6.006- Introduction to Algorithms

1. 6.006- Introduction to Algorithms Lecture14 Prof. Silvio Micali

2. Graphs Undirected • V={a,b,c,d} • E={{a,b}, {a,c}, {b,c}, {b,d}, {c,d}} Directed • V = {a,b,c} • E = {(a,c), (a,b) (b,c), (c,b)} a a b c b c d

3. Computer Representation Four representations with pros/cons Adjacency lists(of neighbors of each vertex) a c b

4. a c b

5. Breadth First Search • Start with vertex v • List all its neighbors (distance 1) • Then all their neighbors (distance 2) • Etc.

6. Augmented Breadth First Search=Shortest Path Alg () Initially, is marked , all other vertices are marked 1. 2. Find all neighbors of at least one vertex marked . If none, STOP. 3. Mark all vertices found in (3) with . 4. Thm:Every vertex is marked with its distance form Complexity:

7. BFS Tree Structure • Spanning Tree with Lots of Structural Information ? z a f s d x v c

8. Depth First Search • Exploring a maze • From current vertex, move to another • Until you get stuck • Then backtrack till you find the first new possibility for exploration

9. DFS 0. Mark all edges “unused”. For all Let and 1. 2. If has no unused edges, go to (4) 3. Choose an unused edge Mark used. If go to (2). Else and go to (1) 4. If HALT 5. and go to (2)

10. DFS Tree Back Edge Tree Edge

11. DFS Tree Directed Case Back Edge Forward Edge Tree Edge

12. Connected Components An equivalence relation Linear with Good Counting!

13. numbering of the vertices of a directed acyclic graph (DAG) such that Topological Sort if then

14. Idea: Topological Reverse numbering of the vertices of a directed acyclic graph (DAG) such that if then

15. More General Shortest Paths for a given node Undirected (Directed in recitation!) graphs with non-negative edge length Picture

16. Dijstra’s Algorithm label. If , then there is a path from to of length , not necessarily minimum Set of temporarily labeled vertices Set of permanently labeled vertices 0. • do: • Choose with minimum label • &

17. Analysis

18. Cycles?