# Suffix Array

## 1. Introduction

Suffix Array is a sorted array of all suffixes of a given (usually long) text string T of length n characters (n can be in order of hundred thousands characters).

Suffix Array is a simple, yet powerful data structure which is used, among others, in full text indices, data compression algorithms, and within the field of bioinformatics.

This data structure is very related to the Suffix Tree data structure. Both data structures are usually studied together.

## 2. Suffix Array Visualization

The visualization of Suffix Array is simply a table where each row represents a suffix and each column represents the attributes of the suffixes.

The four (basic) attributes of each row i are:

1. index i, ranging from 0 to n-1,
2. SA[i]: the i-th lexicographically smallest suffix of T is the SA[i]-th suffix,
3. LCP[i]: the Longest Common Prefix between the i-th and the (i-1)-th lexicographically smallest suffixes of T is LCP[i] (we will see the application of this attribute later), and
4. Suffix T[SA[i]:] - the i-th lexicographically smallest suffix of T is from index SA[i] to the end (index n-1).

Some operations may add more attributes to each row and are explained when that operations are discussed.

## 3. 可用的操作

1. 构建后缀数组 (SA) 是基于 Karp，Miller，和 Rosenberg (1972) 的想法的 O(n log n) 后缀数组构建算法，该算法按照增长长度 (1, 2, 4, 8, ...) 对后缀的前缀进行排序。
2. 搜索 利用后缀数组中的后缀已排序的事实，并在 O(m log n) 中调用两个二进制搜索，以找到模式字符串 P 的长度 m 的第一次和最后一次出现。
3. 最长公共前缀 (LCP) 可以在 O(n) 中使用排列 LCP (PLCP) 定理计算两个相邻后缀（不包括第一个后缀）之间的最长公共前缀。这个算法的名字叫做 Kasai's 算法。
4. 最长重复子串 (LRS) 是一个简单的 O(n) 算法，它找到具有最高 LCP 值的后缀。
5. 最长公共子串 (LCS) 是一个简单的 O(n) 算法，它找到来自两个不同字符串的具有最高 LCP 值的后缀。

### 3-3. Search

After we construct the Suffix Array of T in O(n log n), we can search for the occurrence of Pattern string T in O(m log n) by binary searching the sorted suffixes to find the lower bound (the first occurrence of P as a prefix of any suffix of T) and the upper bound positions (thelast occurrence of P as a prefix of any suffix of T).

Time complexity: O(m log n) and it will return an interval of size k where k is the total number of occurrences.

For example, on the Suffix Array of T = "GATAGACA\$" above, try these scenarios:

1. P returns a range of rows: Search("GA"), occurrences = {4, 0}
2. P returns one row only: Search("CA"), occurrences = {2}

### 3-5. Longest Repeated Substring (LRS)

After we construct the Suffix Array of T in O(n log n) and compute its LCP Array in O(n), we can find the Longest Repeated Substring (LRS) in T by simply iterating through all LCP values and reporting the largest one.

This is because each value LCP[i] the LCP Array means the longest common prefix between two lexicographically adjacent suffixes: Suffix-i and Suffix-(i-1). This corresponds to an internal vertex of the equivalent Suffix Tree of T that branches out to at least two (or more) suffixes, thus this common prefix of these adjacent suffixes are repeated.

The longest common (repeated) prefix is the required answer, which can be found in O(n) by going through the LCP array once.

Without further ado, try LRS("GATAGACA\$"). We have LRS = "GA".

It is possible that T contains more than one LRS, e.g., try LRS("BANANABAN\$").
We have LRS = "ANA" (actually overlap) or "BAN" (without overlap).

### 3-6. Longest Common Substring (LCS)

After we construct the generalized Suffix Array of the concatenation of both strings T1\$T2# of length n = n1+n2 in O(n log n) and compute its LCP Array in O(n), we can find the Longest Repeated Substring (LRS) in T by simply iterating through all LCP values and reporting the largest one that comes from two different strings.

Without further ado, try LCS("GATAGACA\$", "CATA#") on the generalized Suffix Array of string T1 = "GATAGACA\$" and T2 = "CATA#". We have LCS = "ATA".

## 4. Implementation

You are allowed to use/modify our implementation code for fast Suffix Array+LCP: sa_lcp.cpp | py | java | ml to solve programming contest problems that need it.