使用倍增算法(Prefix Doubling)构造后缀数组

如果采用对每个后缀排序的方法来生成后缀数组,即使采用快速排序,由于每次比较的对象是字符串(设输入字符串的长度为n),因此每一个比较操作的复杂度不再是常数,而是O(n),快速排序本身的平均情况下时间复杂度为O(nlgn),因此总的时间复杂度是O(n^2*lgn),如果考虑到采用快速排序最坏情况下复杂度为O(n^2),那么最坏时总的复杂度为O(n^3),显然在输入串长度很大时这种做法不可取。

Prefix Doubling算法(倍增法)是构造后缀数组一个比较实用的算法。其基本思想是先计算出每个后缀的k-前缀的rank值,然后在此基础上计算每个后缀的2k-前缀rank值,k从1开始。直到每个后缀都排出先后顺序为止(任何两个后缀都不会相等,换句话说,每个后缀终究都能排出先后顺序)。在处理2k-前缀时,只需要使用基数排序(radix sort)算法,先后对两位数字排序(可以采用计数排序算法(counting sort)对每一位数字排序)。在最坏情况下,需要做lgn次基数排序,每一次基数排序的操作次数为2*O(n),因此它的时间复杂度是O(nlgn)。倍增法虽然没有达到像DC3算法的线性复杂度,但是它的优点是实现比较简单,因此在实践中常被采用。

在以下实现中,当k=1时,由于只需要对输入串的每个字符排序,因此在采用基数排序时,只有一位数字需要排序。当k>1时,需要对两位数字排序(为考虑通用性,代码中Tuple.digits数组长度可以取>=1的任何整数,而不限于两位数字的基数排序)。如果rank数组中某个后缀具有最大rank值,而且该rank值等于输入串的长度,这时说明每一个后缀都排出了次序,因而可以提前终止程序。另外,假设字母表中最大的字符为MAX_CHAR。

注:rank数组中的rank值对应SA数组的下标+1。例如T=mississippi#,那么SA=(11,10,7,4,1,0,9,8,6,3,5,2),这里第7个后缀(即ippi$)在SA数组中的下标是2,因此Rank[7]=3。


使用倍增算法(Prefix Doubling)构造后缀数组_第1张图片

使用倍增算法(Prefix Doubling)构造后缀数组_第2张图片


实现:

/**
 * 
 * Build Suffix Array using Prefix Doubling Algorithm
 * see also: Udi Manber and Gene Myers' seminal paper(1991): "Suffix arrays: A new method for on-line string searches"
 *  
 * Copyright (c) 2011 ljs (http://blog.csdn.net/ljsspace/)
 * Licensed under GPL (http://www.opensource.org/licenses/gpl-license.php) 
 * 
 * @author ljs
 * 2011-07-17
 *
 */
public class PrefixDoubling {
	public static final char MAX_CHAR = '\u00FF';
	
	class Suffix{
		int[] sa;  
		//Note: the p-th suffix in sa: SA[rank[p]-1]];
		//p is the index of the array "rank", start with 0;
		//a text S's p-th suffix is S[p..n], n=S.length-1.
		int[] rank; 
		boolean done;
	}
	
	//a prefix of suffix[isuffix] represented with digits
	class Tuple{
		int isuffix; //the p-th suffix
		int[] digits;
		public Tuple(int suffix,int[] digits){
			this.isuffix = suffix;
			this.digits = digits;			
		}
		public String toString(){
			StringBuffer sb = new StringBuffer();			
			sb.append(isuffix);
			sb.append("(");
			for(int i=0;i<digits.length;i++){
				sb.append(digits[i]);
				if(i<digits.length-1)
					sb.append("-");
			}
			sb.append(")");
			return sb.toString();
		}
	}
	

