1. Graph Algorithms: Shortest Path
We are given a weighted,
directed graph G = (V, E), with weight function
w: E  R mapping edges to real valued weights. 1 5 10 1 8 3 2 4 3 1 1 3 1 4 6 6 5 2

2. Graph Algorithms: Shortest Path
We are given a weighted,
directed graph G = (V, E), with weight function
w: E  R mapping edges to real valued weights. 1
The weight of a path
p = (v , v , . . ., v ) is
 w(v , v ). 5 10 1 1 2 k
k-1 8 3 2 4 3 i
i+1
i=1 1 1 3
The weight of the path
along the red edges is
= 12. 1 4 6 6 5 2

3. Graph Algorithms: Shortest Path
We are given a weighted,
directed graph G = (V, E), with weight function
w: E  R mapping edges to real valued weights. 1
The weight of a path
p = (v , v , . . ., v ) is
 w(v , v ). 5 10 1 1 2 k
k-1 8 3 2 4 3 i
i+1
i=1 1
Single source shortest path
problem: given a vertex s, for every vertex v  V find a shortest path from s to v. 1 3 1 4 6 6 5 2

4. Graph Algorithms: Shortest Path
Single source shortest
path problem: given vertex s, for every vertex v  V find a shortest path from s to v. 1 5 10 1
Dijkstra’s algorithm solves
this problem efficiently for the case in which all weights are
nonnegative (as in the example graph). 8 3 2 4 3 1 1 3 1 4 6 6 5 2

5. Graph Algorithms: Shortest Path
Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined. 1
It also maintains, for each
vertex v not in S, an upper bound d[v] on the weight of a shortest path from source s to v. 5 10 1 8 3 2 4 3 1 1 3 1 4 6 6 5 2

6. Graph Algorithms: Shortest Path
Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined. 1
It also maintains, for each
vertex v not in S, an upper bound d[v] on the weight of a shortest path from source s to v. 5 10 1 8 3 2 4 3 1
The algorithm repeatedly selects
the vertex u  V – S with minimum
bound d[v], inserts u into S,
and relaxes all edges leaving u. 1 3 1 4 6 6 5 2

7. Graph Algorithms: Shortest Path
Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined.
Suppose vertex 1
is the source. s 1
Set S = { } initially. 5 10 1 v 1 2 3 4 5 6 8 3 2 4 3 d      1 1 3 1 4 6
Also initialize a queue with all
the vertices and their upper
bounds. 6 5 2

8. Graph Algorithms: Shortest Path
Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined.
Suppose vertex 1
is the source. s 1
S = { } 5 10 1 v 1 2 3 4 5 6 8 3 2 4 3 d      1
Select the vertex u  V – S with
minimum bound d[v], insert
u into S, . . . 1 3 1 4 6 6 5 2

9. Graph Algorithms: Shortest Path
Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined.
Suppose vertex 1
is the source. s 1 1
(here we keep the bound
with the vertex in S)
S = 5 10 1 v 2 3 4 5 6 8 3 2 4 3 d      1
. . . remove it from the
queue, insert it in S,. . . 1 3 1 4 6 6 5 2

10. Graph Algorithms: Shortest Path
Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined.
Suppose vertex 1
is the source. s 1 1
S = 5 10 1 v 2 3 4 5 6 8 3 2 4 3 10 1 5  d  1
. . . and relax the edges from
this vertex. 1 3 1 4 6 6 5 2

11. Graph Algorithms: Shortest Path
Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined.
Suppose vertex 1
is the source. s 1 1
S = 5 10 1 v 3 4 2 5 6 8 3 2 4 3 1 5 10  d  1
. . . reordering the queue
according to the new upper
bounds. 1 3 1 4 6 6 5 2

12. Graph Algorithms: Shortest Path
Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined.
Suppose vertex 1
is the source. s 1 1
S = 5 10 1 v 3 4 2 5 6 8 3 2 4 3 1 5 10  d  1
Repeat . . . 1 3 1 4 6 6 5 2

13. Graph Algorithms: Shortest Path
Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined.
Suppose vertex 1
is the source. s 1 1 3
S = 5 1 10 1 v 4 2 5 6 8 3 2 4 3 5 10  d  1
. . . remove from queue and
put in S 1 3 1 4 6 6 5 2

14. Graph Algorithms: Shortest Path
Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined.
Suppose vertex 1
is the source. s 1 1 3
S = 5 1 10 1 v 4 2 5 6 8 3 2 4 3 4 9 2 d  1
. . . relax the edges . . . 1 3 1 4 6 6 5 2

