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

Shortest Common Superstring
Finding a superstring of all the substrings is not difficult, as we can simply concatenate them all together.
It is finding the shortest such string that is problematic. Indeed, shortest common superstring remains NPcomplete
under all reasonable classes of strings.
The problem of finding the shortest common superstring can easily be reduced to that of the traveling
salesman problem (see Section ). Create an overlap graph G where vertex represents string . Edge
will have weight equal to the length of minus the overlap of with . The path visiting all the
vertices of minimum total weight defines the shortest common superstring. The edge weights of this
graph are not symmetric, after all, the overlap of and is not the same as the overlap of
and . Thus only programs capable of solving asymmetric TSPs can be applied to this problem.
The greedy heuristic is the standard approach to approximating the shortest common superstring. Find the
pair of strings with the maximum number of characters of overlap. Replace them by the merged string,
and repeat until only one string remains. Given the overlap graph above, this heuristic can be efficiently
implemented by inserting all of the edge weights into a heap (see Section ) and then merging if the
appropriate ends of the two strings have not yet be used, which can be maintained with an array of
Boolean flags.
The potentially time-consuming part of this heuristic is in building the overlap graph. The brute-force
approach to finding the maximum overlap of two length-l strings takes , which must be repeated
times. Faster times are possible by appropriately using suffix trees (see Section ). Build a tree
containing all suffixes of all reversed strings of S. String overlaps with if a suffix of matches a
suffix of the reverse of . The longest overlap for each fragment can be found in time linear in its
length.
How well does the greedy heuristic perform? If we are unlucky with the input, the greedy heuristic can be
fooled into creating a superstring that is at least twice as long as optimal. Usually, it will be a lot better in
practice. It is known that the resulting superstring can never be more than 2.75 times optimal.
Building superstrings becomes more difficult with positive and negative substrings, where negative
substrings cannot be substrings of the superstring. The problem of deciding whether any such consistent
substring exists is NP-complete, unless you are allowed to add an extra character to the alphabet to use as
a spacer.
Implementations: CAP (Contig Assembly Program) [Hua92] by Xiaoqiu Huang is a C language
program supporting DNA shotgun sequencing by finding the shortest common superstring of a set of
fragments. As to performance, CAP took 4 hours to assemble 1,015 fragments of a total of 252,000
characters on a Sun SPARCstation SLC. 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.
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