1. Reachability as Transitive Closure
Algorithm : Design & Analysis
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2. In the last class… Undirected and Symmetric Digraph
UDF Search Skeleton
Biconnected Components
Articulation Points and Biconnectedness
Biconnected Component Algorithm
Analysis of the Algorithm

3. Reachability as Transitive Closure
Transitive Closure by DFS
Transitive Closure by Shortcuts
Washall’s Algorithm for Transitive Closure
All-Pair Shortest Paths

4. Fundamental Questions
For all pair of vertices in a graph, say, u, v:
Is there a path from u to v?
What is the shortest path from u to v?
Reachability as a (reflexive) transitive closure of the adjacency relation, which can be represented as a bit matrix.

5. Computing the Reachability by DFS
With each vertex vi, constructing a DFS tree rooted at vi.
For each vertex vj encountered, R[i][j] is set to true.
At most, m edges are processed.
It is possible to set one in more than one row during each DFS. (In fact, if vk is on the path from vi to vj, when vj is encountered, not only R[i][j], but also R[k][j] can be set)

6. Computing Reachability by Condensation Graph
Note: the elements of the sub-matrix of R for a strong component are all true.
The steps:
Find the strong components of G, and get G, the condensation graph of G. ((n+m))
Computing reachability for G.
Expand the reachability for G to that for G. (O(n2))

7. Transitive Closure by Shortcuts
The idea: if there are edges sisk, sksj, then an edge sisj, the “shortcut” is inserted. 1 2 5 3 4 1 2 5 3 4 1 2 5 3 4
Pass two
Pass one
Note: the triple (1,5,3) is considered more than once

8. Transitive Closure by Shortcuts: the algorithm
Input: A, an nn boolean matrix that represents a binary relation
Output: R, the boolean matrix for the transitive closure of A
Procedure
void simpleTransitiveClosure(boolean[][] A, int n, boolean[][] R)
int i,j,k;
Copy A to R;
Set all main diagonal entries, rii, to true;
while (any entry of R changed during one complete pass)
for (i=1; in; i++)
for (j=1; jn; j++)
for (k=1; kn; k++)
rij=rij(rikrkj)
The order of (i,j,k) matters

9. Change the order: Washall’s Algorithm
void simpleTransitiveClosure(boolean[][] A, int n, boolean[][] R)
int i,j,k;
Copy A to R;
Set all main diagonal entries, rii, to true;
while (any entry of R changed during one complete pass)
for (k=1; kn; k++)
for (i=1; in; i++)
for (j=1; jn; j++)
rij=rij(rikrkj)
k varys in the outmost loop
Note: “false to true” can not be reversed

10. Highest-numbered intermediate vertex
The highest intermediate vertex in the intervals (sisk), (sksj) are both less than sk sj sk si
A specific order is assumed for all vertices
Vertical position of vertices reflect their vertex numbers

11. Correctness of Washall’s Algorithm
Notation:
The value of rij changes during the execution of the body of the “for k…” loop
After initializations: rij(0)
After the kth time of execution: rij(k)

12. Correctness of Washall’s Algorithm
If there is a simple path from si to sj(ij) for which the highest-numbered intermediate vertex is sk, then rij(k)=true.
Proof by induction:
Base case: rij(0)=true if and only if sisjE
Hypothesis: the conclusion holds for h<k(h0)
Induction: the simple sisj-path can be looked as sisk-path+sksj-path, with the indices h1, h2 of the highest-numbered intermediate vertices of both segment strictly(simple path) less than k. So, rij(h1)=true, rij(h2)=true, then rij(k-1)=true, rij(k-1)=true(Remember, false to true can not be reversed). So, rij(k)=true

13. Correctness of Washall’s Algorithm
If there is no path from si to sj, then rij=false.
Proof
If rij=true, then only two cases:
rij is set by initialization, then sisjE
Otherwise, rij is set during the kth execution of (for k…) when rij(k-1)=true, rij(k-1)=true, which, recursively, leads to the conclusion of the existence of a sisj-path. (Note: If a sisj-path exists, there exists a simple sisj-path)

14. All-pairs Shortest Path
Non-negative weighted graph
Shortest path property: If a shortest path from x to z consisting of path P from x to y followed by path Q from y to z. Then P is a shortest xz-path, and Q, a shortest zy-path.

15. Computing the Distance Matrix
Basic formula:
D(0)[i][j]=wij
D(k)[i][j]=min(D(k-1)[i][j], D(k-1)[i][k]+ D(k-1)[i][j])
Basic property:
D(k)[i][j]dij(k)
where dij(k) is the weight of a shortest path from vi to vj with highest numbered vertex vk.

16. All-Pairs Shortest Paths
Floyd algorithm
Only slight changes on Washall’s algorithm.
Routing table