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Depth First Search and Breadth First SearchUp:7.
Directed GraphsPrevious:7.2.3
All Pairs Shortest Paths Problem: Floyd's Algorithm
7.3 Warshall's Algorithm

REF.

Stephen Warshall. A theorem on boolean matrices.
Journal of the ACM,
Volume 9, Number 1, pp. 1112, 1962.
Def: Given the adjacency matrix C of any digraph C = (v,E), the
matrix A is called the transitive closure of C if
i, jtex2html_image_mark>#tex2html_wrap_inline27764# V,
A[i, j] 
= 1 
if there is a path of length one or more from vertex i to
vertex j 

= 0 
otherwise 
Warshall's algorithm enables to compute the transitive closure of the adjacency
matrix f any digraph. Warshall's algorithm predates Floyd's algorithm and
simple uses the following formula in the kth passes of Floyd's algorithm:
Ak[i, j] = Ak  1[i, j]
(Ak  1[i, k] Ak
 1[k, j])
where the matrix elements are treated as boolean values 0 or 1 and the
symbols
and
denote ``logical or'' and ``logical and'' respectively. It is easy to see
that Warshall's algorithm has a worst case complexity of O(n3) where n
is the number of vertices of the graph.
Next:7.4
Depth First Search and Breadth First SearchUp:7.
Directed GraphsPrevious:7.2.3
All Pairs Shortest Paths Problem: Floyd's Algorithm
eEL,CSA_Dept,IISc,Bangalore