普通RMQ問題轉化爲LCA問題的算法

該基本思想是通過對問題的轉化，最終得到<O(n),O(1)>時間複雜度。

該算法分以下兩大步驟：

1）將RMQ問題轉化爲LCA問題：先構建輸入數列A的笛卡爾樹，構建笛卡爾樹的複雜度爲O(n)。

2）將LCA問題轉化爲RMQ+1/-1問題：通過對笛卡爾樹的DFS遍歷得到歐拉路徑(Euler Tour)，建立三個數組E,L和R。其中E和L大小是2*n-1，E中元素表示笛卡爾樹中每個結點的label值（實際上就是數列A的下標索引值），L中元素對應歐拉路徑上每一個被訪問結點的深度（即從root到該結點的深度，root本身的深度爲0）; R是笛卡爾樹中的每一個結點第一次被訪問時在E中的位置，即E[R[u]] = u，但是E[i] != u （u是笛卡爾樹中的結點label值，0<=i<R[u]）。

顯然，得到歐拉路徑的複雜度爲O(n)。

如果要查詢u和v的LCA，實際上等價於對L數列做RMQ查詢，由於對L數列做RMQ查詢是一個RMQ+1/-1查詢，因此可以使用RMQ+1/-1算法在<O(n),O(1)>時間內完成，具體做法是：先使用RMQ+1/-1算法對L預處理，然後在查詢時利用E,L,R三個數組和預處理好的RMQ+1/-1表得到結果。假設要查詢A[i..j]區間的最小值，首先通過R數組找到i和j在E或L中第一次出現的位置R[i]和R[j]; 然後對L深度數列求最小值（即LCA問題），這是一個RMQ+1/-1查詢，查詢結果返回L數列的下標索引值min，也是E數列的下標索引值; 最後利用返回的最小深度結點的下標索引值min，得到笛卡爾樹中結點的label值，即RMQ[A[i..j]] = E[min]。

該算法的缺點是E和R需要2*n的數組空間，總共4*n，另外構建笛卡爾樹也需要較大的空間開銷。不過在得到E,L,R數組之後，笛卡爾樹可以被銷燬（在java中被GC）。

實現：（以下代碼依賴於MinusOrPlusOne_RMQ類，執行下面程序之前需要將MinusOrPlusOne_RMQ類放到同一個package）

import java.util.Stack;
/**
 * 
 * An algorithm to solve general RMQ problem
 * by converting it into a 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
 * 2011-08-06
 *
 */ 
public class GeneralRMQ_ViaTree {
	static class Node {
		//index
		int index;				
		Node left;
		Node right;
		public Node(int index) {
			this.index = index;
		}
		public String toString(){
			return String.valueOf(index);
		}
	}
	
	//input array
	private int[] A;
	//euler tour
	private int[] E;
	//levels during euler tour
	private int[] L;
	//each node's representation position (first occurence) in E
	private int[] R;
	
	private MinusOrPlusOne_RMQ mpoRMQ; //collaborator for RMQ+1/-1 problem

	public void preprocess(int[] A){
		//prepare
		int n = A.length;
		this.A = A;
		E = new int[2*n-1];
		L = new int[2*n-1];
		R = new int[n];
		
		//build a cartesian tree to convert general RMQ to LCA
		//and then make an euler tour to create arrays E,L,R for later RMQ+1/-1 query
		Node root = this.buildCartesianTree();
		this.eulertour(root); 
		//once E,L,R are created, cartesian tree can be garbage collected!
		
		//create array R by iterating E backward
		for(int i=E.length-1;i>=0;i--){
			R[E[i]] = i;
		}		
		this.reportELR();
		
		//preprocess RMQ+1/-1 for L
		mpoRMQ = new MinusOrPlusOne_RMQ();
		try {
			mpoRMQ.preprocess(L);
		} catch (Exception e) {
			//never get here!
		}
	}

	
	public int query(int[] A,int i,int j){
		int p = R[i];
		int q = R[j];
		//delegate to RMQ+1/-1 query
		int min = mpoRMQ.query(L, p, q);
		return E[min];
	}
  
	//O(n) algorithm for building cartesian tree
	//result: the root of a cartesian tree 
	private Node buildCartesianTree(){
		Stack<Node> rightBranch =  new Stack<Node>();
		Node root = new Node(0);		
		rightBranch.push(root);
		for(int i=1;i<A.length;i++){
			Node node = new Node(i);
			int k = A[i];
			Node rightmost= rightBranch.peek();
			//use >= for building canonical cartesian tree 
			if(k>=A[rightmost.index]){ 
				rightmost.right = node;
				rightBranch.push(node);
			}else{
				rightBranch.pop();
				while(!rightBranch.isEmpty()){
					rightmost = rightBranch.peek();
					if(k>=A[rightmost.index]){						
						Node tmp = rightmost.right;
						rightmost.right = node;
						node.left = tmp;
						rightBranch.push(node);
						break;
					}else{
						rightBranch.pop();
					}
				}
				if(rightBranch.isEmpty()){					
					node.left = root;
					root = node;
					rightBranch.push(node);
				}
			}
		}
		return root;
	}
	//make an euler tour to create arrays: E[0..2n-2],L[0..2n-1],R[0..n-1]
	private void eulertour(Node root) {
		eulertour(root,0);
	}
	