 
	//the plain counting sort algorithm for comparison
	//A: input array
	//B: output array (sorted)
	//max: A value's range is 0...max
	public void countingSort(int[] A,int[] B,int max){
		//init the counter array
		int[] C = new int[max+1];
		for(int i=0;i<=max;i++){
			C[i] = 0;
		}
		//stat the count in A
		for(int j=0;j<A.length;j++){
			C[A[j]]++;
		}
		//process the counter array C
		for(int i=1;i<=max;i++){
			C[i]+=C[i-1];
		}
		//distribute the values in A to array B
		for(int j=A.length-1;j>=0;j--){
			//C[A[j]] <= A.length 
			B[--C[A[j]]]=A[j];			
		}
	}
 
	
	
	//d: the digit to do countingsort
	//max: A value's range is 0...max
	private void countingSort(int d,Tuple[] tA,Tuple[] tB,int max){
		//init the counter array
		int[] C = new int[max+1];
		for(int i=0;i<=max;i++){
			C[i] = 0;
		}
		//stat the count
		for(int j=0;j<tA.length;j++){
			C[tA[j].digits[d]]++;
		}
		//process the counter array C
		for(int i=1;i<=max;i++){
			C[i]+=C[i-1];
		}
		//distribute the values  
		for(int j=tA.length-1;j>=0;j--){
			//C[A[j]] <= A.length 
			tB[--C[tA[j].digits[d]]]=tA[j];			
		}
	}
	
	
	//tA: input
	//tB: output for rank caculation
	private void radixSort(Tuple[] tA,Tuple[] tB,int max,int digitsLen){
		int len = tA.length;
		int digitsTotalLen = tA[0].digits.length;
			
		for(int d=digitsTotalLen-1,j=0;j<digitsLen;d--,j++){
			this.countingSort(d, tA, tB, max);
			//assign tB to tA
			if(j<digitsLen-1){
				for(int i=0;i<len;i++){
					tA[i] = tB[i];
				}		
			}
		}
	}
	//max is the maximum value in any digit of TA.digits[], used for counting sort
	//tA: input
	//tB: the place holder, reused between iterations
	private Suffix rank(Tuple[] tA,Tuple[] tB,int max,int digitsLen){		
		int len = tA.length;		
		radixSort(tA,tB,max,digitsLen);	
		
		int digitsTotalLen = tA[0].digits.length;
		
		//caculate rank and sa	
		int[] sa = new int[len];
		sa[0] = tB[0].isuffix;	
		
		int[] rank = new int[len];		
		int r = 1; //rank starts with 1
		rank[tB[0].isuffix] = r;		
		for(int i=1;i<len;i++){
			sa[i] = tB[i].isuffix;	
			
			boolean equalLast = true;
			for(int j=digitsTotalLen-digitsLen;j<digitsTotalLen;j++){
				if(tB[i].digits[j]!=tB[i-1].digits[j]){
					equalLast = false;
					break;
				}
			}
			if(!equalLast){
				r++;
			}
			rank[tB[i].isuffix] = r;	
		}
				 
		Suffix suffix = new Suffix();
		suffix.rank= rank;		
		suffix.sa = sa;
		//judge if we are done
		if(r==len){
			suffix.done = true;
		}else{
			suffix.done = false;
		}
		return suffix;
		
	}
	

	//Precondition: the last char in text must be less than other chars.
	public Suffix solve(String text){
		if(text == null)return null;
		int len = text.length();
		if(len == 0) return null;
		
		int k=1;
		char base = text.charAt(len-1); //the smallest char
		Tuple[] tA = new Tuple[len];
		Tuple[] tB = new Tuple[len]; //placeholder
		for(int i=0;i<len;i++){
			tA[i] = new Tuple(i,new int[]{0,text.charAt(i)-base});
		}
		Suffix suffix = rank(tA,tB,MAX_CHAR-base,1);		
		while(!suffix.done){ //no need to decide if: k<=len
			k<<=1;
			int offset = k>>1;
			for(int i=0,j=i+offset;i<len;i++,j++){		
				tA[i].isuffix = i;
				tA[i].digits=new int[]{suffix.rank[i],
						(j<len)?suffix.rank[i+offset]:0};				
			}
			int max = suffix.rank[suffix.sa[len-1]];
			suffix = rank(tA,tB,max,2);	
		}
		return suffix;
	}
	
	
	public void report(Suffix suffix){
		int[] sa = suffix.sa;
		int[] rank = suffix.rank;
		int len = sa.length;
		
