最短路径的O(ElgV)的解法。

使用邻接表存储图，使用堆操作选取下一个最小路径点。

本题的难度并不在最短路径本身这个算法，而是在于堆的操作：

1 使用双重指针操作堆的节点，可以省去直接复制操作堆节点，提高效率，并且这才是有效操作动态地址数据的方法，不用双重指针，我思考了下，觉得更加不好做。

2 使用一个数组记录当前顶点在堆中的位置，相当于一个hash表了，可以需要的时候，直接从表中查找表示顶点的堆节点在堆中的位置，要记得更新节点时维护好这个表。

3 释放内存的时候注意，弹出堆的节点可以马上释放，不过注意不要双重释放内存了

记得曾经看到网上有人说堆排序是百万年薪的算法，不过现在看来光是堆排序是非常简单的算法了，会堆排序应该得达不到百万年薪吧，因为现在的算法高手应该都能随手写出堆排序的算法。

但是如本题这样灵活运用堆还是十分困难的。

参考：http://www.geeksforgeeks.org/greedy-algorithms-set-7-dijkstras-algorithm-for-adjacency-list-representation/

#include <stdio.h>
#include <limits>

class Dijsktra_Adjacency_List_and_Head
{
	struct Node
	{
		int des, weight;
		Node *next;
		Node(int d, int w) : des(d), weight(w), next(NULL) {}
		~Node()
		{
			if (next) delete next;
			next = NULL;
		}
	};

	struct AdjList
	{
		Node *head;
		AdjList() : head(NULL) {}
		~AdjList()
		{
			if (head) delete head;
			head = NULL;
		}
	};

	struct Graph
	{
		int V;
		AdjList *arr;
		Graph(int v) : V(v)
		{
			arr = new AdjList[v];
		}
		~Graph()
		{
			if (arr) delete [] arr;
			arr = NULL;
		}
	};

	void addEdge(Graph *graph, int src, int des, int wei)
	{
		Node *n = new Node(des, wei);
		n->next = graph->arr[src].head;
		graph->arr[src].head = n;

		n = new Node(src, wei);
		n->next = graph->arr[des].head;
		graph->arr[des].head = n;
	}

	struct heapNode
	{
		int v, dist;
		heapNode(int v1, int d):v(v1), dist(d){}
	};

	struct minHeap
	{
		int size, capacity;
		int *pos;//记录顶点的位置

		//双重指针：直接操作地址，而不是复制数据
		heapNode **arr;
		explicit minHeap(int c = 0) : capacity(c), size(0)
		{
			pos = new int[capacity];
			arr = new heapNode*[capacity];//创建存储空间
		}
		~minHeap()
		{
			if (pos) delete [] pos;
			for (int i = 0; i < size; i++)
			{
				if (arr[i]) delete arr[i], arr[i] = NULL;
			}
			if (arr) delete [] arr;
		}
	};

	//双重指针的作用:这里只需要交换了地址，并不需要直接交换整个heapNode数据结构
	void swapMinHeapNode(heapNode **a, heapNode **b)
	{
		heapNode *c = *a;
		*a = *b;
		*b = c;
	}

	void minheapify(minHeap *heap, int idx)
	{
		int smallest, left, right;
		smallest = idx;
		left = idx * 2 + 1;
		right = idx * 2 + 2;//错误+2写成+1

		if (left < heap->size && 
			heap->arr[left]->dist < heap->arr[smallest]->dist)
		{
			smallest = left;
		}
		//很难调试的bug：下面不能加else,因为要选择三种中最小的
		if (right < heap->size &&
			heap->arr[right]->dist < heap->arr[smallest]->dist)
		{
			smallest = right;
		}

		if (smallest != idx)
		{
			heapNode *smallestNode = heap->arr[smallest];
			heapNode *idxNode = heap->arr[idx];

			heap->pos[smallestNode->v] = idx;
			heap->pos[idxNode->v] = smallest;

			swapMinHeapNode(&heap->arr[smallest], &heap->arr[idx]);

			minheapify(heap, smallest);
		}
	}

	bool isEmpty(minHeap *heap)
	{
		return heap->size == 0;
	}

	heapNode *extraMin(minHeap *heap)
	{
		if (isEmpty(heap)) return NULL;

		heapNode *root = heap->arr[0];
		heapNode *lastNode = heap->arr[heap->size-1];
		heap->arr[0] = lastNode;
		//swapMinHeapNode(&root, &lastNode);

		heap->pos[root->v] = heap->size-1;
		heap->pos[lastNode->v] = 0;

		--heap->size;
		minheapify(heap, 0);
		return root;
	}

	void decreaseKey(minHeap *heap, int v, int dist)
	{
		int i = heap->pos[v];
		heap->arr[i]->dist = dist;

		while (i && heap->arr[i]->dist < heap->arr[(i-1)/2]->dist)
		{
			heap->pos[heap->arr[i]->v] = (i-1)/2;
			heap->pos[heap->arr[(i-1)/2]->v] = i;

			swapMinHeapNode(&heap->arr[i], &heap->arr[(i-1)/2]);

			i = (i-1)/2;
		}
	}

	bool isInMinHeap(minHeap *heap, int v)
	{
		if (heap->pos[v] < heap->size) return true;
		return false;
	}

	void printArr(int dist[], int n)
	{
		printf("Vertex   Distance from Source\n");
		for (int i = 0; i < n; ++i)
			printf("%d \t\t %d\n", i, dist[i]);
	}

	void dijkstra(Graph *graph, int src, int dist[])
	{
		int V = graph->V;
		minHeap *heap = new minHeap(V);

		for (int i = 0; i < V; i++)
		{
			dist[i] = INT_MAX;
			heap->arr[i] = new heapNode(i, dist[i]);
			heap->pos[i] = i;
		}
		dist[src] = 0;
		decreaseKey(heap, src, dist[src]);
		heap->size = V;

		while (!isEmpty(heap))
		{
			heapNode *hn = extraMin(heap);
			int u = hn->v;
			delete hn;
			hn = NULL;

			Node *pCrawl = graph->arr[u].head;
			while (pCrawl)
			{
				int v = pCrawl->des;
				if (isInMinHeap(heap, v) && dist[u] != INT_MAX
					&& pCrawl->weight + dist[u] < dist[v])
				{
					dist[v] = pCrawl->weight + dist[u];
					decreaseKey(heap, v, dist[v]);
				}
				pCrawl = pCrawl->next;
			}
		}

		delete heap;
	}

public:
	Dijsktra_Adjacency_List_and_Head()
	{
		// create the graph given in above fugure
		int V = 9;
		struct Graph* graph = new Graph(V);
		addEdge(graph, 0, 1, 4);
		addEdge(graph, 0, 7, 8);
		addEdge(graph, 1, 2, 8);
		addEdge(graph, 1, 7, 11);
		addEdge(graph, 2, 3, 7);
		addEdge(graph, 2, 8, 2);
		addEdge(graph, 2, 5, 4);
		addEdge(graph, 3, 4, 9);
		addEdge(graph, 3, 5, 14);
		addEdge(graph, 4, 5, 10);
		addEdge(graph, 5, 6, 2);
		addEdge(graph, 6, 7, 1);
		addEdge(graph, 6, 8, 6);
		addEdge(graph, 7, 8, 7);

		int *dist = (int *) malloc(sizeof(int) * V);

		dijkstra(graph, 0, dist);

		printArr(dist, V);

		free(dist);
		delete graph;
	}
};