's Shortest Path Algorithm
20
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
S = { s, 2, 3, 6, 7 }
PQ = { 4, 5, t }
∞X
∞
∞X
X
44
X
35X
59 XX51
X 34
delmin
∞X 33X
32
24

21. Dijkstra's Shortest Path Algorithm
21
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
S = { s, 2, 3, 5, 6, 7 }
PQ = { 4, t }
∞X
∞
∞X
X
44
X
35X
59 XX51
X 34
24
X50
X45
∞X 33X
32

22. Dijkstra's Shortest Path Algorithm
22
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
S = { s, 2, 3, 5, 6, 7 }
PQ = { 4, t }
∞X
∞
∞X
X
44
X
35X
59 XX51
X 34
24
X50
X45
delmin
∞X 33X
32

23. Dijkstra's Shortest Path Algorithm 23
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
S = { s, 2, 3, 4, 5, 6, 7 }
PQ = { t }
∞X
∞
∞X
X
44
X
35X
59 XX51
X 34
24
X50
X45
∞X 33X
32

24. Dijkstra's Shortest Path Algorithm
24
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
S = { s, 2, 3, 4, 5, 6, 7 }
PQ = { t }
∞X
∞
∞X
X
44
X
35X
59 XX51
X 34
X50
X45
delmin
∞X 33X
32
24

25. Dijkstra's Shortest Path Algorithm 25
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
S = { s, 2, 3, 4, 5, 6, 7, t }
PQ = { }
∞X
∞
∞X
X
44
X
35X
59 XX51
X 34
X50
X45
∞X 33X
32

26. Dijkstra's Shortest Path Algorithm 26
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
S = { s, 2, 3, 4, 5, 6, 7, t }
PQ = { }
∞X
∞
∞X
X
44
X
35X
59 XX51
X 34
X50
X45
∞X 33X
32

28. ASSESSMENT

29. Single Source Shortest Path
1
2
76
5
4
3
5
3
1
7
5
4
9
4
6
8
1
4
1