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数据结构16-图的遍历-Prim算法与Dijkstra算法

Prim算法​

Prim算法用于求解加权无向图的最小生成树。它的基本思想是从一个顶点开始，逐步扩展最小生成树，直到包括所有顶点。

1. 数据结构​

我们需要以下数据结构：

  • 图的邻接矩阵表示
  • 存储最小生成树的数组
  • 标记已包含在最小生成树中的顶点的布尔数组

2. 算法步骤​

  1. 初始化：选择一个顶点作为起点，设置其权重为0，其他顶点的权重为无穷大。
  2. 重复以下步骤直到所有顶点都包含在最小生成树中：
    • 从未包含在最小生成树中的顶点中选择权重最小的顶点。
    • 将该顶点加入最小生成树。
    • 更新与该顶点相邻的顶点的权重。

3. 代码实现​

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

#define V 5  // 图的顶点数

// 找到权值最小且未包含在最小生成树中的顶点
int minKey(int key[], bool mstSet[]) {
    int min = INT_MAX, min_index;

    for (int v = 0; v < V; v++)
        if (mstSet[v] == false && key[v] < min)
            min = key[v], min_index = v;

    return min_index;
}

// 打印构建的最小生成树
void printMST(int parent[], int graph[V][V]) {
    printf("Edge \tWeight\n");
    for (int i = 1; i < V; i++)
        printf("%d - %d \t%d \n", parent[i], i, graph[i][parent[i]]);
}

// 利用Prim算法构建最小生成树
void primMST(int graph[V][V]) {
    int parent[V];  // 存储构建的最小生成树
    int key[V];     // 用于选择最小权重的边
    bool mstSet[V]; // 标记已包含在最小生成树中的顶点

    // 初始化所有键值为无穷大
    for (int i = 0; i < V; i++)
        key[i] = INT_MAX, mstSet[i] = false;

    // 选择第一个顶点作为起点
    key[0] = 0;
    parent[0] = -1; // 起点没有父节点

    // 构建最小生成树
    for (int count = 0; count < V - 1; count++) {
        int u = minKey(key, mstSet);

        mstSet[u] = true;

        for (int v = 0; v < V; v++)
            if (graph[u][v] && mstSet[v] == false && graph[u][v] < key[v])
                parent[v] = u, key[v] = graph[u][v];
    }

    printMST(parent, graph);
}

int main() {
    int graph[V][V] = {{0, 2, 0, 6, 0},
                       {2, 0, 3, 8, 5},
                       {0, 3, 0, 0, 7},
                       {6, 8, 0, 0, 9},
                       {0, 5, 7, 9, 0}};

    primMST(graph);

    return 0;
}

Dijkstra算法​

Dijkstra算法用于求解加权有向图的单源最短路径问题。它的基本思想是从起点开始，逐步扩展最短路径，直到所有顶点都被处理。

1. 数据结构​

我们需要以下数据结构：

  • 图的邻接矩阵表示
  • 存储最短路径的数组
  • 标记已处理顶点的布尔数组

2. 算法步骤​

  1. 初始化：设置起点的距离为0，其他顶点的距离为无穷大。
  2. 重复以下步骤直到所有顶点都被处理：
    • 从未处理的顶点中选择距离最小的顶点。
    • 将该顶点标记为已处理。
    • 更新与该顶点相邻的顶点的距离。

3. 代码实现​

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

#define V 9  // 图的顶点数

// 找到距离最小且未处理的顶点
int minDistance(int dist[], bool sptSet[]) {
    int min = INT_MAX, min_index;

    for (int v = 0; v < V; v++)
        if (sptSet[v] == false && dist[v] <= min)
            min = dist[v], min_index = v;

    return min_index;
}

// 打印从起点到其他所有顶点的最短距离
void printSolution(int dist[]) {
    printf("Vertex \t Distance from Source\n");
    for (int i = 0; i < V; i++)
        printf("%d \t\t %d\n", i, dist[i]);
}

// 利用Dijkstra算法计算从src到所有顶点的最短路径
void dijkstra(int graph[V][V], int src) {
    int dist[V];    // 存储从起点到各顶点的最短距离
    bool sptSet[V]; // 标记已处理顶点

