RMQ+1/1算法 [转]

[转] http://blog.csdn.net/ljsspace/article/details/6659517

　　

RMQ+1/-1问题要求数列中相邻两个元素相差+1或-1。利用这个限定条件可以使该算法复杂度总体上达到<O(n),O(1)>。具体做法是：

1) 设数列A的大小为n，先对数列A分组，每组大小为b=1/2.logn （之所以这样分是为了将预处理复杂度从O(nlogn)降为O(n)），共分为n/b个组；

以下第2到第4步完成RMQ+1/-1问题的预处理阶段(参考以下实现中的preprocess方法)。

2）生成O(sqrt(n))个LU表P[][]和一个block类型数组T[]: 对每个组内部进行预处理(inblock preprocessing)（参考以下实现中的makeLUTable方法），采用的方法是用动态规划加速的蛮立法，组内预处理的结果是生成2^(b- 1)个不同组的lookup table（LU表），这里之所以只需要处理2^(b-1)个不同组(姑且把这些组叫做基本blocks)就是利用了"数列中相邻两个元素相差+1或 -1"这个特点，由于2^(b-1)相当于O(sqrt(n))数量级，因此inblock预处理组的数量是很小的，由于将上三角形状的LU表压缩存储为 一维表L，因此每一个block的LU表占的存储空间可以从b^2降至(b+1)*b/2。需要注意的是，每个block的LU表存储的只是该组的最小元 素在该组内部的下标值，因此在调用queryLUTable方法进行组内查询(inblock query)时还需要加一个偏移量offset，将查询的结果转化为数列A的下标值（参考以下实现中的queryLUTable方法）。

这一步操作的复杂度为O((2^(b-1)) * (b+1)*b/2) = O(n)。

由于在组内查询时需要将n/b个blocks映射到2^(b-1)个基本blocks（多对一映射），因此还需要计算每一个block的类型数组 T[]，计算细节参考以下实现中的computeBlockType方法。计算过程使用Horner规则。这一步操作的复杂度为O(n)。

另外，在第一步的分组中，最后一个block大小可能比b小，这时还需要对该组进行特殊处理。

3）生成每个组的代表元素min值（注：每一个组可以看成一个等价类）的下标数组B[]: 利用inblock预处理的结果找出每一个block的最小元素的下标值（由于区间查询使用的是queryLUTable方法，该下标是数列A的下标值， 而不是block内部的下标值，这正是我们需要得到的下标值），而不需要扫描整个block来找min值。需要注意的是，最后一个block大小小于b的 情况，这时区间查询应该是start...n-1而不是start...end （start和end是该block的起始和结束位置下标值，并且end>n-1）。

这一步生成的结果B[]用于下面一步的组间预处理(outbock preprocessing)中。

4）利用第3步中的数组B生成组间稀疏表M[]: 采用ST(Sparse Table)算法，计算细节参考以下实现中的outBlockPreprocess方法。这一步操作的复杂度为O(n/b*log(n/b))=O(n)。

上面2-4步完成的是预处理，预处理总的复杂度为O(n)。

5）接下来进行查询（参考以下实现中的query方法）。查询分两个cases:

case a)如果查询区间p..q刚好落在一个block中，可以直接调用queryLUTable方法查询得出结果。由于使用lookup table，这一步的复杂度为O(1)。

case b)如果查询区间p..q跨越两个或以上的blocks，则针对第一个和最后一个block使用queryLUTable方法查询得出各自最小值的下标， 这一步的复杂度为O(1)。然后对中间跨越的那些blocks使用组间查询(参考以下实现中的outBlockQuery方法)，这一步的复杂度等于ST 算法中的O(1)。然后比较这三个最小值下标，将他们的最小值对应的下标作为结果返回。

　　

需要注意几点：

1）分组大小为b=1/2.logn，可以将该算法预处理复杂度从普通ST算法的O(nlogn)降到O(n)，即去除对数因子。

2）LU表是基本表，因此该表中存储的下标值是表内部的下标值（即不依赖于任何具体的block），这不同于数列A的下标。因此在查询时需要偏移。

3) 下标数组B[]也叫做概要数组(summary array)，如果在组间查询时将每一个block看成一个等价类，那么B中每一个元素相当于对应block这个等价类中的代表元素。

4）最后一个block可能需要单独处理：将不足一个block的元素扩展为一个整的block。但是在查询阶段，查询区间p...q不会超越数列A的范围0...n-1。

