关于二叉堆（优先队列）的其他操作及其应用

【0】README

0.1）本文总结于数据结构与算法分析；源代码均为原创，旨在了解到我们学习了优先队列后，还能干些什么东西出来，增加学习的interest；
0.2）以下列出了关于二叉堆（优先队列）的其他有用的操作，其内容基本可以囊括大部分的优先队列知识点， 亮点是这里还引入到了堆排序；
0.3）二叉堆的insert 和 deleteMin 基本操作和概念，参见 http://blog.csdn.net/pacosonswjtu/article/details/49498023

【1】二叉堆的操作应用

1.1）降低关键字的值——decreaseKey

1）decreaseKey概述： decreaseKey(P, △， H) 操作降低在位置 P处的关键字的值，降值的幅度为正的量△；由于这可能破坏堆的序，因此必须通过上滤对堆进行调整；
2）decreaseKey应用： 该操作对系统管理程序是有用的，系统管理程序能够使它们的程序以最高的优先级来运行；

1.2）增加关键字的值——increaseKey

1）increaseKey概述： increaseKey(P, △， H) 操作增加在位置 P处的关键字的值，增值的幅度为正的量△；由于这可能破坏堆的序，因此必须通过下滤对堆进行调整；
2）increaseKey应用： 许多os 调度程序自动地降低正在过多地消耗 CPU 时间的进程的优先级；

1.3）删除某个位置上的元素——deleteElement

1）deleteElement概述： delete(P, H)操作删除堆中位置P 上的节点，这通过首先执行 decreaseKey（P， ∞， H），然后再执行 deleteMin 来完成；
2）deleteElement 应用：当一个进程被用户终止（而不是正常终止时），它必须从优先队列中除去；

1.4）构建堆——buildHeap

1）buildHeap 概述： buildHeap(H) 操作把N个关键字作为输入并把他们放入空堆中；显然，这可以使用 N个相继的insert操作来完成；
2）buildHeap 应用：

1.5）优先队列的应用（我这里只列出了选择问题，当然远远不止这一个应用，这个应用我还没有编写源代码实现）

1）我们将要考察的第一个问题是选择问题： 当时的输入是 N 个元素以及一个整数 k，这N 个元素的集可以是全序的，该选择问题是要找出第k个最大的元素；
2）利用优先队列来解决（二叉堆属于优先队列）： 对上述N个元素建立大根堆，然后删除k-1个最大元素，那么二叉堆的根节点的元素就是第k个最大元素；
3）堆排序： 注意，如果我们对 k=N 运行该程序并在元素离开对时记录他们的值，那么实际上已经对输入文件以时间 O（NlogN）做了排序，这样就得到一种快速的排序算法——堆排序；

Attention）

A1）改动的节点的儿子情况只有3种： 要么一个儿子都没有，要么只有一个左儿子，要么有两个儿子（不可能只有一个右儿子）；
A2） 就编程而言，上滤的难度小于下滤的难度；
其他堆操作的源代码（核心是堆的上滤 percolateUp 和下滤percolateDown操作）：

【2】source code+printing

2.0）Warning of percolateDown :

W1）因为涉及到利用下滤操作把随机数组（下标从0开始）建立堆（buildHeap 操作），然而在堆的其他操作，如插入，删除操作中，使用的下标是从1开始以便于编程；所以我们的下滤操作分为下标从0开始的percolateDownFromZero 和下标从1开始的percolateDownFromOne，不过上述percolateDown 的两个版本的编程idea 都是一样的，只是数组的起始下标不一样而已，这是我们应该注意的；
W2） 代码中，只有buildHeap 使用的下滤 percolateDownFromZero 版本，而其他操作使用的是 percolateDownFromOne；
W3）你要知道实现堆操作的代码核心是：上滤percolateUpFromOne 和下滤操作 percolateDownFromOne + percolateDownFromZero；

2.1）download source code ：
https://github.com/pacosonTang/dataStructure-algorithmAnalysis/tree/master/chapter6
2.2）source code at a glance：
[1st file binaryheap.h]

#include <stdio.h>
#include <malloc.h>#define ElementType int
#define Error(str) printf("\n error: %s \n",str)   struct BinaryHeap;
typedef struct BinaryHeap *BinaryHeap;void swap(ElementType *x, ElementType *y);
BinaryHeap initBinaryHeap(int capacity);
void insert(ElementType value, BinaryHeap bh);
ElementType deleteMin(BinaryHeap);
int isFull(BinaryHeap bh);
int isEmpty(BinaryHeap bh);
void percolateUp(int index, BinaryHeap bh);
void percolateDownFromOne(int index, BinaryHeap bh);
void printBinaryHeap(BinaryHeap bh);
void printBinaryHeapFromZero(BinaryHeap bh);struct BinaryHeap 
{int capacity;int size;   ElementType *elements;      
};

