Finding a Longest Increasing Subsequence from the Paths in a Complete Bipartite Graph∗

Let S = s1, s2, . . . , sn be an integer sequence. The longest increasing subsequence problem is to find an increasing subsequence of S with the longest length. By regarding S as a weight sequence of the vertices in a path, we can redefine the longest increasing subsequence problem on graphs as follows. Let G = (V,E) be a graph in which every vertex v ∈ V has a weight w(v). A longest increasing subsequence of G is a longest increasing weight subsequence among all paths in G. Thus, the longest increasing subsequence problem on G is to find a longest increasing subsequence of G. In this paper, we present an O(|V | log |V |)-time algorithm for finding a longest increasing subsequence of complete bipartite graphs Km,n with m > n.