Patricia前缀树(Patricia Trie)及其基本操作

Trie是一种字典树，用于存储文本字符，并利用了单词之间共享前缀的特点，所以叫做前缀树。不像平衡BST，Trie的高度只与最长的文本串的长度s有关系，而与单词的数量n无关。Trie的节点分两种：内部结点和叶子结点，内部结点用来存储单词key的成分字母，如果设字母表大小为d，那么每个内部结点最多有d个孩子，叶子结点存储该单词作为key的数据内容(data)。注意内部结点和叶子结点并不是互斥的，一个内部结点本身可以有儿子结点，同时它也可以是一个叶子结点。例如：

这里ab结点本身就是叶子结点，因为它以#结束符标记了，同时它有两个儿子结点a和c，c是叶子结点。

如果一颗Trie中有很多单词只有一个儿子结点，可以用Patricia Trie（Linux内核中叫做Radix Tree）压缩存储。由于#结束符标记被看作是一个叶子结点，那么一颗Patricia Trie的任何内部结点有2个或以上的孩子结点。

Patricia Trie的基本操作包括：插入、删除和查询。插入操作可能会涉及到split（拆分），删除操作可能会涉及到merge（合并）。基于Patricia Trie的基本性质，split和merge操作都是局部的，这样实现起来比较简单。详细内容请参考以下代码：

