1. Weighted Graphs & Shortest Paths

2. Weighted Graphs
Edges have an associated weight or cost

3. BFS?
BFS only gives shortest path in terms of edge count, not edge weight

4. Dijkstra's Shortest Path Algorithm
Conceptual:
V = all vertices
T = included vertices
Pick starting vertex, include in T
Pick element not in T with minimal cost to reach from T
Move to T
Update costs of remaining vertices

5. Dijkstra's
Find paths from u to all others:

6. Dijkstra's
Find paths from u to all others:

7. Dijkstra's
Find paths from u to all others:

8. Dijkstra's
Find paths from u to all others:

9. Dijkstra's
Find paths from u to all others:

10. Dijkstra's
Find paths from u to all others:

11. Dijkstra's
Find paths from u to all others:

12. Details Implementation Known once visited Cost starts at 
Update all neighbors at each visit
Path marked when cost updated

13. Dijkstra's Algorithm Finds shortest path to all other vertices
Can terminate early once goal is known
Assumption:
No negative edges

14. A*
A-Star : Dijkstra's algorithm, but include estimate of future cost for every vertex
Estimate must never overestimate cost

15. MST Minimum Spanning Tree
Spanning tree with minimal possible total weight

16. Prim's MST Conceptual: Sound familiar? V = all vertices
T = included vertices
Pick starting vertex, include in T
Pick element not in T with minimal cost to reach from T
Move to T
Update costs of remaining vertices
Sound familiar?

17. Prim's MST ~ Dijkstra's Conceptual: One big difference…
V = all vertices
T = included vertices
Pick starting vertex, include in T
Pick element not in T with minimal cost to reach from T
Move to T
Update costs of remaining vertices : cost = just current edge
One big difference…

18. Prim's Implementation Same as Dijkstra's but…
Cost of vertex = min( all known edges to it )

19. Prim's Algorithm Finds a MST Assumption: May be multiple equal cost
Undirected