1. Graph Algorithms
BFS, DFS, Dijkstra’s

2. Distance between u and v
Length of the shortest length path between u and v
Distance between and ? 3

3. Breadth First Search (BFS)
Is s connected to t?
Build layers of vertices connected to s
Lj : all nodes at distance j from s
L0 = {s}
Assume L0,..,Lj have been constructed
Lj+1 set of vertices not chosen yet but are connected to Lj
Stop when new layer is empty

4. BFS Tree BFS naturally defines a tree rooted at s Add non-tree edges
Lj forms the jth “level” in the tree
u in Lj+1 is child of v in Lj from which it was “discovered” 1 7 1 L0 2 3 L1 2 3 8 4 5 4 5 7 8 L2 6 6 L3

5. DFS Data Structures queue<Node> 1 2 3 4 5 7 8 6
vector<bool> 1 2 3 4 5 6 7 8 1 L0 1 7 2 3 L1 2 3 8 4 5 4 5 7 8 L2 6 6 L3

6. DFS(u) Mark u as explored and add u to R For each edge (u,v)
If v is not explored then DFS(v)

7. Depth First Search (DFS)

8. Every non-tree edge is between a node and its ancestor
A DFS run
Every non-tree edge is between a node and its ancestor
DFS(u)
u is explored
For every unexplored
neighbor v of u
DFS(v) 1 1 7 2 2 3 8 4 4 5 5
DFS tree 6 6 3 8 7

9. A DFS run using an explicit stack7 8 1 7 7 6 3 2 3 5 8 4 4 5 5 3 6 2 3 1

11. Finding cycles in O(n+m)
Run BFS or DFS
If an edge is considered and the node is already explored
That’s a cycle! 1 7 2 3 8 4 5 6

12. Shortest Path Problem

13. Driving Directions

14. Shortest Path problem s 100 Input: Directed graph G=(V,E) w15 5 s u w
100
Input: Directed graph G=(V,E)
Edge lengths, le for e in E
“start” vertex s in V 15 5 s u w 5 s u
Output: All shortest paths from s to all nodes in V

15. Naïve Algorithm
Ω(n!) time

16. Dijkstra’s shortest path algorithm
E. W. Dijkstra ( )

17. Dijkstra’s shortest path algorithm1
d’(w) = min e=(u,w) in E, u in R d(u)+le 1 2 4 3 y 4 3 u
d(s) = 0
d(u) = 1 s x 2 4
d(w) = 2
d(x) = 2
d(y) = 3
d(z) = 4 w z 5 4 2 s w
Input: Directed G=(V,E), le ≥ 0, s in V u
R = {s}, d(s) =0
Shortest paths x
While there is a x not in R with (u,x) in E, u in R z y
Pick w that minimizes d’(w)
Add w to R
d(w) = d’(w)

18. Couple of remarks
The Dijkstra’s algo does not explicitly compute the shortest paths
Can maintain “shortest path tree” separately
Dijkstra’s algorithm does not work with negative weights

19. Dijkstra’s shortest path algorithm (runtime)
d’(v) = min e=(u,v) in E, u in S d(u)+le
Input: Directed G=(V,E), le ≥ 0, s in V
S = {s}, d(s) =0
While there is a v not in S with (u,v) in E, u in S
At most n iterations
Pick w that minimizes d’(w)
Add w to S
d(w) = d’(w)
O(m) time
O(mn) time bound is trivial
O(m log n) time implementation is possible
How?