[2st file binaryHeapBasicOperation.c]

#include "binaryheap.h"//judge whether the BinaryHeap is full or not , also 1 or 0 
int isFull(BinaryHeap bh)
{return bh->size == bh->capacity - 1 ? 1 : 0 ;
}//judge whether the BinaryHeap is empty or not , also 1 or 0 
int isEmpty(BinaryHeap bh)
{return bh->size == 0 ? 1 : 0 ;
}void swap(ElementType *x, ElementType *y)
{ElementType temp;temp = *x;*x = *y;*y = temp;
}// get the left child of node under index with startup 1
int leftChildFromOne(int index)
{return index * 2;
}ElementType deleteMin(BinaryHeap bh)
{   ElementType minimum;ElementType *data;  if(isEmpty(bh)){Error("failed deleting minimum , for the BinaryHeap is empty, from func deleteMin !");return -1;  }data = bh->elements;    minimum = data[1];swap(&data[1], &data[bh->size]);bh->size-- ; // size-- occurs prior to percolateDownFromOne percolateDownFromOne(1, bh) ;   return minimum;
} // Attention, the index of the heap starts from 1
void insert(ElementType value, BinaryHeap bh)
{int i;if(isFull(bh)){Error("failed insertion , for the BinaryHeap is full, from func insert!");return ;    }for(i = ++bh->size; bh->elements[i/2] > value; i /= 2)bh->elements[i] = bh->elements[i / 2];bh->elements[i] = value;
}BinaryHeap initBinaryHeap(int capacity)
{BinaryHeap bh;ElementType *temp;bh = (BinaryHeap)malloc(sizeof(struct BinaryHeap));if(!bh) {Error("out of space, from func initBinaryHeap");        return NULL;}  bh->capacity = capacity;bh->size = 0;temp = (ElementType *)malloc(capacity * sizeof(ElementType));if(!temp) {Error("out of space, from func initBinaryHeap");        return NULL;} bh->elements = temp;return bh;
}void printBinaryHeap(BinaryHeap bh)
{int i;ElementType *temp;if(!bh)Error("printing execution failure, for binary heap is null, from func printBinaryHeap");    temp = bh->elements;for(i = 1; i < bh->capacity; i++){printf("\n\t index[%d] = ", i);if(i <= bh->size)printf("%d", bh->elements[i]);elseprintf("NULL");}printf("\n");
}  //print the binary heap who starts from index 0
void printBinaryHeapFromZero(BinaryHeap bh)
{int i;ElementType *temp;if(!bh)Error("printing execution failure, for binary heap is null, from func printBinaryHeap");    temp = bh->elements;for(i = 0; i < bh->capacity; i++){printf("\n\t index[%d] = ", i);if(i < bh->size)printf("%d", bh->elements[i]);elseprintf("NULL");}printf("\n");
}

[3rd file binaryHeapOtherOperation.c]

