** ** Table of content

Teleporter | Description |
---|---|

I. Lyndon Definitions | Definitions of Lyndon word, Lyndon factorization, ... |

II. Duval Algorithm | Talk about the duval algorithm and how it works |

III. Real Life Applications | Motivation to learn the algorithm |

IV. Programming Applications | The code is short and simple to implement |

V. My Questions | Are there any other applications ? |

................................................................... | .......................................................................................................................... |

** ** I. Lyndon Factorization

##### **1)** String Concatenation

**Definition:**The operation of joining character strings end-to-endProperty I: Non-empty string $$$S$$$ is a concatenation of all substrings of itself

Property II: Non-empty string $$$S$$$ is a concatenation of any empty string at any position in itself with itself

Property III: Let $$$A, B, C$$$ the non-empty string then $$$A + B + C = A + (B + C) = (A + B) + C$$$

For convention, let define the operator

**+**is**string concatenation**

##### **2)** Lyndon Word

**Definition:**A nonempty string that is strictly smaller in lexicographic order than all of its rotations.Property I: Lyndon word is nonempty and lexicographically strictly smaller than any of its proper suffixes.

Property II: Let $$$S, T, Z$$$ is nonempty word. $$$S$$$ is Lyndon word if $$$S < T\ \ \ \forall\ S = Z + T$$$

Property III: Lyndon word cant be factorized. It means that the only factor in its factorization is itself.

##### **3)** Lyndon Factorization

**Definition:**Split the string into many lyndon words in such a way that the words in the sequence are in lexicographically non-increasing order.Property I: For $$$s = s_1 + s_2 + \dots + s_k$$$ where $$$s_1, s_2, \dots, s_k$$$ are lyndon words and in non-increasing order ($$$s1 \geq s2 \geq \dots \geq s_k$$$)

Property II: Lyndon factorization is

**unique**.Property III: The last Lyndon Factor is Lexicographically Smallest Suffix of the string

##### **4)** Least Starting Position (LSP)

**Definition:**The minimal position of some substrings that make it**LMSR**.Property I: Let $$$X$$$ the substring of $$$S$$$ that its starting position $$$p$$$ is LSP. Then some Lyndon Factors in $$$X$$$ has the LSP $$$p$$$

Property II: $$$K$$$-repeated String, where each substring has size $$$L$$$ then there are $$$K$$$ LSP: $$$0, L, 2L, \dots, (K-1)L$$$

Property III: The Circular String start from LSP of given string is

**Lexicographically Minimal String Rotation**

** ** II. Duval Algorithm

**Exist Time:**1983**Duval algorithm**is an effecient algorithm for listing the Lyndon words of length at most $$$n$$$ with a given alphabet size $$$s$$$ in lexicographic order in $$$O(n)$$$The position of the last Lyndon factorized word from Duval algorithm provides minimum cyclic string

**The pseudo algorithm**

**Implementation - O(n) Time - O(1) Auxiliary Space**

**Detail Implementation - O(n) Time - O(1) Auxiliary Space**

** ** III. Real Life Applications

##### **1)** Finger print identification:

- We can encode the finger print into many detailed circular strings. How to search such finger print again from those in very huge data base ? Circular comparision using lyndon factorization is requried.

##### **2)** Biological genetics:

- In some cases, we need to know whether these two group's genetics are belonged to the same bacteria, virus.

##### **3)** Games:

- Well, ofcourse we can apply the algorithm in some language/words-related games

** ** IV. Programming Applications

##### **1)** Least rotation to make smallest lexicographical ordered string.

**The problem:**

Given a string $$$S$$$ of size $$$N$$$

A right-rotation is that move the leftmost character to rightmost of $$$S$$$

Find the least right-rotation to make $$$S$$$ become the smallest lexicographical ordered string

**Important Property:** After those right-rotations, the string will be Minimum Acyclic String

**The solution:**

One thing we can come in mind is to use hash or string algorithms in $$$O(n\ log\ n)$$$, but they are kinda complex

Some other approachs can be found here

**Bruteforces Solution:**Let $$$t = s + s$$$. Then for each substring of size $$$|s|$$$, we compare and find the smallest

**Bruteforces Solution - O(n^2) Time - O(n) Auxiliary Space**

**Optimized Bruteforces Solution - O(n) to O(n^2) Time - O(n) Auxiliary Space**

**Duval Solution:**We can apply lyndon factorization with a little bit observation

**Detail Duval Solution - O(n) Time - O(n) Auxiliary Space**

**None-comment Duval Solution - O(n) Time - O(n) Auxiliary Space**

**Optimized Duval Solution - O(n) Time - O(1) Auxiliary Space**

**Practices Problem:**

** ** V. My Question

##### **1)** Are there any other programming applications for **Lyndon Factorization** ?

- The algorithm is simple to code while running pretty fast as its low complexities. It would be kinda sad if there is only one main application, isnt it :(

##### **2)** Are there any other problems for **Lyndon Factorization** ?

- To remember something better and understand the algorithm deeper, we need to practice right :D It would be nice if there are some more problems