Shortest common supersequence explained

In computer science, the shortest common supersequence of two sequences X and Y is the shortest sequence which has X and Y as subsequences. This is a problem closely related to the longest common subsequence problem. Given two sequences X = < x₁,...,x_m > and Y = < y₁,...,y_n >, a sequence U = < u₁,...,u_k > is a common supersequence of X and Y if items can be removed from U to produce X and Y.

A shortest common supersequence (SCS) is a common supersequence of minimal length. In the shortest common supersequence problem, two sequences X and Y are given, and the task is to find a shortest possible common supersequence of these sequences. In general, an SCS is not unique.

For two input sequences, an SCS can be formed from a longest common subsequence (LCS) easily. For example, the longest common subsequence of X

[1..m]=abcbdab

and Y

[1..n]=bdcaba

is Z

[1..L]=bcba

. By inserting the non-LCS symbols into Z while preserving their original order, we obtain a shortest common supersequence U

[1..S]=abdcabdab

. In particular, the equation

L+S=m+n

holds for any two input sequences.

There is no similar relationship between shortest common supersequences and longest common subsequences of three or more input sequences. (In particular, LCS and SCS are not dual problems.) However, both problems can be solved in

O(n^k)

time using dynamic programming, where

is the number of sequences, and

is their maximum length. For the general case of an arbitrary number of input sequences, the problem is NP-hard.^[1]

Shortest common superstring

The closely related problem of finding a minimum-length string which is a superstring of a finite set of strings = is also NP-hard.^[2] Several constant factor approximations have been proposed throughout the years, and the current best known algorithm has an approximation factor of 2.475.^[3] However, perhaps the simplest solution is to reformulate the problem as an instance of weighted set cover in such a way that the weight of the optimal solution to the set cover instance is less than twice the length of the shortest superstring . One can then use the O(log)-approximation for weighted set-cover to obtain an O(log)-approximation for the shortest superstring (note that this is not a constant factor approximation).

For any string in this alphabet, define to be the set of all strings which are substrings of . The instance of set cover is formulated as follows:

Let be empty.
For each pair of strings and , if the last symbols of are the same as the first symbols of , then add a string to that consists of the concatenation with maximal overlap of with .
Define the universe

of the set cover instance to be

Define the set of subsets of the universe to be
Define the cost of each subset (x) to be ||, the length of .

The instance can then be solved using an algorithm for weighted set cover, and the algorithm can output an arbitrary concatenation of the strings for which the weighted set cover algorithm outputs .

Example

Consider the set =, which becomes the universe of the weighted set cover instance. In this case, = . Then the set of subsets of the universe is

$\begin \ &= \ \\ &= \ \} \\ &= \ \} \\\end$ which have costs 3, 3, 3, 5, and 4, respectively.

References

Book: Michael R. . Garey . Michael R. Garey . David S. . Johnson . David S. Johnson . 1979 . Computers and Intractability: A Guide to the Theory of NP-Completeness . W.H. Freeman . 0-7167-1045-5 . 0411.68039 . p. 228 A4.2: SR8 . Computers and Intractability: A Guide to the Theory of NP-Completeness .
Book: Szpankowski, Wojciech . Wojciech Szpankowski

. Wojciech Szpankowski . Average case analysis of algorithms on sequences . With a foreword by Philippe Flajolet . Wiley-Interscience Series in Discrete Mathematics and Optimization . Chichester . Wiley . 2001 . 0-471-24063-X . 0968.68205 .

External links

Notes and References

David Maier. The Complexity of Some Problems on Subsequences and Supersequences. J. ACM. 25. 1978. 322 - 336. 10.1145/322063.322075. ACM Press. 2. 16120634. free.
The shortest common supersequence problem over binary alphabet is NP-complete . Kari-Jouko Räihä, Esko Ukkonen . Theoretical Computer Science . 16 . 2 . 1981 . 187–198 . 10.1016/0304-3975(81)90075-x .
Book: Matthias Englert and Nicolaos Matsakis and Pavel Vesel. Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing. Improved approximation guarantees for shortest superstrings using cycle classification by overlap to length ratios. 2022. 317–330. 10.1145/3519935.3520001. 9781450392648. 243847650. https://dl.acm.org/doi/10.1145/3519935.3520001.