5）总体复杂度是<O(n),O(1)>，因此相比ST算法，该算法是最佳的选择。虽然RMQ+1/-1具有特殊性，但是一般的RMQ 问题也可以在O(n)时间内转化为RMQ+1/-1问题，因此一般的RMQ问题也可以达到<O(n),O(1)>的复杂度。

　　

实现：

　　　　

　　　　

    /** 
     *  
     * An algorithm to solve RMQ+1/-1 problem 
     * time complexity: <O(n),O(1)> 
     *  
     *   
     * Copyright (c) 2011 ljs (http://blog.csdn.net/ljsspace/) 
     * Licensed under GPL (http://www.opensource.org/licenses/gpl-license.php)  
     *  
     * @author ljs 
     * 2011-08-04 
     * 
     */   
    public class MinusOrPlusOne_RMQ {  
        private int blocksize;  
        private int blockcount;  
          
        //block type  
        private int[] T;  
        //block LookUp table  
        private int[][] P;  
          
        //out-of-block sparse table   
        //the first-dimension indices are block IDs  
        private int[][] M;  
          
        //check if A is valid +1/-1 RMQ   
        private void sanityCheck(int[] A) throws Exception{  
            for(int i=1;i<A.length;i++){  
                int diff = A[i] - A[i-1];  
                if(!(diff==1 || diff==-1))  
                    throw new Exception("illegal +1/-1 RMQ problem!");  
            }  
        }  
          
        //output: blocksize, blockcount, T[], P[][], M  
        public void preprocess(int[] A) throws Exception {  
            sanityCheck(A);  
              
            int n = A.length;  
               
            int paddingsize = 0;      
      
            // grouping into logn/2 sized blocks  
            int b = (int) (Math.log(n) / Math.log(2)) >> 1;  
            if(b==0){  
                b = n;  //small problem  
            }  
            int count = (int) Math.ceil(n / (double)b);  
            this.blocksize = b;  
            this.blockcount = count;  
      
            // padding the last block  
            int endblocksize = n - b * (count - 1);  
            if (endblocksize > 0 && endblocksize < b) {  
                paddingsize = b - endblocksize;  
            }  
      
              
              
            //step 1: in-block preprocess  
            //the size of LU table (one-dimensional)  
            int size = b*(b+1)/2;  
              
            int start = 0; // j is the index of A  
            int end = -1;  
            T = new int[count];  
            int blockTypesCount = 1<<(b-1);  
            P = new int[blockTypesCount][size];  
            int[] B = new int[count]; //used in ST algorithm: the min-value array for each block  
            boolean[] pDone =new boolean[blockTypesCount];  
              
            int fullblockscnt = count;  
            if(paddingsize>0){  
                fullblockscnt = count - 1;  
            }         
            for (int i = 0; i < fullblockscnt; i++) {  
                start = end+1;  
                end = (i+1)*b-1;  
                //compute the type of the block   
                int type = computeBlockType(A,start,end);     
                T[i] = type;  
                if(!pDone[type]){  
                    //if LU table is not done yet for this type of block  
                    P[type] = makeLUTable(b,A,start,size);        
                    pDone[type]=true;  
                }             
                B[i] = queryLUTable(b,start,start,end,type);              
            }  
            //the end block  
            if(paddingsize>0){  
                start = end+1;  
                end = count*b-1;  
                  
                //extend the end block  
                int actualsize = n-start;  
                int[] D = new int[b];  
                System.arraycopy(A, start, D, 0, actualsize);   
                for (int k = actualsize; k < b; k++) {  
                    D[k] = D[k - 1] + 1;  
                }             
                  
                int type = computeBlockType(D,0,b-1);     
                T[count-1] = type;  
                if(!pDone[type]){  
                    P[type] = makeLUTable(b,D,0,size);        
                    pDone[type]=true;  
                }     
                //the min-index from start...n-1, not start...end  
                B[count-1] = queryLUTable(b,start,start,n-1,type);        
            }  
              
            //step 2: Sparse table algorithm applied to out-of-blocks         
            this.M = outBlockPreprocess(B,A,count);  
        }  
          
