Efficient auto-complete with a ternary search tree

The original post is http://igoro.com/archive/efficient-auto-complete-with-a-ternary-search-tree/

Over the past couple of years, auto-complete has popped up all over the web. Facebook, YouTube, Google, Bing, MSDN, LinkedIn and lots of other websites all try to complete your phrase as soon as you start typing.

Auto-complete definitely makes for a nice user experience, but it can be a challenge to implement efficiently. In many cases, an efficient implementation requires the use of interesting algorithms and data structures. In this blog post, I will describe one simple data structure that can be used to implement auto-complete: a ternary search tree.

Trie: simple but space-inefficient

Before discussing ternary search trees, let’s take a look at a simple data structure that supports a fast auto-complete lookup but needs too much memory: a trie. A trie is a tree-like data structure in which each node contains an array of pointers, one pointer for each character in the alphabet. Starting at the root node, we can trace a word by following pointers corresponding to the letters in the target word.

Each node could be implemented like this in C#:

class TrieNode
{
public const int ALPHABET_SIZE = 26;
public TrieNode[] m_pointers = new TrieNode[ALPHABET_SIZE];
public bool m_endsString = false;
}

Here is a trie that stores words AB, ABBA, ABCD, and BCD. Nodes that terminate words are marked yellow:

Implementing auto complete using a trie is easy. We simply trace pointers to get to a node that represents the string the user entered. By exploring the trie from that node down, we can enumerate all strings that complete user’s input.

But, a trie has a major problem that you can see in the diagram above. The diagram only fits on the page because the trie only supports four letters {A,B,C,D}. If we needed to support all 26 English letters, each node would have to store 26 pointers. And, if we need to support international characters, punctuation, or distinguish between lowercase and uppercase characters, the memory usage grows becomes untenable.

Our problem has to do with the memory taken up by all the null pointers stored in the node arrays. We could consider using a different data structure in each node, such as a hash map. However, managing thousands and thousands of hash maps is generally not a good idea, so let’s take a look at a better solution.

Ternary search tree to the rescue

A ternary tree is a data structure that solves the memory problem of tries in a more clever way. To avoid the memory occupied by unnecessary pointers, each trie node is represented as a tree-within-a-tree rather than as an array. Each non-null pointer in the trie node gets its own node in a ternary search tree.

For example, the trie from the example above would be represented in the following way as a ternary search tree:

The ternary search tree contains three types of arrows. First, there are arrows that correspond to arrows in the corresponding trie, shown as dashed down-arrows. Traversing a down-arrow corresponds to “matching” the character from which the arrow starts. The left- and right- arrow are traversed when the current character does not match the desired character at the current position. We take the left-arrow if the character we are looking for is alphabetically before the character in the current node, and the right-arrow in the opposite case.

For example, green arrows show how we’d confirm that the ternary tree contains string ABBA:

And this is how we’d find that the ternary string does not contain string ABD:

Ternary search tree on a server

On the web, a significant chunk of the auto-complete work has to be done by the server. Often, the set of possible completions is large, so it is usually not a good idea to download all of it to the client. Instead, the ternary tree is stored on the server, and the client will send prefix queries to the server.

The client will send a query for words starting with “bin” to the server:

And the server responds with a list of possible words:

Implementation

Here is a simple ternary search tree implementation in C#:

public class TernaryTree
{
private Node m_root = null;
private void Add(string s, int pos, ref Node node)
{
if (node == null) { node = new Node(s[pos], false); }
if (s[pos] < node.m_char) { Add(s, pos, ref node.m_left); }
else if (s[pos] > node.m_char) { Add(s, pos, ref node.m_right); }
else
{
if (pos + 1 == s.Length) { node.m_wordEnd = true; }
else { Add(s, pos + 1, ref node.m_center); }
}
}
public void Add(string s)
{
if (s == null || s == "") throw new ArgumentException();
Add(s, 0, ref m_root);
}
public bool Contains(string s)
{
if (s == null || s == "") throw new ArgumentException();
int pos = 0;
Node node = m_root;
while (node != null)
{
int cmp = s[pos] - node.m_char;
if (s[pos] < node.m_char) { node = node.m_left; }
else if (s[pos] > node.m_char) { node = node.m_right; }
else
{
if (++pos == s.Length) return node.m_wordEnd;
node = node.m_center;
}
}
return false;
}
}

And here is the Node class:

class Node
{
internal char m_char;
internal Node m_left, m_center, m_right;
internal bool m_wordEnd;
public Node(char ch, bool wordEnd)
{
m_char = ch;
m_wordEnd = wordEnd;
}
}

Remarks

For best performance, strings should be inserted into the ternary tree in a random order. In particular, do not insert strings in the alphabetical order. Each mini-tree that corresponds to a single trie node would degenerate into a linked list, significantly increasing the cost of lookups. Of course, more complex self-balancing ternary trees can be implemented as well.

And, don’t use a fancier data structure than you have to. If you only have a relatively small set of candidate words (say on the order of hundreds) a brute-force search should be fast enough.