我在过去一天一直在努力完成的编程课上遇到了这个问题……而且我真的不知道该怎么做。
我了解 Prim 算法的基本概念：
1. Start at an arbitrary node (the first node will do) and
add all of its links onto a list.
2. Add the smallest link (which doesn't duplicate an existing path)
in the MST, to the Minimum Spanning Tree.
Remove this link from the list.
3. Add all of the links from the newly linked node onto the list
4. repeat steps 2 & 3 until MST is achieved
(there are no nodes left unconnected).
我已经获得了一个 Graph 的实现（使用邻接列表）来实现 Prim 的算法。问题是我不太了解实现。到目前为止，我对实现的理解如下：
作为一个邻接列表，我们有数组形式的所有节点： 链接到它的是一个链接列表，包含权重、目的地和指向特定节点其余链接的指针的详细信息：
看起来有点像这样的东西：
[0] -> [weight = 1][Destination = 3] -> [weight = 6][Destination = 4][NULL]
[1] -> [weight = 4][Destination = 3][NULL]
and so on...
我们还有一个“Edge”结构，我认为它应该使实现变得更简单，但我并没有真正看到它。
这是给出的代码：
GRAPH.h 界面：
typedef struct {
int v;
int w;
int weight;
} Edge;
Edge EDGE (int, int, int);
typedef struct graph *Graph;
Graph GRAPHinit (int);
void GRAPHinsertE (Graph, Edge);
void GRAPHremoveE (Graph, Edge);
int GRAPHedges (Edge [], Graph g);
Graph GRAPHcopy (Graph);
void GRAPHdestroy (Graph);
int GRAPHedgeScan (Edge *);
void GRAPHEdgePrint (Edge);
int GRAPHsearch (Graph, int[]);
Graph GRAPHmst (Graph);
Graph GRAPHmstPrim (Graph);
#define maxV 8
GRAPH.c 实现：
#include <stdlib.h>
#include <stdio.h>
#include "GRAPH.h"
#define exch(A, B) { Edge t = A; A = B; B = t; }
#define max(A,B)(A>B?A:B)
#define min(A,B)(A<B?A:B)
typedef struct node *link;
struct node {
int v;
int weight;
link next;
};
struct graph {
int V;
int E;
link *adj;
};
static void sortEdges (Edge *edges, int noOfEdges);
static void updateConnectedComponent (Graph g, int from, int to, int newVal, int *connectedComponent);
Edge EDGE (int v, int w, int weight) {
Edge e = {v, w, weight};
return e;
}
link NEW (int v, int weight, link next) {
link x = malloc (sizeof *x);
x->v = v;
x->next = next;
x->weight = weight;
return x;
}
Graph GRAPHinit (int V) {
int v;
Graph G = malloc (sizeof *G);
// Set the size of the graph, = number of verticies
G->V = V;
G->E = 0;
G->adj = malloc (V * sizeof(link));
for (v = 0; v < V; v++){
G->adj[v] = NULL;
}
return G;
}
void GRAPHdestroy (Graph g) {
// not implemented yet
}
void GRAPHinsertE(Graph G, Edge e){
int v = e.v;
int w = e.w;
int weight = e.weight;
G->adj[v] = NEW (w, weight, G->adj[v]);
G->adj[w] = NEW (v, weight, G->adj[w]);
G->E++;
}
void GRAPHremoveE(Graph G, Edge e){
int v = e.v;
int w = e.w;
link *curr;
curr = &G->adj[w];
while (*curr != NULL){
if ((*curr)->v == v) {
(*curr) = (*curr)->next;
G->E--;
break;
}
curr= &((*curr)->next);
}
curr = &G->adj[v];
while (*curr != NULL){
if ((*curr)->v == w) {
(*curr) = (*curr)->next;
break;
}
curr= &((*curr)->next);
}
}
int GRAPHedges (Edge edges[], Graph g) {
int v, E = 0;
link t;
for (v = 0; v < g->V; v++) {
for (t = g->adj[v]; t != NULL; t = t->next) {
if (v < t->v) {
edges[E++] = EDGE(v, t->v, t->weight);
}
}
}
return E;
}
void GRAPHEdgePrint (Edge edge) {
printf ("%d -- (%d) -- %d", edge.v, edge.weight, edge.w);
}
int GRAPHedgeScan (Edge *edge) {
if (edge == NULL) {
printf ("GRAPHedgeScan: called with NULL \n");
abort();
}
if ((scanf ("%d", &(edge->v)) == 1) &&
(scanf ("%d", &(edge->w)) == 1) &&
(scanf ("%d", &(edge->weight)) == 1)) {
return 1;
} else {
return 0;
}
}
// Update the CC label for all the nodes in the MST reachable through the edge from-to
// Assumes graph is a tree, will not terminate otherwise.
