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图论——Dijkstra+prim算法涉及到的优先队列（二叉堆）

pacoson 2015-11-19 09:43 原文

【0】README

0.1）为什么有这篇文章？因为 Dijkstra算法的优先队列实现 涉及到了一种新的数据结构，即优先队列（二叉堆）的操作需要更改以适应这种新的数据结构，我们暂且吧它定义为Distance， 而不是单纯的int类型；
0.2）本文源代码均为原创， int类型的优先队列（二叉堆）的操作实现，参见http://blog.csdn.net/PacosonSWJTU/article/details/49498255， （并比较他们的打印结果，很有必要）

【1】因为 Dijkstra算法的优先队列实现， 需要用到二叉堆的相关操作，但是操作的元素类型（ElementType 不是 单纯的int类型）， 而是如下：

struct Distance
{
 int vertexIndex; //当前顶点下标
 int distance; //初始顶点到当前顶点的distance
};

【2】看个荔枝

2.1）需要特别说明的是： indexOfVertexInHeap 数组记录的是顶点vertex在 heap中的位置， 如 indexOfVertexInHeap [1] = 4；表明heap的第4个位置记录这 编号为1的vertex;
2.2）优先队列的insert和deleteMin 的执行演示（请将我的手动演示结果同我的代码打印结果做对比，经过对比，你发现它们的效果是一致的，恰好说明了我的代码的可行性）：

Attention）

  • A1）其实本文中的二叉堆优先队列的实现源代码和 int类型的优先队列源代码类似，只不过它们操作的数据类型不一样罢了，当然， 这只需要简单的修改即可；
  • A2）打印结果在文末，可以看到，ElementType采用int 和 Distance的打印效果一样，这正证明了我们采用Distance结构体对源码的修改是无误的，相比于单纯的int 类型，只不过Distance又多了一个 顶点下标vertexIndex成员变量而已；

【3】source code + printing results

3.1）download source code：
https://github.com/pacosonTang/dataStructure-algorithmAnalysis/tree/master/chapter9/binaryHeap_dijkstra_prim
3.2）source code at a glance：（for complete code , please click the given link above）

1st file：distance.h

#include <stdio.h>

#define Error(str) printf("\n error: %s \n",str)   

struct Distance;
typedef struct Distance *Distance;
struct Distance
{
	int vertexIndex;
	int distance;
};

Distance makeEmptyDistance();

2nd file：distance.c

#include "distance.h"
#include <malloc.h>

// allocate the memory for Distance struct
Distance makeEmptyDistance()
{
	Distance element;

	element = (Distance)malloc(sizeof(struct Distance));
	if(!element)
	{
		Error("out of space ,from func makeEmptyDistance");
		return NULL;
	}		

	return element;
}

3rd file：binaryheap.h

#include <stdio.h>
#include <malloc.h>
#include "distance.h"

#define ElementType Distance

#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, int*);
ElementType deleteMin(BinaryHeap, int*);
int isFull(BinaryHeap bh);
int isEmpty(BinaryHeap bh);
void percolateUp(int index, BinaryHeap bh);
void percolateDownFromOne(int index, BinaryHeap bh, int*);
void printBinaryHeap(BinaryHeap bh);
void printBinaryHeapFromZero(BinaryHeap bh);

struct BinaryHeap 
{
	int capacity;
	int size;	
	ElementType *elements;		
};

4th file：binaryheap.c

#include "binaryheap.h"
#include <math.h>

#define MaxInt (int)pow(2, 16)
//judge whether the BinaryHeap is full or not , also 1 or 0 
int isFull(BinaryHeap bh)
{
	return bh->size == bh->capacity - 1; 
}

//judge whether the BinaryHeap is empty or not , also 1 or 0 
int isEmpty(BinaryHeap bh)
{
	return bh->size == 0;
}

// get the left child of node under index with startup 1
int leftChildFromOne(int index)
{
	return index * 2;
}

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 heap[%d] = ", i);
		if(i <= bh->size)
			printf("vertex[%d] + distance[%d]", bh->elements[i]->vertexIndex+1, bh->elements[i]->distance);		
		else
			printf("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]->distance);
		else
			printf("NULL");
	}
	printf("\n");
}  

void swap(ElementType x, ElementType y)
{
	struct Distance temp;
	
	temp = *x;
	*x = *y;
	*y = temp;	
}

ElementType deleteMin(BinaryHeap bh, int* heapIndexRecord)
{	
	ElementType minimum;
	ElementType *data;	
	
	if(isEmpty(bh))
	{
		Error("failed deleting minimum , for the BinaryHeap is empty, from func deleteMin !");
		return NULL;	
	}

	data = bh->elements;	 
	minimum = data[1];
	
	swap(data[1], data[bh->size]);		
	bh->size-- ; // size-- occurs prior to percolateDownFromOne 
	percolateDownFromOne(1, bh, heapIndexRecord) ;	
	return minimum;
} 

