构造后缀数组的DC3算法实现
DC3算法(Difference Cover mod 3)是J. Kärkkäinen和P. Sanders在2003年发表的论文 "Simple Linear Work Suffix Array Construction"中描述的线性时间内构造后缀数组的算法。相对Prefix Doubling(前缀倍增)算法而言,虽然它的渐进时间复杂度比较小,但是常数项比较大。DC3算法的思想类似于找中位数的median of medians算法(http://en.wikipedia.org/wiki/Selection_algorithm),它采用分治思想: 先用递归方式对起始下标等于1(mod 3)和2(mod 3)的后缀排序,从而将原始的后缀集合大小缩小为2/3,设这些后缀排好序的结果为S12,然后在S12的基础上对起始下标等于0(mod 3)的后缀排序(这一步只需作两位数的基数排序,一位为0(mod 3)的起始下标,另外一位为S12的rank值),设这一步得到的排好序的后缀数组为S0,最后将S0和S12归并(类似于归并排序算法)。归并过程通过 Difference Cover思想,也是在S12已知的基础上分两个cases得出相邻两个后缀的先后顺序。
实现:
/** * * Build Suffix Array using DC3/KS Algorithm * * * Copyright (c) 2011 ljs (http://blog.csdn.net/ljsspace/) * Licensed under GPL (http://www.opensource.org/licenses/gpl-license.php) * * @author ljs * 2011-07-18 * */ public class DC3 { 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; public Suffix(int[] sa,int[] rank){ this.sa = sa; this.rank = rank; } } //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(); } } //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+2]; //add 2 for sentinel rank[len]=1;rank[len+1] = 1; 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(sa,rank); //judge if we are done if(r==len){ suffix.done = true; }else{ suffix.done = false; } return suffix; } private int[] orderSuffixes(Tuple[] tA,Tuple[] tB,int max,int digitsLen){ int len = tA.length; radixSort(tA,tB,max,digitsLen); //caculate rank and sa int[] sa = new int[len]; for(int i=0;i<len;i++){ sa[i] = tB[i].isuffix; } return sa; } //rank needs sentinel: len+2 public Suffix reduce(int[] rank,int max){ int len = rank.length - 2; int n1 = (len+1)/3; int n2 = len/3; Tuple[] tA = new Tuple[n1+n2]; Tuple[] tB = new Tuple[n1+n2]; for(int i=0,j=1;i<n1;i++,j+=3){ int r1 = rank[j]; int r2 = rank[j+1]; int r3 = rank[j+2]; tA[i] = new Tuple(i,new int[]{r1,r2,r3}); } for(int i=n1,j=2;i<n1+n2;i++,j+=3){ int r1 = rank[j]; int r2 = rank[j+1]; int r3 = rank[j+2]; tA[i] = new Tuple(i,new int[]{r1,r2,r3}); } return rank(tA,tB,max,3); } public int[] skew(int[] rank,int max){ int len = rank.length - 2; //step 1: caculate sa12 Suffix suffixT12 = reduce(rank,max); int[] sa12 = null; if(!suffixT12.done){ int[] rankT12 = suffixT12.rank; int maxT12 = rankT12[suffixT12.sa[suffixT12.sa.length-1]]; sa12 = skew(rankT12,maxT12); // debug for string: GACCCACCACC# //s12 = new Suffix(); //s12.rank = new int[]{3,6,5,4,7,2,1,1,1}; //s12.sa = new int[]{7,6,5,0,3,2,1,4}; //s12.done =true; }else{ sa12 = suffixT12.sa; } //index conversion for sa12 int n1 = (len+1)/3; for(int j=0;j<sa12.length;j++){ if(sa12[j]<n1){ sa12[j] = 1 + 3*sa12[j]; }else{ sa12[j] = 2 + 3*(sa12[j]-n1); } } //recaculate rank for sa12 int[] rank12 = new int[len+2]; rank12[len] = 1;rank12[len+1] = 1; for(int k=0;k<sa12.length;k++){ rank12[sa12[k]] = k+1; } //step 2: caculate sa0 int n0=(len+2)/3; Tuple[] tA = new Tuple[n0]; Tuple[] tB = new Tuple[n0]; for(int i=0,j=0;i<n0;i++,j+=3){ int r1 = rank[j]; int r2 = rank12[j+1]; tA[i] = new Tuple(i,new int[]{r1,r2}); } int max12 = rank12[sa12[sa12.length-1]]; int[] sa0 = orderSuffixes(tA,tB,max<max12?max12:max,2); //index conversion for sa0 for(int j=0;j<n0;j++){ sa0[j] = 3*sa0[j]; } //step 3: merge sa12 and sa0 int[] sa = new int[len]; int i=0,j=0; int k=0; while(i<sa12.length && j<sa0.length){ int p = sa12[i]; int q = sa0[j]; if(p%3==1){ //case 1 if(rank[p]<rank[q]){ sa[k++] = p;i++; }else if(rank[p]>rank[q]){ sa[k++] = q;j++; }else{ if(rank12[p+1]<rank12[q+1]){ sa[k++] = p;i++; }else{ sa[k++] = q;j++; } } }else{ //case 2 if(rank[p]<rank[q]){ sa[k++] = p;i++; }else if(rank[p]>rank[q]){ sa[k++] = q;j++; }else{ if(rank[p+1]<rank[q+1]){ sa[k++] = p;i++; }else if(rank[p+1]>rank[q+1]){ sa[k++] = q;j++; }else{ if(rank12[p+2]<rank12[q+2]){ sa[k++] = p;i++; }else{ sa[k++] = q;j++; } } } } } for(int m=i;m<sa12.length;m++){ sa[k++] = sa12[m]; } for(int m=j;m<sa0.length;m++){ sa[k++] = sa0[m]; } return sa; } //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; 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); int max = suffix.rank[suffix.sa[len-1]]; int[] sa = skew(suffix.rank,max); //caculate rank for result suffix array int[] r = new int[len]; for(int k=0;k<sa.length;k++){ r[sa[k]] = k+1; } return new Suffix(sa,r); } 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) { String text = "GACCCACCACC#"; DC3 dc3 = new DC3(); Suffix suffix = dc3.solve(text); System.out.format("Text: %s%n",text); dc3.report(suffix); text = "mississippi#"; dc3 = new DC3(); suffix = dc3.solve(text); System.out.format("Text: %s%n",text); dc3.report(suffix); text = "abcdefghijklmmnopqrstuvwxyz#"; dc3 = new DC3(); suffix = dc3.solve(text); System.out.format("Text: %s%n",text); dc3.report(suffix); text = "yabbadabbado#"; dc3 = new DC3(); suffix = dc3.solve(text); System.out.format("Text: %s%n",text); dc3.report(suffix); text = "DFDLKJLJldfasdlfjasdfkldjasfldafjdajfdsfjalkdsfaewefsdafdsfa#"; dc3 = new DC3(); suffix = dc3.solve(text); System.out.format("Text: %s%n",text); dc3.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