# Shortest Superstring Problem

Given a set of n strings arr[], 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: arr[] = {"geeks", "quiz", "for"} Output: geeksquizfor Input: arr[] = {"catg", "ctaagt", "gcta", "ttca", "atgcatc"} Output: gctaagttcatgcatc

**Shortest Superstring Greedy Approximate Algorithm**

Shortest Superstring Problem is a NP Hard problem. A solution that always finds shortest superstring takes exponential time. Below is an Approximate Greedy algorithm.

Let arr[] be given set of strings. 1) Create an auxiliary array of strings, temp[]. Copy contents of arr[] to temp[] 2) While temp[] contains more than one strings a) Find the most overlapping string pair in temp[]. Let this pair be 'a' and 'b'. b) Replace 'a' and 'b' with the string obtained after combining them. 3) The only string left in temp[] is the result, return it.

Two strings are overlapping if prefix of one string is same suffix of other string or vice verse. The maximum overlap mean length of the matching prefix and suffix is maximum.

**Working of above Algorithm:**

arr[] = {"catgc", "ctaagt", "gcta", "ttca", "atgcatc"} Initialize: temp[] = {"catgc", "ctaagt", "gcta", "ttca", "atgcatc"} The most overlapping strings are "catgc" and "atgcatc" (Suffix of length 4 of "catgc" is same as prefix of "atgcatc") Replace two strings with "catgcatc", we get temp[] = {"catgcatc", "ctaagt", "gcta", "ttca"} The most overlapping strings are "ctaagt" and "gcta" (Prefix of length 3 of "ctaagt" is same as suffix of "gcta") Replace two strings with "gctaagt", we get temp[] = {"catgcatc", "gctaagt", "ttca"} The most overlapping strings are "catgcatc" and "ttca" (Prefix of length 2 of "catgcatc" as suffix of "ttca") Replace two strings with "ttcatgcatc", we get temp[] = {"ttcatgcatc", "gctaagt"} Now there are only two strings in temp[], after combing the two in optimal way, we get tem[] = {"gctaagttcatgcatc"} Since temp[] has only one string now, return it.

Below is C++ implementation of above algorithm.

`// C++ program to find shortest superstring using Greedy ` `// Approximate Algorithm ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Utility function to calculate minimum of two numbers ` `int` `min(` `int` `a, ` `int` `b) ` `{ ` ` ` `return` `(a < b) ? a : b; ` `} ` ` ` `// Function to calculate maximum overlap in two given strings ` `int` `findOverlappingPair(string str1, string str2, string &str) ` `{ ` ` ` `// max will store maximum overlap i.e maximum ` ` ` `// length of the matching prefix and suffix ` ` ` `int` `max = INT_MIN; ` ` ` `int` `len1 = str1.length(); ` ` ` `int` `len2 = str2.length(); ` ` ` ` ` `// check suffix of str1 matches with prefix of str2 ` ` ` `for` `(` `int` `i = 1; i <= min(len1, len2); i++) ` ` ` `{ ` ` ` `// compare last i characters in str1 with first i ` ` ` `// characters in str2 ` ` ` `if` `(str1.compare(len1-i, i, str2, 0, i) == 0) ` ` ` `{ ` ` ` `if` `(max < i) ` ` ` `{ ` ` ` `//update max and str ` ` ` `max = i; ` ` ` `str = str1 + str2.substr(i); ` ` ` `} ` ` ` `} ` ` ` `} ` ` ` ` ` `// check prefix of str1 matches with suffix of str2 ` ` ` `for` `(` `int` `i = 1; i <= min(len1, len2); i++) ` ` ` `{ ` ` ` `// compare first i characters in str1 with last i ` ` ` `// characters in str2 ` ` ` `if` `(str1.compare(0, i, str2, len2-i, i) == 0) ` ` ` `{ ` ` ` `if` `(max < i) ` ` ` `{ ` ` ` `//update max and str ` ` ` `max = i; ` ` ` `str = str2 + str1.substr(i); ` ` ` `} ` ` ` `} ` ` ` `} ` ` ` ` ` `return` `max; ` `} ` ` ` `// Function to calculate smallest string that contains ` `// each string in the given set as substring. ` `string findShortestSuperstring(string arr[], ` `int` `len) ` `{ ` ` ` `// run len-1 times to consider every pair ` ` ` `while` `(len != 1) ` ` ` `{ ` ` ` `int` `max = INT_MIN; ` `// to store maximum overlap ` ` ` `int` `l, r; ` `// to store array index of strings ` ` ` `// involved in maximum overlap ` ` ` `string resStr; ` `// to store resultant string after ` ` ` `// maximum overlap ` ` ` ` ` `for` `(` `int` `i = 0; i < len; i++) ` ` ` `{ ` ` ` `for` `(` `int` `j = i + 1; j < len; j++) ` ` ` `{ ` ` ` `string str; ` ` ` ` ` `// res will store maximum length of the matching ` ` ` `// prefix and suffix str is passed by reference and ` ` ` `// will store the resultant string after maximum ` ` ` `// overlap of arr[i] and arr[j], if any. ` ` ` `int` `res = findOverlappingPair(arr[i], arr[j], str); ` ` ` ` ` `// check for maximum overlap ` ` ` `if` `(max < res) ` ` ` `{ ` ` ` `max = res; ` ` ` `resStr.assign(str); ` ` ` `l = i, r = j; ` ` ` `} ` ` ` `} ` ` ` `} ` ` ` ` ` `len--; ` `//ignore last element in next cycle ` ` ` ` ` `// if no overlap, append arr[len] to arr[0] ` ` ` `if` `(max == INT_MIN) ` ` ` `arr[0] += arr[len]; ` ` ` `else` ` ` `{ ` ` ` `arr[l] = resStr; ` `// copy resultant string to index l ` ` ` `arr[r] = arr[len]; ` `// copy string at last index to index r ` ` ` `} ` ` ` `} ` ` ` `return` `arr[0]; ` `} ` ` ` `// Driver program ` `int` `main() ` `{ ` ` ` `string arr[] = {` `"catgc"` `, ` `"ctaagt"` `, ` `"gcta"` `, ` `"ttca"` `, ` `"atgcatc"` `}; ` ` ` `int` `len = ` `sizeof` `(arr)/` `sizeof` `(arr[0]); ` ` ` ` ` `cout << ` `"The Shortest Superstring is "` ` ` `<< findShortestSuperstring(arr, len); ` ` ` ` ` `return` `0; ` `} ` `// This code is contributed by Aditya Goel ` |

*chevron_right*

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**Performance of above algorithm:**

The above Greedy Algorithm is proved to be 4 approximate (i.e., length of the superstring generated by this algorithm is never beyond 4 times the shortest possible superstring). This algorithm is conjectured to 2 approximate (nobody has found case where it generates more than twice the worst). Conjectured worst case example is {ab^{k}, b^{k}c, b^{k+1}}. For example {“abb”, “bbc”, “bbb”}, the above algorithm may generate “abbcbbb” (if “abb” and “bbc” are picked as first pair), but the actual shortest superstring is “abbbc”. Here ratio is 7/5, but for large k, ration approaches 2.

There exist better approximate algorithms for this problem. Please refer below link.

Shortest Superstring Problem | Set 2 (Using Set Cover)

**Applications:**

Useful in the genome project since it will allow researchers to determine entire coding regions from a collection of fragmented sections.

**Reference:**

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

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

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

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