实现：

import java.util.LinkedList; import java.util.List; /** * Patricia Trie * * Copyright (c) 2011 ljs (http://blog.csdn.net/ljsspace/) * Licensed under GPL (http://www.opensource.org/licenses/gpl-license.php) * * @author ljs * 2011-06-27 * */ public class PatriciaTrie { private class PatriciaTrieNode { private String key; private Integer data; private List<PatriciaTrieNode> children = new LinkedList<PatriciaTrieNode>(); //use "#" for terminal char private boolean terminal; public PatriciaTrieNode(){ this.key = ""; } public PatriciaTrieNode(String key){ this.key = key; } public String toString(){ return this.key + (this.terminal?"#":"") + "(" + children.size() +")"; } } private PatriciaTrieNode root; //return the value of the external node if found; //otherwise, return null public Integer find(String key){ if(key == null || key.length() == 0) return null; if(root==null){ return null; }else{ return find(root,key); } } private Integer find(PatriciaTrieNode currNode,String key) { for(int i=0;i<currNode.children.size();i++){ PatriciaTrieNode child = currNode.children.get(i); //use min(child.key.length, key.length) int len = child.key.length()<key.length()?child.key.length(): key.length(); int j = 0; for(;j<len;j++){ if(key.charAt(j) != child.key.charAt(j)){ break; } } if(j==0){//this child doesn't match any character with the new key //order keys by lexi-order if(key.charAt(0)<child.key.charAt(0)){ //e.g. child="e", key="c" (currNode="abc") // abc // / / // e h return null; }else{ //e.g. child="e", key="h" (currNode="abc") continue; } }else{//current child's key partially matches with the new key; 0<j<=len if(j==len){ if(key.length()==child.key.length()){ if(child.terminal){ //e.g. child="ab", key="ab" // ab# // / // f# return child.data; }else{ //e.g. child="ab", key="ab" // ab // / / // e f return null; } }else if(key.length()>child.key.length()){ //e.g. child="ab#", key="abc" // ab# // / / // a c# String subkey = key.substring(j); //c //recursion return find(child,subkey); }else{ //key.length()<child.key.length() //e.g. child="abc", key="ab" // abc // / / // e f return null; } }else{//0<j<len //e.g. child="abc", key="abd" // abc // / / // e f return null; } } } return null; } public void delete(String key) throws Exception{ if(key == null || key.length() == 0) return; if(root==null){ return; } delete(root,key); } private void delete(PatriciaTrieNode currNode,String key) throws Exception{ boolean done = false; for(int i=0;i<currNode.children.size();i++){ PatriciaTrieNode child = currNode.children.get(i); //use min(child.key.length, key.length) int len = child.key.length()<key.length()?child.key.length(): key.length(); int j = 0; for(;j<len;j++){ if(key.charAt(j) != child.key.charAt(j)){ break; } } if(j==0){//this child doesn't match any character with the new key //order keys by lexi-order if(key.charAt(0)<child.key.charAt(0)){ //e.g. child="e", key="c" (currNode="abc") // abc // / / // e h done = true; throw new Exception("No such key is found for removal!"); }else{ //e.g. child="e", key="h" (currNode="abc") continue; } }else{//current child's key partially matches with the new key; 0<j<=len if(j==len){ if(key.length()==child.key.length()){ if(child.terminal){ //found key, delete it if(child.children.size()==0){ //e.g. child="ab#", key="ab", currNode="a" // a // / / // d ab# currNode.children.remove(i); //merge node for currNode if(!currNode.terminal && currNode.children.size()==1){ PatriciaTrieNode singleChild = currNode.children.get(0); //d currNode.key += singleChild.key; currNode.data = singleChild.data; currNode.terminal = singleChild.terminal; currNode.children = singleChild.children; } }else{ //child.children.size()>=1 //e.g. child="ab#", key="ab", currNode="a" // a# // / // ab# // / // f# child.terminal = false; //merge node for child if(child.children.size()==1){ PatriciaTrieNode singleChild = child.children.get(0); //f# child.key += singleChild.key; child.data = singleChild.data; child.terminal = singleChild.terminal; //Note: singleChild may not be external node child.children = singleChild.children; } } }else{ //e.g. child="ab", key="ab" // ab // / / // e f throw new Exception("No such key is found for removal!"); } }else if(key.length()>child.key.length()){ //e.g. child="ab#", key="abc" // ab# // / / // a c# String subkey = key.substring(j); //c //recursion delete(child,subkey); }else{ //key.length()<child.key.length() //e.g. child="abc", key="ab" // abc // / / // e f throw new Exception("No such key is found for removal!"); } }else{//0<j<len //e.g. child="abc", key="abd" // abc // / / // e f throw new Exception("No such key is found for removal!"); } done = true; break; } } if(!done) { throw new Exception("No such key is found for removal!"); } } //value is only located at the external node private void insert(PatriciaTrieNode currNode,String key,Integer value) throws Exception{ boolean done = false; for(int i=0;i<currNode.children.size();i++){ PatriciaTrieNode child = currNode.children.get(i); //use min(child.key.length, key.length) int len = child.key.length()<key.length()?child.key.length(): key.length(); int j = 0; for(;j<len;j++){ if(key.charAt(j) != child.key.charAt(j)){ break; } } if(j==0){//this child doesn't match any character with the new key //order keys by lexi-order if(key.charAt(0)<child.key.charAt(0)){ //e.g. child="e" (currNode="abc") // abc abc // / / =========> / | / // e f insert "c" c# e f PatriciaTrieNode node = new PatriciaTrieNode(key); currNode.children.add(i,node); node.terminal = true; node.data = value; done = true; break; }else{ //key.charAt(0)>child.key.charAt(0) //don't forget to add the largest new key after iterating all children continue; } }else{//current child's key partially matches with the new key; 0<j<=len if(j==len){ if(key.length()==child.key.length()){ if(child.terminal){ throw new Exception("Duplicate Key is found when insertion!"); }else{ //e.g. child="ab" // ab ab# // / / =========> / / // e f insert "ab" e f child.terminal = true; child.data = value; } }else if(key.length()>child.key.length()){ //e.g. child="ab#" // ab# ab# // / / ==========> / | / // e f insert "abc" c# e f String subkey = key.substring(j); //recursion insert(child,subkey,value); }else{ //key.length()<child.key.length() //e.g. child="abc#" // abc# ab# // / / =========> / // e f insert "ab" c# // / / // e f String childSubkey = child.key.substring(j); //c PatriciaTrieNode subChildNode = new PatriciaTrieNode(childSubkey); subChildNode.terminal = child.terminal; subChildNode.data = child.data; subChildNode.children = child.children; //inherited from parent child.key = key; //ab child.terminal = true; //ab# child.data = value; child.children = new LinkedList<PatriciaTrieNode>(); child.children.add(subChildNode); } }else{//0<j<len //e.g. child="abc#" // abc# ab // / / ==========> / / // e f insert "abd" c# d# // / / // e f //split at j String childSubkey = child.key.substring(j); //c String subkey = key.substring(j); //d PatriciaTrieNode subChildNode = new PatriciaTrieNode(childSubkey); subChildNode.terminal = child.terminal; subChildNode.data = child.data; subChildNode.children = child.children; //inherited from parent //update child's key child.key = child.key.substring(0,j); //child is not terminal now due to split, it is inherited by subChildNode child.terminal = false; //Note: no need to merge subChildNode PatriciaTrieNode node = new PatriciaTrieNode(subkey); node.terminal = true; node.data = value; child.children = new LinkedList<PatriciaTrieNode>(); if(subkey.charAt(0)<childSubkey.charAt(0)){ child.children.add(node); child.children.add(subChildNode); }else{ child.children.add(subChildNode); child.children.add(node); } } done = true; break; } } if(!done){ PatriciaTrieNode node = new PatriciaTrieNode(key); node.terminal = true; node.data = value; currNode.children.add(node); } } public void insert(String key,Integer value) throws Exception{ if(key == null || key.length() == 0) return; if(root==null){ root = new PatriciaTrieNode(); } insert(root,key,value); } public PatriciaTrieNode getRoot(){ return root; } //for test purpose only public void printTree(){ this.print(0, this.root); } private void print(int level, PatriciaTrieNode node){ for (int i = 0; i < level; i++) { System.out.format(" "); } System.out.format("|"); for (int i = 0; i < level; i++) { System.out.format("-"); } if (node.terminal) System.out.format("%s[%s]#%n", node.key,node.data); else System.out.format("%s%n", node.key); for (PatriciaTrieNode child : node.children) { print(level + 1, child); } } public void testFind(String key){ Integer val = this.find(key); if(val != null) System.out.format("Found key /"%s/" at: %s%n",key,val); else System.out.format("Found no such key: /"%s/"%n",key); } public static void main(String[] args) throws Exception { //test insertion PatriciaTrie ptrie = new PatriciaTrie(); ptrie.insert("ab",1); ptrie.insert("abc",2); ptrie.insert("abde",3); ptrie.insert("abd",4); //ptrie.insert("dc"); ptrie.insert("dce",5); ptrie.insert("dceh",6); ptrie.insert("dceg",7); ptrie.insert("dca",8); ptrie.insert("dcf",9); ptrie.insert("ghk",10); ptrie.insert("gh",11); ptrie.insert("mns",12); ptrie.insert("mnt",13); ptrie.insert("mn",14); ptrie.insert("mg",15); ptrie.printTree(); String key = "dc"; ptrie.testFind(key); key = "d"; ptrie.testFind(key); key = "ab"; ptrie.testFind(key); key = "ef"; ptrie.testFind(key); key = "zz"; ptrie.testFind(key); key = "dk"; ptrie.testFind(key); key = "dcf"; ptrie.testFind(key); key = "dck"; ptrie.testFind(key); key = "abd"; ptrie.delete(key); System.out.format("After delete key: %s%n",key); ptrie.printTree(); System.out.println("****************************"); ptrie = new PatriciaTrie(); ptrie.insert("bear",1); ptrie.insert("bell",2); ptrie.insert("bid",3); ptrie.insert("bull",4); ptrie.insert("buy",5); ptrie.insert("sell",6); ptrie.insert("stock",7); ptrie.insert("stop",8); ptrie.printTree(); System.out.println("****************************"); ptrie = new PatriciaTrie(); ptrie.insert("allot",1); ptrie.insert("alloy",2); ptrie.insert("all",3); ptrie.insert("aloe",4); ptrie.insert("ant",5); ptrie.insert("an",6); ptrie.insert("are",7); ptrie.insert("ate",8); ptrie.insert("be",9); ptrie.printTree(); System.out.println("****************************"); ptrie = new PatriciaTrie(); ptrie.insert("minimize",0); ptrie.insert("mize",4); ptrie.insert("ze",6); ptrie.insert("nimize",2); ptrie.insert("ize",5); ptrie.insert("inimize",1); ptrie.insert("imize",3); ptrie.insert("e",7); ptrie.printTree(); key = "ize"; ptrie.testFind(key); key = "zee"; ptrie.testFind(key); key = "mize"; ptrie.testFind(key); key = "mize"; ptrie.delete(key); System.out.format("After delete key: %s%n",key); ptrie.printTree(); } }