void updateConnectedComponent (Graph g, int from, int to, int newVal, int *connectedComponent) {
link currLink = g->adj[to];
connectedComponent[to] = newVal;
while (currLink != NULL) {
if (currLink->v != from) {
updateConnectedComponent (g, to, currLink->v, newVal, connectedComponent);
}
currLink = currLink->next;
}
}
// insertion sort, replace with O(n * lon n) alg to get
// optimal work complexity for Kruskal
void sortEdges (Edge *edges, int noOfEdges) {
int i;
int l = 0;
int r = noOfEdges-1;
for (i = r-1; i >= l; i--) {
int j = i;
while ((j < r) && (edges[j].weight > edges[j+1].weight)) {
exch (edges[j], edges[j+1]);
j++;
}
}
}
Graph GRAPHmst (Graph g) {
Edge *edgesSorted;
int i;
int *connectedComponent = malloc (sizeof (int) * g->V);
int *sizeOfCC = malloc (sizeof (int) * g->V);
Graph mst = GRAPHinit (g->V);
edgesSorted = malloc (sizeof (*edgesSorted) * g->E);
GRAPHedges (edgesSorted, g);
sortEdges (edgesSorted, g->E);
// keep track of the connected component each vertex belongs to
// in the current MST. Initially, MST is empty, so no vertex is
// in an MST CC, therefore all are set to -1.
// We also keep track of the size of each CC, so that we're able
// to identify the CC with fewer vertices when merging two CCs
for (i = 0; i < g->V; i++) {
connectedComponent[i] = -1;
sizeOfCC[i] = 0;
}
int currentEdge = 0; // the shortest edge not yet in the mst
int mstCnt = 0; // no of edges currently in the mst
int v, w;
// The MST can have at most min (g->E, g->V-1) edges
while ((currentEdge < g->E) && (mstCnt < g->V)) {
v = edgesSorted[currentEdge].v;
w = edgesSorted[currentEdge].w;
printf ("Looking at Edge ");
GRAPHEdgePrint (edgesSorted[currentEdge]);
if ((connectedComponent[v] == -1) ||
(connectedComponent[w] == -1)) {
GRAPHinsertE (mst, edgesSorted[currentEdge]);
mstCnt++;
if (connectedComponent[v] == connectedComponent[w]) {
connectedComponent[v] = mstCnt;
connectedComponent[w] = mstCnt;
sizeOfCC[mstCnt] = 2; // initialise a new CC
} else {
connectedComponent[v] = max (connectedComponent[w], connectedComponent[v]);
connectedComponent[w] = max (connectedComponent[w], connectedComponent[v]);
sizeOfCC[connectedComponent[w]]++;
}
printf (" is in MST\n");
} else if (connectedComponent[v] == connectedComponent[w]) {
printf (" is not in MST\n");
} else {
printf (" is in MST, connecting two msts\n");
GRAPHinsertE (mst, edgesSorted[currentEdge]);
mstCnt++;
// update the CC label of all the vertices in the smaller CC
// (size is only important for performance, not correctness)
if (sizeOfCC[connectedComponent[w]] > sizeOfCC[connectedComponent[v]]) {
updateConnectedComponent (mst, v, v, connectedComponent[w], connectedComponent);
sizeOfCC[connectedComponent[w]] += sizeOfCC[connectedComponent[v]];
} else {
updateConnectedComponent (mst, w, w, connectedComponent[v], connectedComponent);
sizeOfCC[connectedComponent[v]] += sizeOfCC[connectedComponent[w]];
}
}
currentEdge++;
}
free (edgesSorted);
