1. Section 13  Questions  Graphs

2. Graphs  A graph representation: –Adjacency matrix. Pros:? Cons:? –Neighbor lists. Pros:? Cons:?

3. Graphs  A graph representation: –Adjacency matrix. Pros: O(1) check if u is a neighbor of v. Cons:Consumes lots of memory for sparse graphs. Slow in finding all neighbors. –Neighbor lists. Pros:Consume little memory, easy to find all neighbors. Cons:Check if u is a neighbor of v can be expensive.

4. Graphs  In practice, usually neighbor lists are used.

5. Some graphs we often meet.  Acyclic - no cycles.  Directed, undirected.  Directed, acyclic graph = dag.

6. Topological sorting  A problem: Given a dag G determine the ordering of the vertices such that v precedes u iff there is an edge (v, u) in G.  Algorithm: Take out the vertice with no incoming edges along with its edges, put in the beginning of a list, repeat while necessary.  Implementation?

7. Topological sorting  More descriptive description:  1. For each vertice v compute count[v] - the number of incoming edges.  2. Put all the vertices with count[v] = 0 to the stack  3. repeat: Take top element v from the stack, decrease count[u] for its neighbors u.

8. Topological sorting  Implementation:  1. Use DFS/BFS to compute count.  2. Complexity?  O(E + V) - 1. takes E, the loop is executed V times, the number of operations inside loop sums up to E.

9. Topological sorting  Applications: –Job scheduling.

10. The bridges of Konigsberg

11.  In 1735 Leonard Euler asked himself a question.  Is it possible to walk around Konigsberg walking through each bridge exactly once?

12. The brid(g)es of Konigsberg

13. The bridges of Konigsberg  It’s not possible!  It’s a graph problem!

14. Euler’s theorem  The Euler circuit passes through each edge in the graph only once and returns to the starting point.  Theorem: The graph has an Euler circuit iff every vertice has even degree. (the number of adjacent vertices)

15. Euler’s circuit  How to check if there’s one?

16. Hamiltonian’s circuit  A cycle which passes through vertice only once.  How to find one efficiently?

17. Hamiltonian’s cycle  Nobody knows!!!

18. REWARD!!! $1000000 is offered to anyone who finds an efficient algorithm for finding Hamiltonian cycle in a graph. DEAD OR ALIVE