        //return the index  
        public int query(int[] A,int p,int q){  
            if(q<p){  
                //swap    
                int tmp=p;p=q;q=tmp;  
            }  
              
            int start = 0; // j is the index of A  
            int end = -1;  
            int s = 0,t=0; //i..j for ST algorithm  
            int startMin=-1,endMin=-1; //the start block and end block's min index  
            for (int i = 0; i < this.blockcount; i++) {  
                start = end+1;  
                end = (i+1)*this.blocksize-1;  
                if(p>=start && q<=end){  
                    //within a block  
                    return queryLUTable(blocksize,start,p,q,T[i]);  
                }else if(p>=start && p<=end){  
                    startMin = queryLUTable(blocksize,start,p,end,T[i]);  
                    s=i+1;  
                }else if(q<=end && q>=start){  
                    endMin = queryLUTable(blocksize,start,start,q,T[i]);  
                    t=i-1;  
                    break;  
                }  
            }  
            int minIndex = startMin;  
              
            if(s<=t){  
                int outBlocksMin = outBlockQuery(A,s,t);  
                if(A[startMin]>A[outBlocksMin]){  
                    minIndex = outBlocksMin;  
                }  
            }  
            if(A[minIndex]>A[endMin]){  
                minIndex = endMin;  
            }  
            return minIndex;  
        }  
          
        //ST: O(1) for querying  
        //precondition: s<=t  
        private int outBlockQuery(int[] A,int s,int t){  
            int k = (int)(Math.log(t-s+1)/Math.log(2));  
            //the first interval  
            int mina = M[s][k];  
            int minb = M[t-(1<<k)+1][k];  
            if(A[mina]<=A[minb])  
                return mina;  
            else  
                return minb;  
        }  
          
        private int[][] outBlockPreprocess(int[] B,int[] A,int count){  
            //floor value  
            int maxJ=(int)(Math.log(count)/Math.log(2));  
               
            int[][] M = new int[count][maxJ+1];  
              
            //initial condition for dynamic programming: the RMQ for interval length=1  
            for (int i = 0; i < count; i++)  
                  M[i][0] = B[i];  
              
            //dynamic programming: compute values from smaller(j=1) to bigger intervals  
            for (int j = 1; j<=maxJ; j++){  
                for (int i = 0; i + (1 << j) - 1 < count; i++){  
                    int nexti = i + (1 << (j - 1));  
                    if (A[M[i][j - 1]] <= A[M[nexti][j - 1]])  
                        M[i][j] = M[i][j - 1];  
                    else  
                        M[i][j] = M[nexti][j - 1];  
                }  
            }  
            return M;  
        }  
        private int computeBlockType(int[] D,int start,int end){      
            int sum = 0;  
            //use Horner's rule  
            for(int i=start+1;i<=end;i++){  
                int diff = D[i]-D[i-1];  
                if(diff==1){//0 when diff=+1  
                    sum <<= 1;  
                }else{//1 when diff=-1  
                    sum = (sum << 1) + 1;  
                }                 
            }  
            return sum;  
        }  
       
        //return the index    
        //precondition: i<=j  
        private int queryLUTable(int blocksize,int offset,int i,int j,int type){  
            i -= offset; j-=offset;  
            int[] L = P[type];  
            int index = blocksize*i - (i-1)*i/2 + (j-i);  
            return L[index]+offset;  
        }  
        //use naive method to compute the lookup table for a block  
        //the return index is relative to the block itself  
        private int[] makeLUTable(int blocksize, int[] D,int offset,int size) {  
            int[][] Q = new int[blocksize][blocksize];  
            for (int i = 0; i < blocksize; i++)  
                Q[i][i] = i;  
            for (int i = 0; i < blocksize; i++)  
                for (int j = i + 1; j < blocksize; j++)  
                    if (D[Q[i][j - 1]+offset] <= D[j+offset])  
                        Q[i][j] = Q[i][j - 1];  
                    else  
                        Q[i][j] = j;  
            //convert to one-dimension array          
            int[] L = new int[size];  
            int k=0;  
            for(int i=0;i<blocksize;i++){  
                for(int j=i;j<blocksize;j++,k++){  
                    L[k] = Q[i][j];  
                }  
            }  
            return L;  
        }  
          
        private void reportLUTable(int[] A){  
            System.out.format("%n***********************%n");  
            for(int x=0;x<A.length;x++){  
                System.out.format("%d..[%d-%d]",x,x,A.length-1);  
                for(int y=x;y<A.length;y++){  
                    int p = query(A,x,y);                 
                    System.out.format(" %d/%d",A[p],p);  
                }  
                System.out.println();  
            }     
              
        }  
      
        public static void main(String[] args) throws Exception {  
      
            int[] A = new int[]{3,2,3};  
            MinusOrPlusOne_RMQ mpoRMQ = new MinusOrPlusOne_RMQ();  
            mpoRMQ.preprocess(A);  
            mpoRMQ.reportLUTable(A);  
              