free (connectedComponent);
free (sizeOfCC);
return mst;
}
// my code so far
Graph GRAPHmstPrim (Graph g) {
// Initializations
Graph mst = GRAPHinit (g->V); // graph to hold the MST
int i = 0;
int nodeIsConnected[g->V];
// initially all nodes are not connected, initialize as 0;
for(i = 0; i < g->V; i++) {
nodeIsConnected[i] = 0;
}
// extract the first vertex from the graph
nodeIsConnected[0] = 1;
// push all of the links from the first node onto a temporary list
link tempList = newList();
link vertex = g->adj[0];
while(vertex != NULL) {
tempList = prepend(tempList, vertex);
vertex = vertex->next;
}
// find the smallest link from the node;
mst->adj[0] =
}
// some helper functions I've been writing
static link newList(void) {
return NULL;
}
static link prepend(link list, link node) {
link temp = list;
list = malloc(sizeof(list));
list->v = node->v;
list->weigth = node->weight;
list->next = temp;
return list;
}
static link getSmallest(link list, int nodeIsConnected[]) {
link smallest = list;
while(list != NULL){
if((list->weight < smallest->weight)&&(nodeIsConnected[list->v] == 0)) {
smallest = list;
}
list = list->next;
}
if(nodeIsConnected[smallest->v] != 0) {
return NULL;
} else {
return smallest;
}
}
为清楚起见，file 以从文件中获取测试数据：
#include <stdlib.h>
#include <stdio.h>
#include "GRAPH.h"
// call with graph_e1.txt as input, for example.
//
int main (int argc, char *argv[]) {
Edge e, *edges;
Graph g, mst;
int graphSize, i, noOfEdges;
if (argc < 2) {
printf ("No size provided - setting max. no of vertices to %d\n", maxV);
graphSize = maxV;
} else {
graphSize = atoi (argv[1]);
}
g = GRAPHinit (graphSize);
printf ("Reading graph edges (format: v w weight) from stdin\n");
while (GRAPHedgeScan (&e)) {
GRAPHinsertE (g, e);
}
edges = malloc (sizeof (*edges) * graphSize * graphSize);
noOfEdges = GRAPHedges (edges, g);
printf ("Edges of the graph:\n");
for (i = 0; i < noOfEdges; i++) {
GRAPHEdgePrint (edges[i]);
printf ("\n");
}
mst = GRAPHmstPrim (g);
noOfEdges = GRAPHedges (edges, mst);
printf ("\n MST \n");
for (i = 0; i < noOfEdges; i++) {
GRAPHEdgePrint (edges[i]);
printf ("\n");
}
GRAPHdestroy (g);
GRAPHdestroy (mst);
free (edges);
return EXIT_SUCCESS;
}
提前致谢。
卢克
完整文件：http ://www.cse.unsw.edu.au/~cs1927/12s2/labs/13/MST.html
更新：我对这个问题进行了另一次尝试。这是更新的代码（上面的一个编辑将graph_client.c更改为使用我编写的“GRAPHmstPrim”函数。
GRAPH_adjlist.c::
#include <stdlib.h>
#include <stdio.h>
#include "GRAPH.h"
#define exch(A, B) { Edge t = A; A = B; B = t; }
#define max(A,B)(A>B?A:B)
#define min(A,B)(A<B?A:B)
typedef struct _node *link;
struct _node {
int v;
int weight;
link next;
}node;
struct graph {
int V;
int E;
link *adj;
};
typedef struct _edgeNode *edgeLink;
struct _edgeNode {
int v;
int w;
int weight;
edgeLink next;
}edgeNode;
static void sortEdges (Edge *edges, int noOfEdges);
static void updateConnectedComponent (Graph g, int from, int to, int newVal, int *connectedComponent);
Edge EDGE (int v, int w, int weight) {