15. Graph Algorithms: Shortest Path
Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined.
Suppose vertex 1
is the source. s 1 1 3
S = 5 1 10 1 v 5 4 2 6 8 3 2 4 3 d 2 4 9  1
. . . and reorder the queue 1 3 1 4 6 6 5 2

16. Graph Algorithms: Shortest Path
Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined.
Suppose vertex 1
is the source. s 1 1 3
S = 5 1 10 1 v 5 4 2 6 8 3 2 4 3 d 2 4 9  1
Repeat . . . 1 3 1 4 6 6 5 2

17. Graph Algorithms: Shortest Path
Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined.
Suppose vertex 1
is the source. s 1 1 3 5
S = 5 1 2 10 1 v 4 2 6 8 3 2 4 3 4 9 d  1
. . . remove from queue and
put in S 1 3 1 4 6 6 5 2

18. Graph Algorithms: Shortest Path
Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined.
Suppose vertex 1
is the source. s 1 1 3 5
S = 5 1 2 10 1 v 4 2 6 8 3 2 4 3 4 9 d 4 1
. . . relax the edges . . . 1 3 1 4 6 6 5 2

19. Graph Algorithms: Shortest Path
Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined.
Suppose vertex 1
is the source. s 1 1 3 5
S = 5 1 2 10 1 v 4 6 2 8 3 2 4 3 d 4 4 9 1
. . . and reorder the queue 1 3 1 4 6 6 5 2

20. Graph Algorithms: Shortest Path
Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined.
Suppose vertex 1
is the source. s 1 1 3 5
S = 5 1 2 10 1 v 4 6 2 8 3 2 4 3 d 4 4 9 1
Repeat . . . 1 3 1 4 6 6 5 2

21. Graph Algorithms: Shortest Path
Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined.
Suppose vertex 1
is the source. s 1 1 3 4 5
S = 5 1 4 2 10 1 v 6 2 8 3 2 4 3 d 4 9 1
. . . remove from queue and
put in S 1 3 1 4 6 6 5 2

22. Graph Algorithms: Shortest Path
Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined.
Suppose vertex 1
is the source. s 1 1 3 4 5
S = 5 1 4 2 10 1 v 6 2 8 3 2 4 3 d 4 9 1
. . . relax the edges (no change
in this case) . . . 1 3 1 4 6 6 5 2

23. Graph Algorithms: Shortest Path
Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined.
Suppose vertex 1
is the source. s 1 1 3 4 5
S = 5 1 4 2 10 1 v 6 2 8 3 2 4 3 d 4 9 1
. . . and reorder the queue
(no change in this case) . . . 1 3 1 4 6 6 5 2

24. Graph Algorithms: Shortest Path
Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined.
Suppose vertex 1
is the source. s 1 1 3 4 5
S = 5 1 4 2 10 1 v 6 2 8 3 2 4 3 d 4 9 1
Repeat . . . 1 3 1 4 6 6 5 2

25. Graph Algorithms: Shortest Path
Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined.
Suppose vertex 1
is the source. s 1 6 1 3 4 5
S = 5 1 4 2 4 10 1 v 2 8 3 2 4 3 d 9 1
. . . remove from queue and
put in S 1 3 1 4 6 6 5 2

26. Graph Algorithms: Shortest Path
Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined.
Suppose vertex 1
is the source. s 1 6 1 3 4 5
S = 5 1 4 2 4 10 1 v 2 8 3 2 4 3 d 9 1
. . . relax the edges (no change
in this case) . . . 1 3 1 4 6 6 5 2

27. Graph Algorithms: Shortest Path
Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined.
Suppose vertex 1
is the source. s 1 6 1 3 4 5
S = 5 1 4 2 4 10 1 v 2 8 3 2 4 3 9 d 1
Repeat . . . 1 3 1 4 6 6 5 2

28. Graph Algorithms: Shortest Path
Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined.
Suppose vertex 1
is the source. s 1 6 1 2 3 4 5
S = 5 9 1 4 2 4 10 1 v 8 3 2 4 3 d 1
Done! 1 3 1 4 6 6 5 2

29. Graph Algorithms: Shortest Path
Dijkstra’s algorithm maintains a set S of vertices whose final shortest path weights have already been determined.
Suppose vertex 1
is the source. s v 1 6 1 2 3 4 5
S = 5 d 9 1 4 2 4 10 1 8 3 2 4 3
The result is the bottom row
which contains the length of
the shortest path from s to the
vertex above it. 1 1 3 1 4 6 6 5 2

