网的构建
typedef struct Net{
int** weights;
int numNodes;
} *NetPtr;
创造一个范例网
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 datarn");
tempPtr = (int**) malloc (numNodes * sizeof (int*));
for(i = 0; i < numNodes; i++){
tempPtr[i] = (int*)malloc(numNodes * sizeof(int));
}
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];
}
}
}
printf("Data readyrn");
NetPtr resultNetPtr = initNet(numNodes, tempPtr);
return resultNetPtr;
}
初始化网
NetPtr initNet(int paraSize, int** paraData){
int i, j;
NetPtr resultPtr = (NetPtr) malloc (sizeof (Net));
resultPtr -> numNodes = paraSize;
//开辟二维空间
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++){
//储存值
resultPtr -> weights[i][j] = paraData[i][j];
}
}
return resultPtr;
}
生成树的Prim算法，或最近路径的Dijkstra算法
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));
//本质上是布尔型的
int *visitedArray = (int*) malloc (numNodes * sizeof (int));
//初始化。任何结点都可当作起点
for(i = 0; i < numNodes; i++){
distanceArray[i] = paraPtr->weights[source][i];
parentArray[i] = source;
visitedArray[i] = 0;
}
distanceArray[source] = 0;
parentArray[source] = -1;
visitedArray[source] = 1;
//主回路
tempBestNode = -1;
for(i = 0; i < numNodes - 1; i++){
//找到最近的结点
minDistance = MAX_DISTANCE;
for(j = 0; j < numNodes; j++){
//判断结点是否已访问
if(visitedArray[j]){
continue;
}
if(minDistance > distanceArray[j]){
minDistance = distanceArray[j];
tempBestNode = j;
}
}
visitedArray[tempBestNode] = 1;
//准备下一轮循环
for(j = 0; j < numNodes; j++){
//判断结点是否已访问
if(visitedArray[j]){
continue;
}
//判断结点是否能到达
if(paraPtr->weights[tempBestNode][j] >= MAX_DISTANCE){
continue;
}
//注意：Dijkstra和Prim算法之间的差异
if(paraAlgorithm == 0){
if(distanceArray[j] > distanceArray[tempBestNode] + paraPtr->weights[tempBestNode][j]){
//更改与起点距离
distanceArray[j] = distanceArray[tempBestNode] + paraPtr->weights[tempBestNode][j];
//更改父级
parentArray[j] = tempBestNode;
}
} else {
if(distanceArray[j] > paraPtr->weights[tempBestNode][j]){
//更改与新起点距离
distanceArray[j] = paraPtr->weights[tempBestNode][j];
//更改父级
parentArray[j] = tempBestNode;
}
}
}
}
printf("the parent of each node: ");
for(i = 0; i < numNodes; i++){
printf("%d, ", parentArray[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]);
}
} else {
resultCost = 0;
for(i = 0; i < numNodes; i++){
resultCost += distanceArray[i];
printf("cost of node %d is %d, total = %drn", i, distanceArray[i], resultCost);
}
printf("Finally, the total cost is %d.rn ", resultCost);
}
printf("rn");
return resultCost;
}
二、全部代码
// // Prim 算法与 Dijkstra 算法.c // 网 // // Created on 2022/6/1. // #include#include #define MAX_DISTANCE 10000 //网的构建 typedef struct Net{ int** weights; int numNodes; } *NetPtr; //创造一个范例网 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 datarn"); tempPtr = (int**) malloc (numNodes * sizeof (int*)); for(i = 0; i < numNodes; i++){ tempPtr[i] = (int*)malloc(numNodes * sizeof(int)); } 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]; } } } printf("Data readyrn"); NetPtr resultNetPtr = initNet(numNodes, tempPtr); return resultNetPtr; } //初始化网 NetPtr initNet(int paraSize, int** paraData){ int i, j; NetPtr resultPtr = (NetPtr) malloc (sizeof (Net)); resultPtr -> numNodes = paraSize; //开辟二维空间 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++){ //储存值 resultPtr -> weights[i][j] = paraData[i][j]; } } return resultPtr; } //生成树的Prim算法，或最近路径的Dijkstra算法 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)); //本质上是布尔型的 int *visitedArray = (int*) malloc (numNodes * sizeof (int)); //初始化。任何结点都可当作起点 for(i = 0; i < numNodes; i++){ distanceArray[i] = paraPtr->weights[source][i]; parentArray[i] = source; visitedArray[i] = 0; } distanceArray[source] = 0; parentArray[source] = -1; visitedArray[source] = 1; //主回路 tempBestNode = -1; for(i = 0; i < numNodes - 1; i++){ //找到最近的结点 minDistance = MAX_DISTANCE; for(j = 0; j < numNodes; j++){ //判断结点是否已访问 if(visitedArray[j]){ continue; } if(minDistance > distanceArray[j]){ minDistance = distanceArray[j]; tempBestNode = j; } } visitedArray[tempBestNode] = 1; //准备下一轮循环 for(j = 0; j < numNodes; j++){ //判断结点是否已访问 if(visitedArray[j]){ continue; } //判断结点是否能到达 if(paraPtr->weights[tempBestNode][j] >= MAX_DISTANCE){ continue; } //注意：Dijkstra和Prim算法之间的差异 if(paraAlgorithm == 0){ if(distanceArray[j] > distanceArray[tempBestNode] + paraPtr->weights[tempBestNode][j]){ //更改与起点距离 distanceArray[j] = distanceArray[tempBestNode] + paraPtr->weights[tempBestNode][j]; //更改父级 parentArray[j] = tempBestNode; } } else { if(distanceArray[j] > paraPtr->weights[tempBestNode][j]){ //更改与新起点距离 distanceArray[j] = paraPtr->weights[tempBestNode][j]; //更改父级 parentArray[j] = tempBestNode; } } } } printf("the parent of each node: "); for(i = 0; i < numNodes; i++){ printf("%d, ", parentArray[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]); } } else { resultCost = 0; for(i = 0; i < numNodes; i++){ resultCost += distanceArray[i]; printf("cost of node %d is %d, total = %drn", i, distanceArray[i], resultCost); } printf("Finally, the total cost is %d.rn ", resultCost); } printf("rn"); return resultCost; } //测试 void testPrim(){ NetPtr tempNetPtr = constructSampleNet(); printf("=====Dijkstra algorithm=====rn"); dijkstraOrPrim(tempNetPtr, 0); printf("=====Prim algorithm=====rn"); dijkstraOrPrim(tempNetPtr, 1); } //main int main(){ testPrim(); return 0; }
三、小结
都是使用贪心算法和线性规划，每一步都是选择权值/花费最小的边。
贪心算法：一个局部最有解也是全局最优解；
线性规划：主问题包含n个子问题，而且其中有重叠的子问题。
Dijkstra算法通过线性规划缓存了最优子路径的解，每一步也通过贪心算法来选择最小的边。
Prim算法通过贪心算法来选择最小的边，而Prim的每个子树都是最小生成树说明满足线性规划的两个条件。
