大-小顶混合堆的实现与应用(a min-max heap)

一般情况下我们使用的堆都是大顶堆或者小顶堆，其能实现在常数时间内获得数组的最大值或者最小值，同时满足在对数时间内删除和插入元素。但是如果要同时实现即能在常数时间内获得最大值和最小值，又能在对数时间内删除和插入元素，通常情况下的堆就不能满足上述要求了。为此介绍一种新的数据结构min-max heap

min-max heap 是一颗完全二叉树，但是二叉树的奇数层存的是max元素，偶数层存的是min元素，也即在以偶数层的某节点为根节的子树中，根节点最大，若在以奇数层为根节点的子树中，根节点最小。根据上述的思想构造出相应的min-max heap。

算法实现：

#include "min_max_heap.h"#include <iostream>#include<vector>using namespace std;bool min_max_heap::is_min_level(int index){int res = 0;index = index+1;while(index>1){res = res + 1;index = index>>1;}if(res % 2 == 0)return true;return false;}bool min_max_heap::has_child(int index) const{int size = data.size();if(2*index<size-1)return true;return false;}int min_max_heap::min_child(int index) const{int size = data.size();int res=index*2+1;if(res<size-1 && data[res]>data[res+1])res++;return res;}int min_max_heap::max_child(int index) const{int size = data.size();int res = 2*index +1;if(res<size-1 && data[res]<data[res+1])res++;return res;}bool min_max_heap::has_grandchild(int index) const{int size = data.size();int k=2*index+1;if(2*k<size-1)return true;return false;}int min_max_heap::min_grandchild(int index) const{int size = data.size();int res = 2*index+1;int left_res = 2*res+1;if(left_res < size-1 && data[left_res]>data[left_res + 1])left_res++;int right_res=-1;if(has_child(res+1))right_res = 2*(res+1)+1;if(right_res == -1)res = left_res;else{if(right_res < size-1 && data[right_res]>data[right_res + 1])right_res++;if(data[left_res] > data[right_res])res = right_res;elseres = left_res;}return res;}int min_max_heap::max_grandchild(int index) const{int size = data.size();int res = 2*index+1;int left_res = 2*res+1;if(left_res<size-1 && data[left_res] < data[left_res+1])left_res++;int right_res = -1;if(has_child(res+1))right_res = 2*(res+1)+1;if(right_res == -1)res = left_res;else{if(right_res<size-1 && data[right_res]<data[right_res+1])right_res++;if(data[left_res] > data[right_res])res = left_res;elseres = right_res;}return res;}bool min_max_heap::has_grandfather(int index) const{if(has_parent(index)){int res = parent(index);if(has_parent(res))return true;}return false;}int min_max_heap::grandfather(int index) const{int p = parent(index);return parent(p);}bool min_max_heap::has_parent(int index) const{if(index == 0)return false;int res = (index-1)/2;if(res >=0)return true;return false;}int min_max_heap::parent(int index) const{int res = (index-1)/2;return res;}min_max_heap::min_max_heap(const int* array, const int n){for(int i=0; i<n; i++)data.push_back(array[i]);for(int i=(n-1)/2; i>=0; i--){if(is_min_level(i))min_shift_down(i);elsemax_shift_down(i);}}min_max_heap::~min_max_heap(){}void min_max_heap::swap(int i, int j){int temp = data[i];data[i] = data[j];data[j] = temp;}void min_max_heap::min_shift_up(int index){if(!has_parent(index))return;else if(!has_grandfather(index)){int p = parent(index);if(data[p] < data[index])swap(p,index);return;}else{int grand = grandfather(index);if(data[grand] > data[index]){swap(index,grand);min_shift_up(grand);}else{int p = parent(index);if(data[p] > data[index])return;else{swap(p,index);max_shift_up(p);}}}}void min_max_heap::max_shift_up(int index){if(!has_parent(index))return;else if(!has_grandfather(index)){int p = parent(index);if(data[p] > data[index])swap(p,index);return;}else{int grand = grandfather(index);if(data[grand] < data[index]){swap(grand,index);max_shift_up(grand);}else{int p = parent(index);if(data[p] < data[index])return;else{swap(index,p);min_shift_up(p);}}}}void min_max_heap::min_shift_down(int index){if(!has_child(index))return;else if(!has_grandchild(index)){int c = min_child(index);if(data[c] <data[index])swap(c,index);return;}else{int c = min_child(index);if(data[c] < data[index]){swap(index,c);max_shift_down(c);}int grand = min_grandchild(index);if(data[grand] > data[index])return;else {swap(grand,index);index = grand;int p = parent(index);if(data[p] < data[index])swap(p,index);min_shift_down(index);}}}void min_max_heap::max_shift_down(int index){if(!has_child(index))return;else if(!has_grandchild(index)){int c = max_child(index);if(data[c] > data[index])swap(c,index);return;}else{int c = max_child(index);if(data[c] > data[index]){swap(c,index);min_shift_down(c);}int grand = max_grandchild(index);if(data[grand] < data[index])return;else{swap(grand,index);index = grand;int p = parent(index);if(data[p] > data[index])swap(p,index);max_shift_down(index);}}}void min_max_heap::insert(int item){data.push_back(item);int index = data.size()-1;if(is_min_level(index))min_shift_up(index);elsemax_shift_up(index);}int min_max_heap::delmin(){int res = -1;int n = data.size();if(n == 0)return -1;res = data[0];swap(0,n-1);data.pop_back();min_shift_down(0);return res;}int min_max_heap::delmax(){int n = data.size();int res = -1;if(n == 0)return res;if(n==1){res = data[0];data.pop_back();}else{int c = max_child(0);res = data[c];swap(c,n-1);data.pop_back();max_shift_down(c);}return res;}int min_max_heap::min(){if(data.size()==0)return -1;return data[0];}int min_max_heap::max(){int n = data.size();if(n==0)return -1;if(n==1)return data[0];return data[max_child(0)];}ostream& operator<<(ostream& os, const min_max_heap& hp){for(unsigned i=0; i<hp.data.size(); i++)os<<hp.data[i]<<" ";return os;}

由于存在奇数层和偶数层之分，也即max层和min层之分，因此在堆的“上浮”和“下沉”的过程中要依据节点所在的层次选择不同的“上浮”和“下层”方法

测试代码：

#include <iostream>#include "min_max_heap.h"#include <time.h>#include <stdlib.h>using namespace std;int* create_array(const int n);void main(){int n;cin>>n;while(n>0){int* a = create_array(n);cout<<"The array: ";for(int i=0; i<n; i++)cout<<a[i]<<" ";cout<<endl;min_max_heap hp(a,n);cout<<"The min-max heap: "<<hp<<endl;cout<<"delmax(): ";for(int i=0; i<n; i++)cout<<hp.delmax()<<" ";cout<<endl;for(int i=0; i<n; i++)hp.insert(a[i]);cout<<"The min-max heap: "<<hp<<endl;cout<<"delmin(): ";for(int i=0; i<n; i++)cout<<hp.delmin()<<" ";cout<<endl;cin>>n;}}int* create_array(const int n){int* res = new int[n];for(int i=0; i<n; i++)res[i] = 0;for(int i=0; i<n; i++){srand((unsigned)time(0));while(1){int m=rand()%n;if(res[m] ==0){res[m] = i;break;}}}return res;}

上述代码只是我在学习min-max heap 时使用的测试代码，如有什么不明白的地方欢迎讨论。