		System.out.println("suffix array:");
		for(int i=0;i<len;i++){
			System.out.format(" %s", sa[i]);			
		}
		System.out.println();
		System.out.println("rank array:");
		for(int i=0;i<len;i++){
			System.out.format(" %s", rank[i]);			
		}		
		System.out.println();
	}
	public static void main(String[] args){
		/*
		//plain counting sort test:
		
		int[] A= {2,5,3,0,2,3,0,3};
		PrefixDoubling pd = new PrefixDoubling();
		int[] B = new int[A.length];
		pd.countingSort(A,B,5);
		for(int i=0;i<B.length;i++)
			System.out.format(" %d", B[i]);
		System.out.println();
		*/		
		
		String text = "GACCCACCACC#";
		PrefixDoubling pd = new PrefixDoubling();
		Suffix suffix = pd.solve(text);
		System.out.format("Text: %s%n",text);
		pd.report(suffix);
		
		System.out.println("********************************");
		text = "mississippi#";
		pd = new PrefixDoubling();
		suffix = pd.solve(text);
		System.out.format("Text: %s%n",text);
		pd.report(suffix);
		
		System.out.println("********************************");
		text = "abcdefghijklmmnopqrstuvwxyz#";
		pd = new PrefixDoubling();
		suffix = pd.solve(text);
		System.out.format("Text: %s%n",text);
		pd.report(suffix);
		
		System.out.println("********************************");
		text = "yabbadabbado#";
		pd = new PrefixDoubling();
		suffix = pd.solve(text);
		System.out.format("Text: %s%n",text);
		pd.report(suffix);
		
		System.out.println("********************************");
		text = "DFDLKJLJldfasdlfjasdfkldjasfldafjdajfdsfjalkdsfaewefsdafdsfa#";
		pd = new PrefixDoubling();
		suffix = pd.solve(text);
		System.out.format("Text: %s%n",text);
		pd.report(suffix);
		
	}
}

测试:

Text: GACCCACCACC#
suffix array:
 11 8 5 1 10 7 4 9 6 3 2 0
rank array:
 12 4 11 10 7 3 9 6 2 8 5 1
********************************
Text: mississippi#
suffix array:
 11 10 7 4 1 0 9 8 6 3 5 2
rank array:
 6 5 12 10 4 11 9 3 8 7 2 1
********************************
Text: abcdefghijklmmnopqrstuvwxyz#
suffix array:
 27 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
rank array:
 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 1
********************************
Text: yabbadabbado#
suffix array:
 12 1 6 4 9 3 8 2 7 5 10 11 0
rank array:
 13 2 8 6 4 10 3 9 7 5 11 12 1
********************************
Text: DFDLKJLJldfasdlfjasdfkldjasfldafjdajfdsfjalkdsfaewefsdafdsfa#
suffix array:
 60 0 2 1 5 7 4 6 3 59 47 54 30 34 41 17 11 25 53 29 33 9 19 23 13 56 44 37 50 48 58 46 10 55 36 39 15 31 20 27 51 40 16 24 32 35 43 21 28 8 22 14 42 52 18 12 57 45 38 26 49
rank array:
 2 4 3 9 7 5 8 6 50 22 33 17 56 25 52 37 43 16 55 23 39 48 51 24 44 18 60 40 49 20 13 38 45 21 14 46 35 28 59 36 42 15 53 47 27 58 32 11 30 61 29 41 54 19 12 34 26 57 31 10 1

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