1. Single-Source Shortest Path & Minimum Spanning Trees
Dijkstra’s, Prim’s, and Kruskal’s Algorithms

2. Single-Source Shortest Path Defined
Given
G = (V, E)
Source vertex s  V
Compute
{SP} = set of shortest paths from s to all vertices in V

3. Path Weight Defined Weight of path p = {v1, v2, … vk}
Shortest-path weight from u to v

4. Determining Path Lengths1 3 s b 5 2 4 c

5. Determining Path Lengths3 1 3 s b 5 5 2 4 c 2

6. Determining Path Lengths3 1 3 s b 5 5 2 4 c
d[b] <= d[c] + w(c,b)
5 <= 2

7. Determining Path Lengths
d[b] > d[a] + w(a,b)
5 > 3 1 3 s b 5 5 2 4 c 2

8. Determining Path Lengths
d[b] > d[a] + w(a,b)
5 > 3 1 3 s b 5 4 2 4 c 2
Reduce

9. Path Lengths - FIN a
All vertices are in S 3 1 3 s b 5 4 2 4 c 2

10. Cycles in Paths a b -4 3 -1 3 4 s 6 c d g 5 8 5 11 - -3 2 3 7 e f -5 11 - -3 2 3 7 e f - - -6

11. Properties of Shortest Paths
Cannot contain a negative-weight cycle
No need to contain 0-weight cycle
No need to contain positive-weight cycle
Thus contains no cycles
Will contain at most V-1 edges
Assumes no cycles (as stated above)
Maintain predecessor information
E & V

12. Shortest Paths Not Uniquex 6 3 9 3 s 4 2 1 2 7 3 5 5 11 6 y z t x t x 3 6 9 3 6 9 3 3 s 4 s 4 2 1 2 7 2 1 2 7 3 3 5 11 5 11 5 5 6 6 y y z z

13. Shortest Path Distance Is Uniquex s u z v y
[z] must be either x or y, but not both
This is true even if d[x] + w(x,z) = d[y] + w(y,z)
establishing that if s is source, d[z] is well-defined and unique

14. Setting up the Solution
For each vertex v in V[G]
Predecessor of v ([v]) = NIL
Distance to v from source (d[v]) = 
RELAX: as we process the graph, we’ll
“Improve” [v] & d[v] by “relaxing”
if (d[v] > d[u] + w(u,v))
d[v] = d[u] + w(u,v)
[v] = u

15. Init & Relax Initialize(G, s) for each v in V[G] d[v] = INFINITY
[v] = NULL d[s] = 0
Relax(u, v, w)
if (d[v] > d[u] + w(u,v))
d[v] = d[u] + w(u,v)
[v] = u

16. Dijkstra’s Algorithm (1959)
Assume G = (V, E), source s, & weight function w
Also assume all edges are non-negative
Dijkstra(G, w, s)
Initialize(G, s)
S = {}
Q = V[G]
while Q != {}
u = Extract_Min(Q)
S.Add(u)
for each vertex v in u.Adjacency()
Relax(u, v, w)

17. Example: Dijkstra’s t x 1   10 s 9 2 3 4 6 7 5   y 2 z

18. Example: Dijkstra’s t x 1 10  10 s 9 2 3 4 6 7 5 5  y 2 z

19. Example: Dijkstra’s t x 1 8 14 10 s 9 2 3 4 6 7 5 5 7 y 2 z

20. Example: Dijkstra’s t x 1 8 13 10 s 9 2 3 4 6 7 5 5 7 y 2 z

21. Example: Dijkstra’s t x 1 8 9 10 s 9 2 3 4 6 7 5 5 7 y 2 z

22. Dijkstra’s - FIN t x 1 8 9 10 s 9 2 3 4 6 7 5 5 7 y 2 z

23. Runtime of Dijkstra’s O(V lgV + E lgV) O(E lgV) O(V lgV + E)
If G is connected
O(V lgV + E)
If we use a Fibonacci heap
Better
if V << E
(i.e. sparsely
connected)

24. MST Defined

25. Minimum Spanning Tree Connect all nodes at minimum cost.
Can start anywhere?
Why?
One solution.
i.e. There is just one minimum
Two algorithms
Prim’s
Kruskal’s

26. A Graph 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

27. Prim's
start 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

28. Prim's 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

29. Prim's 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

30. Prim's 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

31. Prim's 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

32. Prim's 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

33. Prim's 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

34. Prim's 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

35. Prim's 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

36. Prim's 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

37. Prim's 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

38. Prim's 5 4 4 2 5 2 5 6 6 44 5

39. The Same Graph 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

40. Kruskal's 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

41. Kruskal's 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

42. Kruskal's 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

43. Kruskal's 5 4 9 4 2 5 12 6 8 7 2 5 8 6 8 7 9 12 6 5 11

44. Kruskal's 5 4 4 2 5 2 5 6 6 44 5