Trie is a type of k-ary search tree used for storing and searching a specific key from a set. Using Trie, search complexities can be brought to optimal limit (key length).
Definition
A trie (derived from retrieval) is a multiway tree data structure used for storing strings over an alphabet. It is used to store a large amount of strings. The pattern matching can be done efficiently using tries.
The trie shows words like allot, alone, ant, and, are, bat, bad. The idea is that all strings sharing common prefix should come from a common node. The tries are used in spell checking programs.
- Preprocessing pattern improves the performance of pattern matching algorithm. But if a text is very large then it is better to preprocess text instead of pattern for efficient search.
- A trie is a data structure that supports pattern matching queries in time proportional to the pattern size.
If we store keys in a binary search tree, a well balanced BST will need time proportional to M * log N, where M is the maximum string length and N is the number of keys in the tree. Using Trie, the key can be searched in O(M) time. However, the penalty is on Trie storage requirements (Please refer to Applications of Trie for more details).
Trie is also known as digital tree or prefix tree. Or 字典树 in Chinese language.
Template
int son[N][26], cnt[N], idx;
// 0号点既是根节点，又是空节点
// son[][]存储树中每个节点的子节点
// cnt[]存储以每个节点结尾的单词数量
// 插入一个字符串
void insert(char *str)
{
int p = 0;
for (int i = 0; str[i]; i ++ )
{
int u = str[i] - 'a';
if (!son[p][u]) son[p][u] = ++ idx;
p = son[p][u];
}
cnt[p] ++ ;
}
// 查询字符串出现的次数
int query(char *str)
{
int p = 0;
for (int i = 0; str[i]; i ++ )
{
int u = str[i] - 'a';
if (!son[p][u]) return 0;
p = son[p][u];
}
return cnt[p];
}TrieST.java from algs4
/******************************************************************************
* Compilation: javac TrieST.java
* Execution: java TrieST < words.txt
* Dependencies: StdIn.java
* Data files: https://algs4.cs.princeton.edu/52trie/shellsST.txt
*
* A string symbol table for extended ASCII strings, implemented
* using a 256-way trie.
*
* % java TrieST < shellsST.txt
* by 4
* sea 6
* sells 1
* she 0
* shells 3
* shore 7
* the 5
*
******************************************************************************/
package edu.princeton.cs.algs4;
/**
* The {@code TrieST} class represents a symbol table of key-value
* pairs, with string keys and generic values.
* It supports the usual <em>put</em>, <em>get</em>, <em>contains</em>,
* <em>delete</em>, <em>size</em>, and <em>is-empty</em> methods.
* It also provides character-based methods for finding the string
* in the symbol table that is the <em>longest prefix</em> of a given prefix,
* finding all strings in the symbol table that <em>start with</em> a given prefix,
* and finding all strings in the symbol table that <em>match</em> a given pattern.
* A symbol table implements the <em>associative array</em> abstraction:
* when associating a value with a key that is already in the symbol table,
* the convention is to replace the old value with the new value.
* Unlike {@link java.util.Map}, this class uses the convention that
* values cannot be {@code null}—setting the
* value associated with a key to {@code null} is equivalent to deleting the key
* from the symbol table.
* <p>
* This implementation uses a 256-way trie.
* The <em>put</em>, <em>contains</em>, <em>delete</em>, and
* <em>longest prefix</em> operations take time proportional to the length
* of the key (in the worst case). Construction takes constant time.
* The <em>size</em>, and <em>is-empty</em> operations take constant time.
* Construction takes constant time.
* <p>
* For additional documentation, see <a href="https://algs4.cs.princeton.edu/52trie">Section 5.2</a> of
* <i>Algorithms, 4th Edition</i> by Robert Sedgewick and Kevin Wayne.
*/
public class TrieST<Value> {
private static final int R = 256; // extended ASCII
private Node root; // root of trie
private int n; // number of keys in trie
// R-way trie node
private static class Node {
private Object val;
private Node[] next = new Node[R];
}
/**
* Initializes an empty string symbol table.
