1. Graph Theory

2. undirected graph node: a, b, c, d, e, f edge: (a, b), (a, c), (b, c), (b, e), (c, d), (c, f), (d, e), (d, f), (e, f) subgraph

3. adjacency matrix:

4. directed graph ( digraph)

5. weighted graph weight matrix

6. path cycle

7. connectivity matrix C C jk = 1 if there is a path of length 0 or more from v j to v k 0otherwise C: reflexive and transitive closure of G.

8. How to compute the connectivity matrix? Matrix multiplication ﹡ →  (AND) + →  (OR) e. g. Z = X ‧ Y Zij= (xi1  y1j)  (xi2  y2j)  …  (xin  ynj)

9. the algorithm: matrix B b jk =a jk for j  k b jj =1 that is, b jk =1 if there is a path of length 0 or 1 from v j to v k vj to vk 0otherwise B 2 represents paths of length 2 or less. B 4 … B n

10. complexity: O(logn) matrix multiplications

11. Connected components

12. Assign each vertex a component number which is the smallest vertex label in the same component. Step 1: Compute the connectivity matrix.

13. Step 2: In each row, find the smallest label. component 0: v0, v1, v5 component 2: v2, v3 component 4: v4

14. complexity: same as that for computing the connectivity matrix. e. g. on hypercubes time: O(log 2 n) processor: n 3

15. All pairs shortest path

16. a sequential method — dynamic programming for k=0 to n-1 do for i=0 to n-1 do for j=0 to n-1 do : the length of a shortest path from v i to v j going through no vertex of label greater than k.

17. ←shortest paths time: O(n 3 )

18. a parallel method matrix multiplication an algorithm similar to that computes the connectivity matrix. in matrix multiplication: * → + + →min

19. We can compute D 1, D 2, D 4, D 8, …, D n. complexity: O(log n) matrix multiplications e.g. on hypercubes time: O(log 2 n) processor: n 3

20. Minimum spanning trees 1 3 6 2 9 16 9 1 2 9