	private void eulertour(Node node,int level) {
		visit(node,level);
		if (node.left != null) {
			eulertour(node.left, level + 1);
			visit(node, level);
		}
		if (node.right != null) {
			eulertour(node.right, level + 1);
			visit(node, level);
		}
	}
	private int i=0; //the position in E
	private void visit(Node node, int level) {
		this.E[i] = node.index;
		this.L[i] = level;		
		i++;		
	}
	
	private void reportELR(){
		System.out.format("E:");
		for(int i=0;i<E.length;i++){
			System.out.format(" %d",E[i]);
		}
		System.out.println();
		System.out.format("L:");
		for(int i=0;i<L.length;i++){
			System.out.format(" %d",L[i]);
		}
		System.out.println();
		System.out.format("R:");
		for(int i=0;i<R.length;i++){
			System.out.format(" %d",R[i]);
		}
	}
	
	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) {
		int[] A=new int[]{1,3,4,5,6,2};
		GeneralRMQ_ViaTree gRMQTree = new GeneralRMQ_ViaTree();
		gRMQTree.preprocess(A);
		gRMQTree.reportLUTable(A);
		
		System.out.format("%n***********************%n");
		A=new int[]{2,4,3,1,6,7,8,9,1,7};
		gRMQTree = new GeneralRMQ_ViaTree();
		gRMQTree.preprocess(A);
		
		int i=0,j=6;
		gRMQTree.query(A, i, j);
		int min = gRMQTree.query(A,i,j);
		System.out.format("%n%nRMQ for A[%d..%d]: A[%d]=%d", i,j,min,A[min]);
		
		
		gRMQTree.reportLUTable(A);
		
		System.out.format("%n***********************%n");
		A=new int[]{10,15,34,20,7,5,18,68,29,40, //0..9
					24,3,45,26,7,23,43,12,68,34,  //10..19
					26,34,33,12,80,57,24,42,77,27, //20..29
					56,33,23,32,54,13,79,65,19,33,  //30..39
					15,24,43,73,55,13,63,8,23,17};  //40..49
		gRMQTree = new GeneralRMQ_ViaTree();
		gRMQTree.preprocess(A);
		gRMQTree.reportLUTable(A);
	}

}

測試輸出：

E: 0 5 1 2 3 4 3 2 1 5 0
L: 0 1 2 3 4 5 4 3 2 1 0
R: 0 2 3 4 5 1
***********************
0..[0-5] 1/0 1/0 1/0 1/0 1/0 1/0
1..[1-5] 3/1 3/1 3/1 3/1 2/5
2..[2-5] 4/2 4/2 4/2 2/5
3..[3-5] 5/3 5/3 2/5
4..[4-5] 6/4 2/5
5..[5-5] 2/5

***********************
E: 3 0 2 1 2 0 3 8 4 5 6 7 6 5 4 8 9 8 3
L: 0 1 2 3 2 1 0 1 2 3 4 5 4 3 2 1 2 1 0
R: 1 3 2 0 8 9 10 11 7 16

RMQ for A[0..6]: A[3]=1
***********************
0..[0-9] 2/0 2/0 2/0 1/3 1/3 1/3 1/3 1/3 1/3 1/3
1..[1-9] 4/1 3/2 1/3 1/3 1/3 1/3 1/3 1/3 1/3
2..[2-9] 3/2 1/3 1/3 1/3 1/3 1/3 1/3 1/3
3..[3-9] 1/3 1/3 1/3 1/3 1/3 1/3 1/3
4..[4-9] 6/4 6/4 6/4 6/4 1/8 1/8
5..[5-9] 7/5 7/5 7/5 1/8 1/8
6..[6-9] 8/6 8/6 1/8 1/8
7..[7-9] 9/7 1/8 1/8
8..[8-9] 1/8 1/8
9..[9-9] 7/9

