The.Algorithm.Design.Manual.Springer-Verlag.1998

Longest Common Substring
, maintain the minimum y-coordinate of any path going through exactly k points.
Inserting a new point will change exactly one of these paths by reducing the y-coordinate of the
path whose last point is barely greater than the new point.
● What if the strings are permutations? - If the strings are permutations, then there are exactly n
pairs of matching characters, and the above algorithm runs in time. A particularly
important case of this occurs in finding the longest increasing subsequence of a sequence of
numbers. Sorting this sequence and then replacing each number by its rank in the total order gives
us a permutation p. The longest common subsequence of p and gives the longest
increasing subsequence.
● What if we have more than two strings to align? - The basic dynamic programming algorithm can
be generalized to k strings, taking time, where n is the length of the longest string. This
algorithm is exponential in the number of strings k, and so it will likely be too expensive for more
than 3 to 4 strings. Further, the problem is NP-complete, so no better exact algorithm is destined
to come along.
This problem of multiple sequence alignment has received considerable attention, and numerous
heuristics have been proposed. Many heuristics begin by computing the pairwise alignment
between each of the pairs of strings, and then work to merge these alignments. One approach is
to build a graph with a vertex for each character of each string. There will be an edge between
and if the corresponding characters are matched in the alignment between S and T. Any k-
clique (see Section ) in this graph describes a commonly aligned character, and all such cliques
can be found efficiently because of the sparse structure of this graph.
Although these cliques will define a common subsequence, there is no reason to believe that it
will be the longest such substring. Appropriately weakening the clique requirement provides a
way to increase it, but still there can be no promises.
Implementations: MAP (Multiple Alignment Program) [Hua94] by Xiaoqiu Huang is a C language
program that computes a global multiple alignment of sequences using an iterative pairwise method.
Certain parameters will need to be tweaked to make it accommodate non-DNA data. It is available by
anonymous ftp from cs.mtu.edu in the pub/huang directory.
Combinatorica [Ski90] provides a Mathematica implementation of an algorithm to construct the longest
increasing subsequence of a permutation, which is a special case of longest common subsequence. This
algorithm is based on Young tableaux rather than dynamic programming. See Section .
Notes: Good expositions on longest common subsequence include [AHU83, CLR90]. A survey of
algorithmic results appears in [GBY91]. The algorithm for the case where all the characters in each
sequence are distinct or infrequent is due to Hunt and Szymanski [HS77]. Expositions of this algorithm
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