 #include "binaryheap.h"#define Infinity 10000// switch the elements void swap(ElementType *x, ElementType *y)
{ElementType temp;temp = *x;*x = *y;*y = temp;
}// get the parent of the element with index 
int parentFromOne(int index)
{return index / 2;
}// percolating up the element when its value is greater than children (minimal heap)//Attention: all of bh->elements starts from index 1void percolateUpFromOne(int index, BinaryHeap bh){  ElementType *data;ElementType temp;int size;   int parent;data = bh->elements;size = bh->size;for(temp = data[index]; parentFromOne(index) > 0; index = parent){parent = parentFromOne(index);if(parent == 0 || temp > data[parent])              break;else        data[index] = data[parent];                                         }data[index] = temp;}// get the left child of node under index with startup one
int leftChildFromOne(int index)
{return index * 2;
}// percolating down the element when its value is greater than children (minimal heap)//Attention: all of bh->elements starts from index 1void percolateDownFromOne(int index, BinaryHeap bh){  ElementType *data;int size;ElementType temp;int child;data = bh->elements;size = bh->size;for(temp = data[index]; leftChildFromOne(index) <= size; index = child){child = leftChildFromOne(index);if(child < size && data[child] > data[child+1])child++;if(temp > data[child])data[index] = data[child];elsebreak;}data[index] = temp;}//decreasing value of the element under index by increment
void decreaseKey(int index, ElementType decrement, BinaryHeap bh)
{   if(index > bh->size || index < 1){Error(" failed decreaseKey, since overstep the boundary! ");return ;}bh->elements[index] -= decrement; // update the element under given indexpercolateUpFromOne(index, bh);
}//increasing value of the element under index by increment
void increaseKey(int index, ElementType increment, BinaryHeap bh)
{   if(index > bh->size || index < 1){Error(" failed increaseKey, since overstep the boundary! ");return ;}bh->elements[index] += increment; // update the element under given indexpercolateDownFromOne(index, bh);
}//deleting the element under index 
void deleteElement(int index, BinaryHeap bh)
{decreaseKey(index, Infinity, bh); // 1st step, decreaseKey operation placing the element under index upto the root  deleteMin(bh); //2nd step, deleteMin deleting the element under the root;
} // get the left child of node under index with startup zero
int leftChildFromZero(int index)
{return index * 2 + 1;
}// percolating down the element when its value is greater than children (minimal heap)//Attention: all of bh->elements starts from index 0void percolateDownFromZero(int index, BinaryHeap bh){  ElementType *data;ElementType temp;int size;   int child;data = bh->elements;size = bh->size;for(temp = data[index]; leftChildFromZero(index) < size; index = child){child = leftChildFromZero(index);if(child < size - 1 && data[child] > data[child+1])child++;if(temp > data[child])data[index] = data[child];elsebreak;}data[index] = temp;
}// building the heap with data in array randomly
void buildHeap(BinaryHeap bh)
{int i;  ElementType *data;data = bh->elements;for(i = bh->size/2; i >= 0; i--)percolateDownFromZero(i, bh);       
}int main()
{ElementType data[] = {85, 80, 40, 30, 10, 70, 110}; // P141 ElementType buildHeapData[] = {150, 80, 40, 30, 10, 70, 110, 100, 20, 90, 60, 50, 120, 140, 130};BinaryHeap bh;  int size;int i;  int capacity;printf("\n\t=== test for inserting the binary heap with {85, 80, 40, 30, 10, 70, 110} in turn ===\n");capacity = 14;bh = initBinaryHeap(capacity);size = 7;   for(i = 0; i < size; i++)insert(data[i], bh);printBinaryHeap(bh);printf("\n\t=== test for inserting the binary heap with {100, 20, 90} in turn ===\n");insert(100, bh);insert(20, bh);insert(90, bh);printBinaryHeap(bh);printf("\n\t=== test for inserting the binary heap with 5 ===\n");insert(5, bh);      printBinaryHeap(bh);printf("\n\t=== test for 3 deletings towards the minimum in binary heap ===\n");deleteMin(bh);  printBinaryHeap(bh);deleteMin(bh);      printBinaryHeap(bh);deleteMin(bh);  printBinaryHeap(bh);// other operations in bianry heap  printf("\n\t====== test for other operations in bianry heap as follows ======\n");printf("\n\t=== test for increaseKey(4, 120, bh) ===\n");increaseKey(4, 120, bh);printBinaryHeap(bh);printf("\n\t=== test for increaseKey(2, 120, bh) ===\n");increaseKey(2, 120, bh);printBinaryHeap(bh);printf("\n\t=== test for decreaseKey(9, 195, bh) ===\n");decreaseKey(9, 195, bh);printBinaryHeap(bh);printf("\n\t=== test for decreaseKey(4, 90, bh) ===\n");decreaseKey(4, 90, bh);printBinaryHeap(bh);printf("\n\t=== test for decreaseKey(7, 50, bh) ===\n");decreaseKey(7, 50, bh);printBinaryHeap(bh);printf("\n\t=== test for decreaseKey(5, 155, bh) ===\n");decreaseKey(5, 155, bh);printBinaryHeap(bh);printf("\n\t=== test for deleteElement(4, bh) ===\n");deleteElement(4, bh);printBinaryHeap(bh);printf("\n\t=== test for deleteElement(1, bh) ===\n");deleteElement(1, bh);printBinaryHeap(bh);printf("\n\t=== test for deleteElement(3, bh) ===\n");deleteElement(3, bh);printBinaryHeap(bh); // test over , Bingo!// as you know, the build heap operation is identical with other operationsprintf("\n\t=== test for building heap with {150, 80, 40, 30, 10, 70, 110, 100, 20, 90, 60, 50, 120, 140, 130} ===\n");capacity = 16;bh = initBinaryHeap(capacity);bh->size = 15;bh->elements = buildHeapData; buildHeap(bh);printBinaryHeapFromZero(bh);return 0;
}