# Shortest Superstring Problem | Set 2 (Using Set Cover)

Given a set of n strings S, find the smallest string that contains each string in the given set as substring. We may assume that no string in arr[] is substring of another string.

Examples:

Input: S = {"001", "01101", "010"} Output: 0011010 Input: S = {"geeks", "quiz", "for"} Output: geeksquizfor Input: S = {"catg", "ctaagt", "gcta", "ttca", "atgcatc"} Output: gctaagttcatgcatc

In the previous post, we have discussed a solution that is proved to be 4 approximate (conjectured as 2 approximate).

In this post, a solution is discussed that can be proved as 2H_{n} approximate. where H_{n} = 1 + 1/2 + 1/3 + … 1/n. The idea is to transform Shortest Superstring problem into Set Cover problem (The Set cover problem is given some subsets of a universe and every give subset has an associated cost. The task is to find the lowest cost set of given subsets such that all elements of universe are covered). For a Set Cover problem, we need to have a universe and subsets of universe with their associated costs.

Below are steps to transform Shortest Superstring into Set Cover.

1) Let S be the set of given strings. S = {s_{1}, s_{2}, ... s_{n}} 2) Universe for Set Cover problem is S (We need to find a superstring that has every string as substring) 3) Let us initialize subsets to be considered for universe as Subsets = {{s_{1}}, {s_{2}}, ... {s_{n}}} Cost of every subset is length of string in it. 3) For all pairs of strings s_{i}and s_{j}in S, If s_{i}and s_{j}overlap a) Construct a string r_{ijk}where k is the maximum overlap between the two. b) Add the set represented by r_{ijk}to Subsets, i.e., Subsets = Subsets U Set(r_{ijk}) The set represented by r_{ijk}is the set of all strings which are substring of it. Cost of the subset is length of r_{ijk}. 4) Now problem is transformed to Set Cover, we can run Greedy Set Cover approximate algorithm to find set cover of S using Subsets. Cost of every element in Subsets is length of string in it.

**Example:**

S = {s_{1}, s_{2}, s_{3}}. s_{1}= "001" s_{2}= "01101" s_{3}= "010" [Combination of s_{1}and s_{2}with 2 overlapping characters] r_{122}= 001101 [Combination of s_{1}and s_{3}with 2 overlapping characters] r_{132}= 0010 Similarly, r_{232}= 011010 r_{311}= 01001 r_{321}= 0101101Now set cover problem becomes as following:Universeto cover is {s_{1}, s_{2}, s_{3}}Subsets of the universe and their costs :{s_{1}}, cost 3 (length of s_{1}) {s_{2}}, cost 5 (length of s_{2}) {s_{3}}, cost 5 (length of s_{3}) set(r_{122}), cost 6 (length of r_{122}) The set r_{122}represents all strings which are substrings of r_{122}. Therefore set(r_{122}) = {s_{1}, s_{2}} set(r_{132}), cost 3 (length of r_{132}) The subset r_{132}represents all strings which are substrings of r_{132}Therefore set(r_{132}) = {s_{1}, s_{3}} Similarly there are more subsets for set(r_{232}), set(r_{311}), and set(r_{321}). So we have a set cover problem with universe and subsets of universe with costs associated with every subset.

We have discussed that an instance of Shortest Superstring problem can be transformed into an instance of Set Cover problem in polynomial time.

Refer this for proof of the fact that Set Cover based algorithm is 2H_{n} approximate.

**Reference:**

http://www.cs.dartmouth.edu/~ac/Teach/CS105-Winter05/Notes/wan-ba-notes.pdf

http://fileadmin.cs.lth.se/cs/Personal/Andrzej_Lingas/superstring.pdf

http://math.mit.edu/~goemans/18434S06/superstring-lele.pdf

This article is contributed **Dheeraj Gupta**. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

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