*/
public TrieST() {
}
/**
* Returns the value associated with the given key.
* @param key the key
* @return the value associated with the given key if the key is in the symbol table
* and {@code null} if the key is not in the symbol table
* @throws IllegalArgumentException if {@code key} is {@code null}
*/
public Value get(String key) {
if (key == null) throw new IllegalArgumentException("argument to get() is null");
Node x = get(root, key, 0);
if (x == null) return null;
return (Value) x.val;
}
/**
* Does this symbol table contain the given key?
* @param key the key
* @return {@code true} if this symbol table contains {@code key} and
* {@code false} otherwise
* @throws IllegalArgumentException if {@code key} is {@code null}
*/
public boolean contains(String key) {
if (key == null) throw new IllegalArgumentException("argument to contains() is null");
return get(key) != null;
}
private Node get(Node x, String key, int d) {
if (x == null) return null;
if (d == key.length()) return x;
char c = key.charAt(d);
return get(x.next[c], key, d+1);
}
/**
* Inserts the key-value pair into the symbol table, overwriting the old value
* with the new value if the key is already in the symbol table.
* If the value is {@code null}, this effectively deletes the key from the symbol table.
* @param key the key
* @param val the value
* @throws IllegalArgumentException if {@code key} is {@code null}
*/
public void put(String key, Value val) {
if (key == null) throw new IllegalArgumentException("first argument to put() is null");
if (val == null) delete(key);
else root = put(root, key, val, 0);
}
private Node put(Node x, String key, Value val, int d) {
if (x == null) x = new Node();
if (d == key.length()) {
if (x.val == null) n++;
x.val = val;
return x;
}
char c = key.charAt(d);
x.next[c] = put(x.next[c], key, val, d+1);
return x;
}
/**
* Returns the number of key-value pairs in this symbol table.
* @return the number of key-value pairs in this symbol table
*/
public int size() {
return n;
}
/**
* Is this symbol table empty?
* @return {@code true} if this symbol table is empty and {@code false} otherwise
*/
public boolean isEmpty() {
return size() == 0;
}
/**
* Returns all keys in the symbol table as an {@code Iterable}.
* To iterate over all of the keys in the symbol table named {@code st},
* use the foreach notation: {@code for (Key key : st.keys())}.
* @return all keys in the symbol table as an {@code Iterable}
*/
public Iterable<String> keys() {
return keysWithPrefix("");
}
/**
* Returns all of the keys in the set that start with {@code prefix}.
* @param prefix the prefix
* @return all of the keys in the set that start with {@code prefix},
* as an iterable
*/
public Iterable<String> keysWithPrefix(String prefix) {
Queue<String> results = new Queue<String>();
Node x = get(root, prefix, 0);
collect(x, new StringBuilder(prefix), results);
return results;
}
private void collect(Node x, StringBuilder prefix, Queue<String> results) {
if (x == null) return;
if (x.val != null) results.enqueue(prefix.toString());
for (char c = 0; c < R; c++) {
prefix.append(c);
collect(x.next[c], prefix, results);
prefix.deleteCharAt(prefix.length() - 1);
}
}
/**
* Returns all of the keys in the symbol table that match {@code pattern},
* where the character '.' is interpreted as a wildcard character.
* @param pattern the pattern
* @return all of the keys in the symbol table that match {@code pattern},
* as an iterable, where . is treated as a wildcard character.