    // 初始化所有距离为无穷大
    for (int i = 0; i < V; i++)
        dist[i] = INT_MAX, sptSet[i] = false;

    // 起点的距离为0
    dist[src] = 0;

    // 找到从起点到所有顶点的最短路径
    for (int count = 0; count < V - 1; count++) {
        int u = minDistance(dist, sptSet);

        sptSet[u] = true;

        for (int v = 0; v < V; v++)
            if (!sptSet[v] && graph[u][v] && dist[u] != INT_MAX && dist[u] + graph[u][v] < dist[v])
                dist[v] = dist[u] + graph[u][v];
    }

    printSolution(dist);
}

int main() {
    int graph[V][V] = {{0, 4, 0, 0, 0, 0, 0, 8, 0},
                       {4, 0, 8, 0, 0, 0, 0, 11, 0},
                       {0, 8, 0, 7, 0, 4, 0, 0, 2},
                       {0, 0, 7, 0, 9, 14, 0, 0, 0},
                       {0, 0, 0, 9, 0, 10, 0, 0, 0},
                       {0, 0, 4, 14, 10, 0, 2, 0, 0},
                       {0, 0, 0, 0, 0, 2, 0, 1, 6},
                       {8, 11, 0, 0, 0, 0, 1, 0, 7},
                       {0, 0, 2, 0, 0, 0, 6, 7, 0}};

    dijkstra(graph, 0);

    return 0;
}

闵帆老师写的​

代码​

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

#define MAX_DISTANCE 10000

/**
 * The structure of a Net.
 */
typedef struct Net {
	int** weights;
	int numNodes;
} Net, *NetPtr;

/**
 * Initialize a Net.
 */
NetPtr initNet(int paraSize, int** paraData) {
	int i, j;
	NetPtr resultPtr = (NetPtr)malloc(sizeof(Net));
	resultPtr -> numNodes = paraSize;

	//Two stage space allocation.
	resultPtr->weights = (int**)malloc(paraSize * sizeof(int*));
	for (i = 0; i < paraSize; i ++) {
		resultPtr -> weights[i] = (int*)malloc(paraSize * sizeof(int));
		for (j = 0; j < paraSize; j ++) {
			resultPtr -> weights[i][j] = paraData[i][j];
		}//Of for j
	}//Of for i
	
	return resultPtr;
}//Of initNet

/**
 * The Prim algorithm for spanning tree, or the Dijkstra algorithm for nearest path.
 * @param paraAlgorithm 0 for Dijkstra, 1 for Prim
 * @return The total cost of the tree.
 */
int dijkstraOrPrim(NetPtr paraPtr, int paraAlgorithm) {
	int i, j, minDistance, tempBestNode, resultCost;
	int source = 0;
	int numNodes = paraPtr->numNodes;
	int *distanceArray = (int*)malloc(numNodes * sizeof(int));
	int *parentArray = (int*)malloc(numNodes * sizeof(int));
	//Essentially boolean
	int *visitedArray = (int*)malloc(numNodes * sizeof(int)); 

	// Step 1. Initialize. Any node can be the source.
	for (i = 0; i < numNodes; i++) {
		distanceArray[i] = paraPtr->weights[source][i];
		parentArray[i] = source;
		visitedArray[i] = 0;
	} // Of for i
	distanceArray[source] = 0;
	parentArray[source] = -1;
	visitedArray[source] = 1;

	// Step 2. Main loops.
	tempBestNode = -1;
	for (i = 0; i < numNodes - 1; i++) {
		// Step 2.1 Find out the best next node.
		minDistance = MAX_DISTANCE;
		for (j = 0; j < numNodes; j++) {
			// This node is visited.
			if (visitedArray[j]) {
				continue;
			} // Of if

			if (minDistance > distanceArray[j]) {
				minDistance = distanceArray[j];
				tempBestNode = j;
			} // Of if
		} // Of for j

		visitedArray[tempBestNode] = 1;

		// Step 2.2 Prepare for the next round.
		for (j = 0; j < numNodes; j++) {
			// This node is visited.
			if (visitedArray[j]) {
				continue;
			} // Of if