测试输出：

| |-ab[1]# |--c[2]# |--d[4]# |---e[3]# |-dc |--a[8]# |--e[5]# |---g[7]# |---h[6]# |--f[9]# |-gh[11]# |--k[10]# |-m |--g[15]# |--n[14]# |---s[12]# |---t[13]# Found no such key: "dc" Found no such key: "d" Found key "ab" at: 1 Found no such key: "ef" Found no such key: "zz" Found no such key: "dk" Found key "dcf" at: 9 Found no such key: "dck" After delete key: abd | |-ab[1]# |--c[2]# |--de[3]# |-dc |--a[8]# |--e[5]# |---g[7]# |---h[6]# |--f[9]# |-gh[11]# |--k[10]# |-m |--g[15]# |--n[14]# |---s[12]# |---t[13]# **************************** | |-b |--e |---ar[1]# |---ll[2]# |--id[3]# |--u |---ll[4]# |---y[5]# |-s |--ell[6]# |--to |---ck[7]# |---p[8]# **************************** | |-a |--l |---l[3]# |----o |-----t[1]# |-----y[2]# |---oe[4]# |--n[6]# |---t[5]# |--re[7]# |--te[8]# |-be[9]# **************************** | |-e[7]# |-i |--mize[3]# |--nimize[1]# |--ze[5]# |-mi |--nimize[0]# |--ze[4]# |-nimize[2]# |-ze[6]# Found key "ize" at: 5 Found no such key: "zee" Found key "mize" at: 4 After delete key: mize | |-e[7]# |-i |--mize[3]# |--nimize[1]# |--ze[5]# |-minimize[0]# |-nimize[2]# |-ze[6]#

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