A bipartite graph is a graph G E such that the set can be pa

A bipartite graph is a graph G=( ,E) such that the set can be partitioned into two (disjoint) sets and ( = ) where every edge in the graph is between a node a and a node in . A complete bipartite graph is a bipartite graph where every node in is connected to every node in . Based on a graph traversal technique, write an algorithm that takes as input an undirected graph G, and returns whether or not G is a complete bipartite; if G is complete bipartite, the algorithm should also return and . Analyze the time complexity of your algorithm.

Solution

iven an array \'\'A\'\' of \'\'n\'\' elements with values or [[Record (computer science)|records]] \'\'A\'\'₀ ... \'\'A\'\'_\'\'n\'\'1, sorted such that \'\'A\'\'₀ ≤ ... ≤ \'\'A\'\'_\'\'n\'\'1, and target value \'\'T\'\', the following [[subroutine]] uses binary search to find the index of \'\'T\'\' in \'\'A\'\'.{{Sfn|Knuth|1998|loc=§6.2.1 (\"Searching an ordered table\"), subsection \"Algorithm B\"}}
# Set \'\'L\'\' to 0 and \'\'R\'\' to \'\'n\'\' − 1.
# If \'\'L\'\' > \'\'R\'\', the search terminates as unsuccessful.
# Set \'\'m\'\' (the position of the middle element) to the [[Floor and ceiling functions|floor]] of (\'\'L\'\' + \'\'R\'\') / 2.
# If A_\'\'m\'\' < \'\'T\'\', set L to \'\'m\'\' + 1 and go to step 2.
# If A_\'\'m\'\' > T, set R to \'\'m\'\' – 1 and go to step 2.
# Now A_\'\'m\'\' = \'\'T\'\', the search is done; return \'\'m\'\'.

This iterative procedure keeps track of the search boundaries via two variables. Some implementations may place the comparison for equality at the end of the algorithm, resulting in a faster comparison loop but costing one more iteration on average.<ref name=\"bottenbruch\">{{cite journal | title=Structure and Use of ALGOL 60 | author=Bottenbruch, Hermann | journal=[[Journal of the ACM]] | year=1962 | volume=9 | issue=2 | pages=161–211}} Procedure is described at p. 214 (§43), titled \"Program for Binary Search\".</ref>