## Introduction

In this post I talk about three data structures I implemented to compare their performance in different scenarios. The three data structures are **Trie**, **Ternary Search Tree** and **Radix Tree**.

**All the code for the data structures as well as the tested scenarios are available in this repo in GitHub.**

## What is a Prefix Tree?

It is an **ordered** tree data structure that is used to store a dynamic set of elements where the key is a string. All the data structures that I implemented have the following methods:

Operation | Description |
---|---|

clear | Removes all the content of the data structure |

find | Returns the value stored in the data structure for a specific key provided |

insert | Adds a new pair of key and value to the data structure |

size | Returns the amount of elements stored in the data structure |

show | Prints the content of the data structure in the dot format |

erase | Removes the value stored in the data structure for a specific key provided |

contains | Returns true if there is a value associated to the key provided |

keys | Returns a std::vectorstd::string with all the keys in the data structure that have the provided prefix. If no prefix is provided, it returns a std::vectorstd::string with all the keys in the data structure. |

lcp | Returns the longest common prefix of all keys stored in the data structure. |

## Trie

Unlike a binary search tree, none of the nodes in the tree stores the key associated with that node. Instead, **its position in the tree defines the key** with which it is associated. All the descendants of a node have a common prefix of the string associated with that node, and the root is associated with the empty string. Values are not necessarily associated with every node. Rather, **values tend only to be associated with leaves**, and with some inner nodes that correspond to keys of interest.

Example of a **Trie** that contains the keys: “He”, “Hello”, “Kthulu”, “No”, “Wololo”, “World”, “Worst”, “he”:

## Ternary Search Tree

In a **Ternary Search Tree** nodes are arranged in a manner **similar to a binary search tree**, but with up to **three children** rather than the binary tree’s limit of two. Like other prefix trees, a ternary search tree can be used as an **associative map structure with the ability for incremental string search**. However, ternary search trees are more **space efficient** compared to standard prefix trees, at the cost of speed.

Example of a **Ternary Search Tree** that contains the keys: “He”, “Hello”, “Kthulu”, “No”, “Wololo”, “World”, “Worst”, “he”:

## Radix Tree

**Radix Tree** is a space-optimized **Trie** in which each node that is the **only child is merged with its parent**. The result is that the **number of children of every internal node is at least the radix r** of the radix trie, where

*r*is a positive integer and a power

*x*of 2, having

*x*>= 1. Unlike in regular tries,

**edges can be labeled with sequences of elements**as well as single elements. This makes radix trees much

**more efficient for small sets**(especially if the strings are long) and for

**sets of strings that share long prefixes**.

Example of a **Radix Tree** that contains the keys: “He”, “Hello”, “Kthulu”, “No”, “Wololo”, “World”, “Worst”, “he”:

## Comparison

In this section I show a comparison of the performance for the three data structures in different scenarios. **Note: sometimes the Trie runs out of memory. Therefore, there is no data on these cases.**

### Insert

### Search

#### Found

#### Not found

### Erase

#### Found

#### Not found

### Get keys

### Get keys prefix

### Longest common path

## Conclusion

The **Radix Tree** is in average the best of the three data structures. Nevertheless, I found out a paper where it explains how to implement a **Balanced Ternary Search Tree**, because one of the weakness of the **Ternary Search Tree** is that its shape depends of the order of the insertions. **I have plenty of work to do regarding this topic** but I wanted to share with you some of the preliminar results. Anyway, I am working on writing a paper explaining with more detail this experiment and the way I conducted the expertiment.