Reachability as Transitive Closure

1
Reachability as Transitive Closure

• Algorithm Design Analysis
• 17

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
• Washalls 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, Rij 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
  Rij, but also Rkj 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. (?(nm))
• 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.

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 (i1 i?n i)
• for (j1 j?n j)
• for (k1 k?n k)
• rijrij?(rik?rkj)

The order of (i,j,k) matters
9
Change the order Washalls 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 (k1 k?n k)
• for (i1 i?n i)
• for (j1 j?n j)
• rijrij?(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 Washalls 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 Washalls 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 hltk(h?0)
• Induction the simple sisj-path can be looked as
  sisk-pathsksj-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 Washalls Algorithm

• If there is no path from si to sj, then
  rijfalse.
• Proof
• If rijtrue, 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)ijwij
• D(k)ijmin(D(k-1)ij, D(k-1)ik
  D(k-1)ij)
• Basic property
• D(k)ij?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 Washalls algorithm.
• Routing table