Data Structures and Algorithms in Java[1].pdf - Fulvio Frisone

Directed Breadth-First Search
As with DFS, we can extend breadth-first search (BFS) to work for directed
graphs. The algorithm still visits vertices level by level and partitions the set of
edges into tree edges (or discovery edges), which together form a directed
breadth-first search tree rooted at the start vertex, and nontree edges. Unlike the
directed DFS method, however, the directed BFS method only leaves two kinds of
nontree edges: back edges, which connect a vertex to one of its ancestors, and
cross edges, which connect a vertex to another vertex that is neither its ancestor
nor its descendent. There are no forward edges, which is a fact we explore in an
Exercise (C-13.10).
13.4.2 Transitive Closure
In this section, we explore an alternative technique for computing the transitive
closure of a digraph. Let
be a digraph with n vertices and m edges. We compute
the transitive closure of in a series of rounds. We initialize = . We also
arbitrarily number the vertices of as v 1 , v 2 ,…, v n . We then begin the
computation of the rounds, beginning with round 1. In a generic round k, we
cons truct digraph starting with = and adding to the directed
edge (vi, v j ) if digraph contains both the edges (v i ,v K ) and (v k ,v j ). In this
way, we will enforce a simple rule embodied in the proposition that follows.
Proposition 13.18: For i=1,…,n, digraph
only if digraph
has an edge (v i , v j ) if and
has a directed path from v i to v j , whose intermediate vertices (if
any) are in the set{v 1 ,…,v k }. In particular, is equal to , the transitive
closure of .
Proposition 13.18 suggests a simple algorithm for computing the transitive closure
of
that is based on the series of rounds we described above. This algorithm is
known as the Floyd-Warshall algorithm, and its pseudo-code is given in Code
Fragment 13.12. From this pseudo-code, we can easily analyze the running time of
the Floyd-Warshall algorithm assuming that the data structure representing G
supports methods areAdjacent and insertDirectedEdge in O(1) time. The main loop
is executed n times and the inner loop considers each of O(n 2 ) pairs of vertices,
performing a constant-time computation for each one. Thus, the total running time
of the Floyd-Warshall algorithm is O(n 3 ).
Code Fragment 13.12: Pseudo-code for the Floyd-
Warshall algorithm. This algorithm computes the
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