*/
public Iterable<String> keysThatMatch(String pattern) {
Queue<String> results = new Queue<String>();
collect(root, new StringBuilder(), pattern, results);
return results;
}
private void collect(Node x, StringBuilder prefix, String pattern, Queue<String> results) {
if (x == null) return;
int d = prefix.length();
if (d == pattern.length() && x.val != null)
results.enqueue(prefix.toString());
if (d == pattern.length())
return;
char c = pattern.charAt(d);
if (c == '.') {
for (char ch = 0; ch < R; ch++) {
prefix.append(ch);
collect(x.next[ch], prefix, pattern, results);
prefix.deleteCharAt(prefix.length() - 1);
}
}
else {
prefix.append(c);
collect(x.next[c], prefix, pattern, results);
prefix.deleteCharAt(prefix.length() - 1);
}
}
/**
* Returns the string in the symbol table that is the longest prefix of {@code query},
* or {@code null}, if no such string.
* @param query the query string
* @return the string in the symbol table that is the longest prefix of {@code query},
* or {@code null} if no such string
* @throws IllegalArgumentException if {@code query} is {@code null}
*/
public String longestPrefixOf(String query) {
if (query == null) throw new IllegalArgumentException("argument to longestPrefixOf() is null");
int length = longestPrefixOf(root, query, 0, -1);
if (length == -1) return null;
else return query.substring(0, length);
}
// returns the length of the longest string key in the subtrie
// rooted at x that is a prefix of the query string,
// assuming the first d character match and we have already
// found a prefix match of given length (-1 if no such match)
private int longestPrefixOf(Node x, String query, int d, int length) {
if (x == null) return length;
if (x.val != null) length = d;
if (d == query.length()) return length;
char c = query.charAt(d);
return longestPrefixOf(x.next[c], query, d+1, length);
}
/**
* Removes the key from the set if the key is present.
* @param key the key
* @throws IllegalArgumentException if {@code key} is {@code null}
*/
public void delete(String key) {
if (key == null) throw new IllegalArgumentException("argument to delete() is null");
root = delete(root, key, 0);
}
private Node delete(Node x, String key, int d) {
if (x == null) return null;
if (d == key.length()) {
if (x.val != null) n--;
x.val = null;
}
else {
char c = key.charAt(d);
x.next[c] = delete(x.next[c], key, d+1);
}
// remove subtrie rooted at x if it is completely empty
if (x.val != null) return x;
for (int c = 0; c < R; c++)
if (x.next[c] != null)
return x;
return null;
}
/**
* Unit tests the {@code TrieST} data type.
*
* @param args the command-line arguments
*/
public static void main(String[] args) {
// build symbol table from standard input
TrieST<Integer> st = new TrieST<Integer>();
for (int i = 0; !StdIn.isEmpty(); i++) {
String key = StdIn.readString();
st.put(key, i);
}
// print results
if (st.size() < 100) {
StdOut.println("keys(\"\"):");
for (String key : st.keys()) {
StdOut.println(key + " " + st.get(key));
}
StdOut.println();
}
StdOut.println("longestPrefixOf(\"shellsort\"):");
StdOut.println(st.longestPrefixOf("shellsort"));
StdOut.println();
StdOut.println("longestPrefixOf(\"quicksort\"):");
StdOut.println(st.longestPrefixOf("quicksort"));
StdOut.println();
StdOut.println("keysWithPrefix(\"shor\"):");
for (String s : st.keysWithPrefix("shor"))
StdOut.println(s);
StdOut.println();
StdOut.println("keysThatMatch(\".he.l.\"):");
for (String s : st.keysThatMatch(".he.l."))
StdOut.println(s);
}
}
/******************************************************************************
* Copyright 2002-2022, Robert Sedgewick and Kevin Wayne.
*
* This file is part of algs4.jar, which accompanies the textbook
*
* Algorithms, 4th edition by Robert Sedgewick and Kevin Wayne,
* Addison-Wesley Professional, 2011, ISBN 0-321-57351-X.
* http://algs4.cs.princeton.edu
*
*
* algs4.jar is free software: you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation, either version 3 of the License, or
* (at your option) any later version.
*
* algs4.jar is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with algs4.jar. If not, see http://www.gnu.org/licenses.
******************************************************************************/