			// This node cannot be reached.
			if (paraPtr->weights[tempBestNode][j] >= MAX_DISTANCE) {
				continue;
			} // Of if

			// Attention: the difference between Dijkstra and Prim algorithms.
			if (paraAlgorithm == 0) {
				if (distanceArray[j] > distanceArray[tempBestNode] + paraPtr->weights[tempBestNode][j]) {
					// Change the distance.
					distanceArray[j] = distanceArray[tempBestNode] + paraPtr->weights[tempBestNode][j];
					// Change the parent.
					parentArray[j] = tempBestNode;
				} // Of if
			} else {
				if (distanceArray[j] > paraPtr->weights[tempBestNode][j]) {
					// Change the distance.
					distanceArray[j] = paraPtr->weights[tempBestNode][j];
					// Change the parent.
					parentArray[j] = tempBestNode;
				} // Of if
			}//Of if
		} // Of for j
	} // Of for i

	printf("the parent of each node: ");
	for (i = 0; i < numNodes; i++) {
		printf("%d, ", parentArray[i]);
	} // Of for i

	if (paraAlgorithm == 0) {
		printf("From node 0, path length to all nodes are: ");
		for (i = 0; i < numNodes; i++) {
			printf("%d (%d), ", i, distanceArray[i]);
		} // Of for i
	} else {
		resultCost = 0;
		for (i = 0; i < numNodes; i++) {
			resultCost += distanceArray[i];
			printf("cost of node %d is %d, total = %d\r\n", i, distanceArray[i], resultCost);
		} // Of for i
		printf("Finally, the total cost is %d.\r\n ", resultCost);
	}//Of if

	// Step 3. Output for debug.
	printf("\r\n");

	return resultCost;
}// Of dijkstraOrPrim

/**
 * Construct a sample net.
 * Revised from testGraphTranverse().
 */
NetPtr constructSampleNet() {
	int i, j;
	int myGraph[6][6] = { 
		{0, 6, 1, 5, 0, 0},
		{6, 0, 5, 0, 3, 0}, 
		{1, 5, 0, 5, 6, 4}, 
		{5, 0, 5, 0, 0, 2}, 
		{0, 3, 6, 0, 0, 6},
		{0, 0, 4, 2, 6, 0}};
	int** tempPtr;
	int numNodes = 6;
	printf("Preparing data\r\n");
		
	tempPtr = (int**)malloc(numNodes * sizeof(int*));
	for (i = 0; i < numNodes; i ++) {
		tempPtr[i] = (int*)malloc(numNodes * sizeof(int));
	}//Of for i
	 
	for (i = 0; i < numNodes; i ++) {
		for (j = 0; j < numNodes; j ++) {
			if (myGraph[i][j] == 0) {
				tempPtr[i][j] = MAX_DISTANCE;
			} else {
				tempPtr[i][j] = myGraph[i][j];
			}//Of if
		}//Of for j
	}//Of for i
 
	printf("Data ready\r\n");
	
	NetPtr resultNetPtr = initNet(numNodes, tempPtr);
	return resultNetPtr;
}//Of constructSampleNet

/**
 * Test the Prim algorithm.
 */
void testPrim() {
	NetPtr tempNetPtr = constructSampleNet();
	printf("=====Dijkstra algorithm=====\r\n");
	dijkstraOrPrim(tempNetPtr, 0);
	printf("=====Prim algorithm=====\r\n");
	dijkstraOrPrim(tempNetPtr, 1);
}//Of testPrim

/**
 * The entrance.
 */
int main(){
	testPrim();
	return 1;
}//Of main

运行结果​

Preparing data
Data ready
=====Dijkstra algorithm=====
the parent of each node: -1, 0, 0, 0, 2, 2, From node 0, path length to all nodes are: 0 (0), 1 (6), 2 (1), 3 (5), 4 (7), 5 (5),
=====Prim algorithm=====
the parent of each node: -1, 2, 0, 5, 1, 2, cost of node 0 is 0, total = 0
cost of node 1 is 5, total = 5
cost of node 2 is 1, total = 6
cost of node 3 is 2, total = 8
cost of node 4 is 3, total = 11
cost of node 5 is 4, total = 15
Finally, the total cost is 15.

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