Sign in Subscribe

CS (Cyclic suffix)

Last updated on 

Cyclic suffix (CS) is a data structure used for solving the problem of finding the lexicographically smallest rotation of a string. The problem of finding the smallest rotation is important in various applications such as text compression, bioinformatics, and pattern matching. The CS data structure is a compact representation of the rotations of a string that can be used to solve the problem efficiently.

In this article, we will discuss the CS data structure in detail, including its construction, properties, and applications.

Overview of the Problem

Given a string S of length n, the problem is to find the lexicographically smallest rotation of S. A rotation of S is obtained by moving the first k characters to the end of the string, where k is any integer between 0 and n-1. For example, if S = "abcd", then the rotations of S are "abcd", "bcda", "cdab", and "dabc".

The lexicographically smallest rotation of S is the rotation that comes first in lexicographic order among all the rotations of S. For example, the lexicographically smallest rotation of "abcd" is "abcd" itself, since it is the smallest rotation in lexicographic order among all the rotations of "abcd".

The problem of finding the lexicographically smallest rotation can be solved using various algorithms such as the brute-force approach and the suffix array approach. However, the CS data structure provides a more efficient solution to the problem.

Construction of Cyclic Suffix

The CS data structure is constructed by first computing the suffix array of the string S. The suffix array is an array of integers that represents the lexicographic order of all the suffixes of S. Specifically, the ith element of the suffix array represents the starting index of the ith smallest suffix of S. For example, the suffix array of "banana" is [5, 3, 1, 0, 4, 2], which means that the smallest suffix of "banana" is "a" (starting at index 5), the second smallest suffix is "ana" (starting at index 3), and so on.

Using the suffix array, the CS data structure is constructed as follows:

  1. Create an array C of size n, where n is the length of S.
  2. Initialize C[sa[0]] to 0 and set k to 0.
  3. For i from 1 to n-1: a. If S[sa[i]] == S[sa[i-1]] and S[sa[i]+k] == S[sa[i-1]+k], set C[sa[i]] to C[sa[i-1]]. b. Otherwise, set C[sa[i]] to C[sa[i-1]] + 1 and set k to the maximum of 0 and k-1.

In the above algorithm, sa[i] represents the ith element of the suffix array, and k is the length of the common prefix of the suffixes sa[i] and sa[i-1] in the current iteration.

Properties of Cyclic Suffix

The CS data structure has several important properties that make it useful for solving the problem of finding the lexicographically smallest rotation of a string.

  1. Compactness: The CS data structure requires only O(n) space, where n is the length of the input string. This is because the CS data structure only stores information about the suffix array and the common prefixes of adjacent suffixes.
  2. Efficiency: The time complexity of constructing the CS data structure is O(n log n), which is the same as the time complexity of computing the suffix array. However, once the CS data structure is constructed, finding the lexicographically smallest rotation of a string can be done in O(log n) time using a binary search algorithm.
  3. Binary search: The CS data structure supports binary search operations, which allows us to efficiently find the lexicographically smallest rotation of a string. Specifically, given an index i, we can use binary search to find the index j such that the rotation S[j...n-1]S[0...j-1] is lexicographically smallest among all rotations of S. This can be done by comparing the common prefixes of S[i...n-1] and S[j...n-1] with the help of the C array.
  4. Equivalence classes: The C array can be used to group together the rotations of S that have the same common prefix. Specifically, the ith element of the C array represents the equivalence class of the ith rotation of S, which contains all rotations that have the same common prefix as the ith rotation. This can be useful for various applications such as pattern matching.

Applications of Cyclic Suffix

The CS data structure has several applications in various fields such as text compression, bioinformatics, and pattern matching. Some of the important applications are discussed below.

  1. Text compression: The CS data structure can be used to compress a given text by finding the lexicographically smallest rotation of the text and then encoding the difference between this rotation and the original text. This can be useful for reducing the storage requirements of large texts.
  2. Bioinformatics: The CS data structure can be used to efficiently compare DNA sequences by finding the lexicographically smallest cyclic shift of the sequences. This can be useful for various applications such as sequence alignment and genome assembly.
  3. Pattern matching: The C array of the CS data structure can be used to efficiently search for a pattern in a text. Specifically, the pattern can be encoded as a rotation of the text, and then binary search can be used to find the starting index of the pattern in the text. This can be useful for various applications such as text indexing and information retrieval.

Conclusion

In conclusion, the cyclic suffix (CS) data structure is a compact representation of the rotations of a string that can be used to efficiently find the lexicographically smallest rotation of the string. The CS data structure is constructed by computing the suffix array of the string and then using it to calculate the C array, which represents the equivalence classes of the rotations of the string. The CS data structure has several important properties such as compactness, efficiency, binary search support, and equivalence class grouping, which make it useful for various applications such as text compression, bioinformatics, and pattern matching.