              
            A = new int[]{1,0,1,2};  
            mpoRMQ = new MinusOrPlusOne_RMQ();  
            mpoRMQ.preprocess(A);  
            mpoRMQ.reportLUTable(A);  
              
            A = new int[]{1,0,1,0};  
            mpoRMQ = new MinusOrPlusOne_RMQ();  
            mpoRMQ.preprocess(A);  
            mpoRMQ.reportLUTable(A);  
              
              
            A = new int[] { 0, 1, 0, 1, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 2, 1, 0 };  
            mpoRMQ = new MinusOrPlusOne_RMQ();  
            mpoRMQ.preprocess(A);  
              
            System.out.format("%n***********************%n");         
            int i=5,j=11;         
            int min = mpoRMQ.query(A,i,j);  
            System.out.format("RMQ for A[%d..%d]: A[%d]=%d", i,j,min,A[min]);  
              
            System.out.format("%n***********************%n");         
            j=5;  
            i=11;         
            min = mpoRMQ.query(A,i,j);  
            System.out.format("RMQ for A[%d..%d]: A[%d]=%d", i,j,min,A[min]);  
              
            System.out.format("%n***********************%n");         
            i=4;  
            j=16;         
            min = mpoRMQ.query(A,i,j);  
            System.out.format("RMQ for A[%d..%d]: A[%d]=%d", i,j,min,A[min]);  
              
            System.out.format("%n***********************%n");  
            mpoRMQ.reportLUTable(A);  
              
              
              
            A = new int[]{   
                    10,11,12,13,12,11,10,9,10,11,  //0..9  
                    12,13,14,15,14,15,14,15,16,17, //10..19  
                    16,15,14,13,12,11,10,9,8,7,    //20..29  
                    8,9,10,11,12,13,12,11,10,9,    //30..39  
                    10,11,12,13,14,15,14,15,14,15, //40..49  
                    16,17,16,15,14,13,12,11,10,9,  //50..59  
                    8,7,8,9,10,11,12,13,12,11,     //60..69  
                    10,9,10,11,12,13,14,15,14,15,  //70..79  
                    14,15,16,17,16,15,14,13,12,11, //80..89  
                    10,9,8,7,8,9,10,11,12,13,14};     //90..100  
            mpoRMQ = new MinusOrPlusOne_RMQ();  
            mpoRMQ.preprocess(A);  
            mpoRMQ.reportLUTable(A);  
        }  
      
    }  

　　　　　　

　　　　

　　　　　　