30. Graph Algorithms: Shortest Pathv 1 6 1 2 3 4 5
S = 5 d 9 1 4 2 4 10 1 p 3 1 3 3 5 8 3 2 4 3
To compute the corresponding
paths, we augment the data
structure with an additional
attribute, p(v), which is the
vertex that precedes v in a
shortest path. 1 1 3 1 4 6 6 5 2

31. Graph Algorithms: Shortest Pathv 1 6 1 2 3 4 5
S = 5 d 9 1 4 2 4 10 1 p 3 1 3 3 5 8 3 2 4 3
Using the predecessor
attribute, p(v), it’s easy to
construct the path to v, working
backwards from v to s. 1 1 3 1 4 6 6 5 2

32. Graph Algorithms: Shortest Pathv 1 6 1 2 3 4 5
S = 5 d 9 1 4 2 4 10 1 p 3 1 3 3 5 8 3 2 4 3
E.g., v = 6. 1 1 3 1 4 6 6 5 2

33. Graph Algorithms: Shortest Pathv 1 6 1 2 3 4 5
S = 5 d 9 1 4 2 4 10 1 p 3 1 3 3 5 8 3 2 4 3
E.g., v = 6.
p(6) = 5, 1 1 3 1 4 6 6 5 2

34. Graph Algorithms: Shortest Pathv 1 6 1 2 3 4 5
S = 5 d 9 1 4 2 4 10 1 p 3 1 3 3 5 8 3 2 4 3
E.g., v = 6.
p(6) = 5,
p(5) = 3, 1 1 3 1 4 6 6 5 2

35. Graph Algorithms: Shortest Pathv 1 6 1 2 3 4 5
S = 5 d 9 1 4 2 4 10 1 p 3 1 3 3 5 8 3 2 4 3
E.g., v = 6.
p(6) = 5,
p(5) = 3,
p(3) = 1. 1 1 3 1 4 6 6 5 2

36. Graph Algorithms: Shortest Pathv 1 6 1 2 3 4 5
S = 5 d 9 1 4 2 4 10 1 p 3 1 3 3 5 8 3 2 4 3
E.g., v = 6.
p(6) = 5,
p(5) = 3,
p(3) = 1. 1 1 3 1 4 6 6 5
Path: 1, 3, 5, 6.
Weight: 4. 2

37. Graph Algorithms: Shortest Path
Computing time analysis: look at different implementations
of the priority queue, with different costs for the queue
operations.
Delete_min executed how many times?

38. Graph Algorithms: Shortest Path
Computing time analysis: look at different implementations
of the priority queue, with different costs for the queue
operations.
Delete_min executed |V| times.

39. Graph Algorithms: Shortest Path
Computing time analysis: look at different implementations
of the priority queue, with different costs for the queue
operations.
Delete_min executed |V| times.
Decrease_key executed how many times?

40. Graph Algorithms: Shortest Path
Computing time analysis: look at different implementations
of the priority queue, with different costs for the queue
operations.
Delete_min executed |V| times.
Decrease_key executed |E| times.
Total time = |V| T |E| T
Delete_min
Decrease_key

41. Graph Algorithms: Shortest Path
Computing time analysis: look at different implementations
of the priority queue, with different costs for the queue
operations.
Delete_min executed |V| times.
Decrease_key executed |E| times.
Total time = |V| T |E| T
Delete_min
Decrease_key
Priority queue T T Total time
Array
Binary heap
Decrease_key
Delete_min

42. Graph Algorithms: Shortest Path
Computing time analysis: look at different implementations
of the priority queue, with different costs for the queue
operations.
Delete_min executed |V| times.
Decrease_key executed |E| times.
Total time = |V| T |E| T
Delete_min
Decrease_key
Priority queue T T Total time
Array O(|V|) O(1)
Binary heap
Decrease_key
Delete_min

43. Graph Algorithms: Shortest Path
Computing time analysis: look at different implementations
of the priority queue, with different costs for the queue
operations.
Delete_min executed |V| times.
Decrease_key executed |E| times.
Total time = |V| T |E| T
Delete_min
Decrease_key
Priority queue T T Total time
Array O(|V|) O(1) O(|V| )
Binary heap O(log |V|) O(log |V|)
Decrease_key
Delete_min 2

44. Graph Algorithms: Shortest Path
Computing time analysis: look at different implementations
of the priority queue, with different costs for the queue
operations.
Delete_min executed |V| times.
Decrease_key executed |E| times.
Total time = |V| T |E| T
Delete_min
Decrease_key
Priority queue T T Total time
Unsorted Array O(|V|) O(1) O(|V| )
Binary heap O(log |V|) O(log |V|) O(E log |V|)
Decrease_key
Delete_min 2