// percolating down the element when its value is greater than children (minimal heap)
 //Attention: all of bh->elements starts from index 1
 void percolateDownFromOne(int index, BinaryHeap bh, int* heapIndexRecord)
 {	
	ElementType *data;
	int size;
	struct Distance 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]->distance > data[child+1]->distance)
			child++;
		if(temp.distance > data[child]->distance)
		{			
			*data[index] = *data[child];			
			heapIndexRecord[bh->elements[index]->vertexIndex] = index; //update the heapIndexRecord
		}
		else
			break;
	}	
	*data[index] = temp;	
	heapIndexRecord[bh->elements[index]->vertexIndex] = index; //update the heapIndexRecord
 }

// Attention, the index of the heap starts from 1
// return the index the element inserted into the binary heap
void insert(ElementType value, BinaryHeap bh, int* heapIndexRecord)
{
	int i;
	
	if(isFull(bh))
	{
		Error("failed insertion , for the BinaryHeap is full, from func insert!");
		return ;	
	}	
	if(!isEmpty(bh))
		for(i = ++bh->size; bh->elements[i/2]->distance > value->distance; i /= 2)		 			
		{
			//copyElement(bh->elements[i/2], bh->elements[i]);		 
			*bh->elements[i] = *bh->elements[i/2];
			heapIndexRecord[bh->elements[i]->vertexIndex] = i; //update the heapIndexRecord
		}
	else
		i = ++bh->size;		
	*bh->elements[i] = *value;
	heapIndexRecord[bh->elements[i]->vertexIndex] = i; //update the heapIndexRecord
}

BinaryHeap initBinaryHeap(int capacity)
{
	BinaryHeap bh;
	ElementType *temp;
	int i;

	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(Distance));
	if(!temp) {
        Error("out of space, from func initBinaryHeap");        
        return NULL;
    } 
	bh->elements = temp;
	
	for(i=0; i < capacity; i++)
	{
		temp[i] = (ElementType)malloc(sizeof(struct Distance));
		if(!temp[i]) {
			Error("out of space, from func initBinaryHeap");        
			return NULL;
		} 		
	}

	return bh;
}

// allocate the memory for storing index of  vertex in heap and let every element -1
int *makeEmptyArray(int size)
{
	int *array;
	int i;

	array = (int*)malloc(size * sizeof(int));
	if(!array)
	{
		Error("out of space ,from func makeEmptyArray");
		return NULL;
	}		
	for(i=0; i<size; i++)
		array[i] = -1;

	return array;
} 


void printIndexOfVertexInHeap(int size, int *array)
{
	int i;

	for(i=0; i<size; i++)	 
		printf("\tindexOfVertexInHeap[%d] = %d\n", i+1, array[i]);	 
}

int main()
{
	int data[] = {85, 80, 40, 30, 10, 70, 110}; // P141	
	int buildHeapData[] = {150, 80, 40, 30, 10, 70, 110, 100, 20, 90, 60, 50, 120, 140, 130};
	BinaryHeap bh;	
	int size;
	int i;	
	int capacity;
	Distance tempDisStruct;
	int *indexOfVertexInHeap;
	
	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;	
	
	tempDisStruct = makeEmptyDistance(); 
	indexOfVertexInHeap = makeEmptyArray(size);

	for(i = 0; i < size; i++) 
	{
		tempDisStruct->distance = data[i];
		tempDisStruct->vertexIndex = i;
		insert(tempDisStruct, bh, indexOfVertexInHeap);
	}	
	printBinaryHeap(bh);
	printIndexOfVertexInHeap(bh->size, indexOfVertexInHeap);

	printf("\n\t=== test for inserting the binary heap with element {100, 20, 90} in turn ===\n");
	
	tempDisStruct->distance = 100;
	tempDisStruct->vertexIndex = size;
	insert(tempDisStruct, bh, indexOfVertexInHeap);	
	printBinaryHeap(bh);

	tempDisStruct->distance = 20;
	tempDisStruct->vertexIndex = size+1;
	insert(tempDisStruct, bh, indexOfVertexInHeap);	
	printBinaryHeap(bh);

	tempDisStruct->distance = 90;
	tempDisStruct->vertexIndex = size+2;
	insert(tempDisStruct, bh, indexOfVertexInHeap);	
	printBinaryHeap(bh);

	printIndexOfVertexInHeap(bh->size, indexOfVertexInHeap);

	printf("\n\t=== test for inserting the binary heap with 5 ===\n");	
	tempDisStruct->distance = 5;
	tempDisStruct->vertexIndex = size+3;
	insert(tempDisStruct, bh, indexOfVertexInHeap);	
	printBinaryHeap(bh);

	printf("\n\t=== test for 3 deletings towards the minimum in binary heap ===\n");
	deleteMin(bh, indexOfVertexInHeap);	
	printBinaryHeap(bh);
	deleteMin(bh, indexOfVertexInHeap);		
	printBinaryHeap(bh);
	deleteMin(bh, indexOfVertexInHeap);	
	printBinaryHeap(bh);
}

3.3）printing results：
这里写图片描述
这里写图片描述
这里写图片描述

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