测试输出：

　　　　
***********************
0..[0-3] 1/0 1/0 1/0 1/0
1..[1-3] 2/1 2/1 2/1
2..[2-3] 3/2 3/2
3..[3-3] 4/3
　　
***********************
0..[0-3] 1/0 0/1 0/1 0/1
1..[1-3] 0/1 0/1 0/1
2..[2-3] 1/2 1/2
3..[3-3] 2/3
　　
***********************
0..[0-3] 1/0 0/1 0/1 0/1
1..[1-3] 0/1 0/1 0/1
2..[2-3] 1/2 0/3
3..[3-3] 0/3
　　
***********************
RMQ for A[5..11]: A[9]=1
***********************
RMQ for A[11..5]: A[9]=1
***********************
RMQ for A[4..16]: A[16]=0
***********************
　　
***********************
0..[0-16] 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0 0/0
1..[1-16] 1/1 0/2 0/2 0/2 0/2 0/2 0/2 0/2 0/2 0/2 0/2 0/2 0/2 0/2 0/2 0/2
2..[2-16] 0/2 0/2 0/2 0/2 0/2 0/2 0/2 0/2 0/2 0/2 0/2 0/2 0/2 0/2 0/2
3..[3-16] 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 0/16
4..[4-16] 2/4 2/4 2/4 2/4 2/4 1/9 1/9 1/9 1/9 1/9 1/9 1/9 0/16
5..[5-16] 3/5 2/6 2/6 2/6 1/9 1/9 1/9 1/9 1/9 1/9 1/9 0/16
6..[6-16] 2/6 2/6 2/6 1/9 1/9 1/9 1/9 1/9 1/9 1/9 0/16
7..[7-16] 3/7 2/8 1/9 1/9 1/9 1/9 1/9 1/9 1/9 0/16
8..[8-16] 2/8 1/9 1/9 1/9 1/9 1/9 1/9 1/9 0/16
9..[9-16] 1/9 1/9 1/9 1/9 1/9 1/9 1/9 0/16
10..[10-16] 2/10 2/10 2/10 2/10 2/10 1/15 0/16
11..[11-16] 3/11 2/12 2/12 2/12 1/15 0/16
12..[12-16] 2/12 2/12 2/12 1/15 0/16
13..[13-16] 3/13 2/14 1/15 0/16
14..[14-16] 2/14 1/15 0/16
15..[15-16] 1/15 0/16
16..[16-16] 0/16
　　
***********************
0..[0-100] 10/0 10/0 10/0 10/0 10/0 10/0 10/0 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 8/28 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29
1..[1-100] 11/1 11/1 11/1 11/1 11/1 10/6 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 8/28 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29
2..[2-100] 12/2 12/2 12/2 11/5 10/6 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 8/28 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29
3..[3-100] 13/3 12/4 11/5 10/6 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 8/28 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29
4..[4-100] 12/4 11/5 10/6 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 8/28 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29
5..[5-100] 11/5 10/6 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 8/28 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29
6..[6-100] 10/6 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 8/28 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29
7..[7-100] 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 9/7 8/28 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29
8..[8-100] 10/8 10/8 10/8 10/8 10/8 10/8 10/8 10/8 10/8 10/8 10/8 10/8 10/8 10/8 10/8 10/8 10/8 10/8 10/8 9/27 8/28 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29
9..[9-100] 11/9 11/9 11/9 11/9 11/9 11/9 11/9 11/9 11/9 11/9 11/9 11/9 11/9 11/9 11/9 11/9 11/9 10/26 9/27 8/28 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29
10..[10-100] 12/10 12/10 12/10 12/10 12/10 12/10 12/10 12/10 12/10 12/10 12/10 12/10 12/10 12/10 12/10 11/25 10/26 9/27 8/28 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29
11..[11-100] 13/11 13/11 13/11 13/11 13/11 13/11 13/11 13/11 13/11 13/11 13/11 13/11 13/11 12/24 11/25 10/26 9/27 8/28 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29
12..[12-100] 14/12 14/12 14/12 14/12 14/12 14/12 14/12 14/12 14/12 14/12 14/12 13/23 12/24 11/25 10/26 9/27 8/28 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29
13..[13-100] 15/13 14/14 14/14 14/14 14/14 14/14 14/14 14/14 14/14 14/14 13/23 12/24 11/25 10/26 9/27 8/28 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29
14..[14-100] 14/14 14/14 14/14 14/14 14/14 14/14 14/14 14/14 14/14 13/23 12/24 11/25 10/26 9/27 8/28 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29
15..[15-100] 15/15 14/16 14/16 14/16 14/16 14/16 14/16 14/16 13/23 12/24 11/25 10/26 9/27 8/28 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29
16..[16-100] 14/16 14/16 14/16 14/16 14/16 14/16 14/16 13/23 12/24 11/25 10/26 9/27 8/28 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29
17..[17-100] 15/17 15/17 15/17 15/17 15/17 14/22 13/23 12/24 11/25 10/26 9/27 8/28 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29
18..[18-100] 16/18 16/18 16/18 15/21 14/22 13/23 12/24 11/25 10/26 9/27 8/28 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29
19..[19-100] 17/19 16/20 15/21 14/22 13/23 12/24 11/25 10/26 9/27 8/28 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29
20..[20-100] 16/20 15/21 14/22 13/23 12/24 11/25 10/26 9/27 8/28 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29
21..[21-100] 15/21 14/22 13/23 12/24 11/25 10/26 9/27 8/28 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29
22..[22-100] 14/22 13/23 12/24 11/25 10/26 9/27 8/28 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29
23..[23-100] 13/23 12/24 11/25 10/26 9/27 8/28 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29
24..[24-100] 12/24 11/25 10/26 9/27 8/28 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29
25..[25-100] 11/25 10/26 9/27 8/28 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29
26..[26-100] 10/26 9/27 8/28 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29
27..[27-100] 9/27 8/28 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29
28..[28-100] 8/28 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29
29..[29-100] 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29 7/29
30..[30-100] 8/30 8/30 8/30 8/30 8/30 8/30 8/30 8/30 8/30 8/30 8/30 8/30 8/30 8/30 8/30 8/30 8/30 8/30 8/30 8/30 8/30 8/30 8/30 8/30 8/30 8/30 8/30 8/30 8/30 8/30 8/30 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61 7/61
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