***********************
E: 11 5 4 0 1 3 2 3 1 0 4 5 6 10 8 7 8 9 8 10 6 5 11 14 13 12 13 14 47 17 15 16 15 17 23 20 19 18 19 20 22 21 22 20 23 35 32 26 25 24 25 26 29 27 28 27 29 31 30 31 29 26 32 33 34 33 32 35 45 40 38 37 36 37 38 39 38 40 41 42 44 43 44 42 41 40 45 46 45 35 23 17 47 49 48 49 47 14 11
L: 0 1 2 3 4 5 6 5 4 3 2 1 2 3 4 5 4 5 4 3 2 1 0 1 2 3 2 1 2 3 4 5 4 3 4 5 6 7 6 5 6 7 6 5 4 5 6 7 8 9 8 7 8 9 10 9 8 9 10 9 8 7 6 7 8 7 6 5 6 7 8 9 10 9 8 9 8 7 8 9 10 11 10 9 8 7 6 7 6 5 4 3 2 3 4 3 2 1 0
R: 3 4 6 5 2 1 12 15 14 17 13 0 25 24 23 30 31 29 37 36 35 41 40 34 49 48 47 53 54 52 58 57 46 63 64 45 72 71 70 75 69 78 79 81 80 68 87 28 94 93
***********************
0..[0-49] 10/0 10/0 10/0 10/0 7/4 5/5 5/5 5/5 5/5 5/5 5/5 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11
1..[1-49] 15/1 15/1 15/1 7/4 5/5 5/5 5/5 5/5 5/5 5/5 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11
2..[2-49] 34/2 20/3 7/4 5/5 5/5 5/5 5/5 5/5 5/5 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11
3..[3-49] 20/3 7/4 5/5 5/5 5/5 5/5 5/5 5/5 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11
4..[4-49] 7/4 5/5 5/5 5/5 5/5 5/5 5/5 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11
5..[5-49] 5/5 5/5 5/5 5/5 5/5 5/5 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11
6..[6-49] 18/6 18/6 18/6 18/6 18/6 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11
7..[7-49] 68/7 29/8 29/8 24/10 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11
8..[8-49] 29/8 29/8 24/10 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11
9..[9-49] 40/9 24/10 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11
10..[10-49] 24/10 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11
11..[11-49] 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11 3/11
12..[12-49] 45/12 26/13 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14
13..[13-49] 26/13 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14
14..[14-49] 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14 7/14
15..[15-49] 23/15 23/15 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 8/47 8/47 8/47
16..[16-49] 43/16 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 8/47 8/47 8/47
17..[17-49] 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 12/17 8/47 8/47 8/47
18..[18-49] 68/18 34/19 26/20 26/20 26/20 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 8/47 8/47 8/47
19..[19-49] 34/19 26/20 26/20 26/20 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 8/47 8/47 8/47
20..[20-49] 26/20 26/20 26/20 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 8/47 8/47 8/47
21..[21-49] 34/21 33/22 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 8/47 8/47 8/47
22..[22-49] 33/22 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 8/47 8/47 8/47
23..[23-49] 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 12/23 8/47 8/47 8/47
24..[24-49] 80/24 57/25 24/26 24/26 24/26 24/26 24/26 24/26 23/32 23/32 23/32 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 8/47 8/47 8/47
25..[25-49] 57/25 24/26 24/26 24/26 24/26 24/26 24/26 23/32 23/32 23/32 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 8/47 8/47 8/47
26..[26-49] 24/26 24/26 24/26 24/26 24/26 24/26 23/32 23/32 23/32 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 8/47 8/47 8/47
27..[27-49] 42/27 42/27 27/29 27/29 27/29 23/32 23/32 23/32 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 8/47 8/47 8/47
28..[28-49] 77/28 27/29 27/29 27/29 23/32 23/32 23/32 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 8/47 8/47 8/47
29..[29-49] 27/29 27/29 27/29 23/32 23/32 23/32 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 8/47 8/47 8/47
30..[30-49] 56/30 33/31 23/32 23/32 23/32 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 8/47 8/47 8/47
31..[31-49] 33/31 23/32 23/32 23/32 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 8/47 8/47 8/47
32..[32-49] 23/32 23/32 23/32 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 8/47 8/47 8/47
33..[33-49] 32/33 32/33 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 8/47 8/47 8/47
34..[34-49] 54/34 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 8/47 8/47 8/47
35..[35-49] 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 13/35 8/47 8/47 8/47
36..[36-49] 79/36 65/37 19/38 19/38 15/40 15/40 15/40 15/40 15/40 13/45 13/45 8/47 8/47 8/47
37..[37-49] 65/37 19/38 19/38 15/40 15/40 15/40 15/40 15/40 13/45 13/45 8/47 8/47 8/47
38..[38-49] 19/38 19/38 15/40 15/40 15/40 15/40 15/40 13/45 13/45 8/47 8/47 8/47
39..[39-49] 33/39 15/40 15/40 15/40 15/40 15/40 13/45 13/45 8/47 8/47 8/47
40..[40-49] 15/40 15/40 15/40 15/40 15/40 13/45 13/45 8/47 8/47 8/47
41..[41-49] 24/41 24/41 24/41 24/41 13/45 13/45 8/47 8/47 8/47
42..[42-49] 43/42 43/42 43/42 13/45 13/45 8/47 8/47 8/47
43..[43-49] 73/43 55/44 13/45 13/45 8/47 8/47 8/47
44..[44-49] 55/44 13/45 13/45 8/47 8/47 8/47
45..[45-49] 13/45 13/45 8/47 8/47 8/47
46..[46-49] 63/46 8/47 8/47 8/47
47..[47-49] 8/47 8/47 8/47
48..[48-49] 23/48 17/49
49..[49-49] 17/49