RMQ+1/-1算法

RMQ+1/-1問題要求數列中相鄰兩個元素相差+1或-1。利用這個限定條件可以使該算法複雜度總體上達到<O(n),O(1)>。具體做法是：

1) 設數列A的大小爲n，先對數列A分組，每組大小爲b=1/2.logn （之所以這樣分是爲了將預處理複雜度從O(nlogn)降爲O(n)），共分爲n/b個組；

以下第2到第4步完成RMQ+1/-1問題的預處理階段(參考以下實現中的preprocess方法)。

2）生成O(sqrt(n))個LU表P[][]和一個block類型數組T[]: 對每個組內部進行預處理(inblock preprocessing)（參考以下實現中的makeLUTable方法），採用的方法是用動態規劃加速的蠻立法，組內預處理的結果是生成2^(b-1)個不同組的lookup table（LU表），這裏之所以只需要處理2^(b-1)個不同組(姑且把這些組叫做基本blocks)就是利用了"數列中相鄰兩個元素相差+1或-1"這個特點，由於2^(b-1)相當於O(sqrt(n))數量級，因此inblock預處理組的數量是很小的，由於將上三角形狀的LU表壓縮存儲爲一維表L，因此每一個block的LU表佔的存儲空間可以從b^2降至(b+1)*b/2。需要注意的是，每個block的LU表存儲的只是該組的最小元素在該組內部的下標值，因此在調用queryLUTable方法進行組內查詢(inblock query)時還需要加一個偏移量offset，將查詢的結果轉化爲數列A的下標值（參考以下實現中的queryLUTable方法）。

這一步操作的複雜度爲O((2^(b-1)) * (b+1)*b/2) = O(n)。

由於在組內查詢時需要將n/b個blocks映射到2^(b-1)個基本blocks（多對一映射），因此還需要計算每一個block的類型數組T[]，計算細節參考以下實現中的computeBlockType方法。計算過程使用Horner規則。這一步操作的複雜度爲O(n)。

另外，在第一步的分組中，最後一個block大小可能比b小，這時還需要對該組進行特殊處理。

3）生成每個組的代表元素min值（注：每一個組可以看成一個等價類）的下標數組B[]: 利用inblock預處理的結果找出每一個block的最小元素的下標值（由於區間查詢使用的是queryLUTable方法，該下標是數列A的下標值，而不是block內部的下標值，這正是我們需要得到的下標值），而不需要掃描整個block來找min值。需要注意的是，最後一個block大小小於b的情況，這時區間查詢應該是start...n-1而不是start...end （start和end是該block的起始和結束位置下標值，並且end>n-1）。

這一步生成的結果B[]用於下面一步的組間預處理(outbock preprocessing)中。

4）利用第3步中的數組B生成組間稀疏表M[]: 採用ST(Sparse Table)算法，計算細節參考以下實現中的outBlockPreprocess方法。這一步操作的複雜度爲O(n/b*log(n/b))=O(n)。

上面2-4步完成的是預處理，預處理總的複雜度爲O(n)。

5）接下來進行查詢（參考以下實現中的query方法）。查詢分兩個cases:

case a)如果查詢區間p..q剛好落在一個block中，可以直接調用queryLUTable方法查詢得出結果。由於使用lookup table，這一步的複雜度爲O(1)。

case b)如果查詢區間p..q跨越兩個或以上的blocks，則針對第一個和最後一個block使用queryLUTable方法查詢得出各自最小值的下標，這一步的複雜度爲O(1)。然後對中間跨越的那些blocks使用組間查詢(參考以下實現中的outBlockQuery方法)，這一步的複雜度等於ST算法中的O(1)。然後比較這三個最小值下標，將他們的最小值對應的下標作爲結果返回。

需要注意幾點：

1）分組大小爲b=1/2.logn，可以將該算法預處理複雜度從普通ST算法的O(nlogn)降到O(n)，即去除對數因子。

2）LU表是基本表，因此該表中存儲的下標值是表內部的下標值（即不依賴於任何具體的block），這不同於數列A的下標。因此在查詢時需要偏移。

3) 下標數組B[]也叫做概要數組(summary array)，如果在組間查詢時將每一個block看成一個等價類，那麼B中每一個元素相當於對應block這個等價類中的代表元素。

4）最後一個block可能需要單獨處理：將不足一個block的元素擴展爲一個整的block。但是在查詢階段，查詢區間p...q不會超越數列A的範圍0...n-1。

5）總體複雜度是<O(n),O(1)>，因此相比ST算法，該算法是最佳的選擇。雖然RMQ+1/-1具有特殊性，但是一般的RMQ問題也可以在O(n)時間內轉化爲RMQ+1/-1問題，因此一般的RMQ問題也可以達到<O(n),O(1)>的複雜度。

實現：

/**
 * 
 * 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);
	}

}

測試輸出：

***********************
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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