poj_1442 Treap

Treap是一种动态平衡二叉树结构，具有期望的O(log2n)的复杂度。适用于动态区间数据的查询、更改、维护等操作。

题目大意

    一组数从前向后插入队列中，插入的过程中会有查询，查询当前队列中的第k小的数。

题目分析

    对于数据的查询，可以考虑使用treap这种平衡二叉树来实现。而且treap这种动态平衡树结构，可以很方便的实现第k大的查询。 
    需要注意的是，每个节点保存与该节点相同元素的个数count对查询第k个数据时候的影响，见代码。

实现(c++)

#define _CRT_SECURE_NO_WARNINGS
#include<stdio.h>
#include<algorithm>
#define MAX_NUM 30010
struct TreapNode{
	int key;
	int priority;
	int size;
	int count;
	TreapNode* child[2];
	TreapNode(int val){
		key = val;
		priority = rand();
		child[0] = child[1] = NULL;
		count = size = 1;
	}
	void Update(){
		size = count;
		if (child[0]){
			size += child[0]->size;
		}
		if (child[1]){
			size += child[1]->size;
		}
	}
};

struct Treap{
	TreapNode* root;
	Treap() :root(NULL){};
	void Rotate(TreapNode*& node, int dir){
		TreapNode* ch = node->child[dir];
		node->child[dir] = ch->child[!dir];
		ch->child[!dir] = node;

		node->Update();
		node = ch; //reference
	}
	void Insert(TreapNode*& node, int k){
		if (!node){
			node = new TreapNode(k);
		}
		else if (node->key == k){
			node->count++;
		}
		else{
			int dir = node->key < k;
			Insert(node->child[dir], k);
			if (node->priority < node->child[dir]->priority){
				Rotate(node, dir);
			}
		}
		node->Update();
	}
	int GetKth(TreapNode* root, int k){
		TreapNode* node = root;
		while (node){
			if (!node->child[0]){
				if (k <= node->count){
					return node->key;
				}
				else{
					k -= (node->count);
					node = node->child[1];
				}
			}
			else{
				if (node->child[0]->size < k && node->child[0]->size + node->count >= k){
					return node->key;
				}
				else if (node->child[0]->size >= k){
					node = node->child[0];
				}
				else{
					k -= (node->child[0]->size + node->count);
					node = node->child[1];
					
				}
			}
		}
		return 0;
	}
};
int gNumber[MAX_NUM];
Treap gTreap;
int main(){
	int number_count, query_count;
	scanf("%d%d", &number_count, &query_count);
	for (int i = 0; i < number_count; i++){
		scanf("%d", &gNumber[i]);
	}

	int query_old = 0;
	int query_new;
	for (int i = 1; i <= query_count; i++){
		scanf("%d", &query_new);
		for (int k = query_old; k < query_new; k++){
			gTreap.Insert(gTreap.root, gNumber[k]);
		}
		query_old = query_new;
		int result = gTreap.GetKth(gTreap.root, i);
		printf("%d
", result);
	}
	return 0;
}