Edge e = {v, w, weight};
return e;
}
link NEW (int v, int weight, link next) {
link x = malloc (sizeof *x);
x->v = v;
x->next = next;
x->weight = weight;
return x;
}
Graph GRAPHinit (int V) {
int v;
Graph G = malloc (sizeof *G);
G->V = V;
G->E = 0;
G->adj = malloc (V * sizeof(link));
for (v = 0; v < V; v++){
G->adj[v] = NULL;
}
return G;
}
void GRAPHdestroy (Graph g) {
// not implemented yet
}
void GRAPHinsertE(Graph G, Edge e){
int v = e.v;
int w = e.w;
int weight = e.weight;
G->adj[v] = NEW (w, weight, G->adj[v]);
G->adj[w] = NEW (v, weight, G->adj[w]);
G->E++;
}
void GRAPHremoveE(Graph G, Edge e){
int v = e.v;
int w = e.w;
link *curr;
curr = &G->adj[w];
while (*curr != NULL){
if ((*curr)->v == v) {
(*curr) = (*curr)->next;
G->E--;
break;
}
curr= &((*curr)->next);
}
curr = &G->adj[v];
while (*curr != NULL){
if ((*curr)->v == w) {
(*curr) = (*curr)->next;
break;
}
curr= &((*curr)->next);
}
}
int GRAPHedges (Edge edges[], Graph g) {
int v, E = 0;
link t;
for (v = 0; v < g->V; v++) {
for (t = g->adj[v]; t != NULL; t = t->next) {
if (v < t->v) {
edges[E++] = EDGE(v, t->v, t->weight);
}
}
}
return E;
}
void GRAPHEdgePrint (Edge edge) {
printf ("%d -- (%d) -- %d", edge.v, edge.weight, edge.w);
}
int GRAPHedgeScan (Edge *edge) {
if (edge == NULL) {
printf ("GRAPHedgeScan: called with NULL \n");
abort();
}
if ((scanf ("%d", &(edge->v)) == 1) &&
(scanf ("%d", &(edge->w)) == 1) &&
(scanf ("%d", &(edge->weight)) == 1)) {
return 1;
} else {
return 0;
}
}
// Update the CC label for all the nodes in the MST reachable through the edge from-to
// Assumes graph is a tree, will not terminate otherwise.
void updateConnectedComponent (Graph g, int from, int to, int newVal, int *connectedComponent) {
link currLink = g->adj[to];
connectedComponent[to] = newVal;
while (currLink != NULL) {
if (currLink->v != from) {
updateConnectedComponent (g, to, currLink->v, newVal, connectedComponent);
}
currLink = currLink->next;
}
}
// insertion sort, replace with O(n * lon n) alg to get
// optimal work complexity for Kruskal
void sortEdges (Edge *edges, int noOfEdges) {
int i;
int l = 0;
int r = noOfEdges-1;
for (i = r-1; i >= l; i--) {
int j = i;
while ((j < r) && (edges[j].weight > edges[j+1].weight)) {
exch (edges[j], edges[j+1]);
j++;
}
}
}
Graph GRAPHmst (Graph g) {
Edge *edgesSorted;
int i;
int *connectedComponent = malloc (sizeof (int) * g->V);
int *sizeOfCC = malloc (sizeof (int) * g->V);
Graph mst = GRAPHinit (g->V);
edgesSorted = malloc (sizeof (*edgesSorted) * g->E);
GRAPHedges (edgesSorted, g);
sortEdges (edgesSorted, g->E);
// keep track of the connected component each vertex belongs to
// in the current MST. Initially, MST is empty, so no vertex is
// in an MST CC, therefore all are set to -1.
// We also keep track of the size of each CC, so that we're able
// to identify the CC with fewer vertices when merging two CCs
for (i = 0; i < g->V; i++) {
connectedComponent[i] = -1;
sizeOfCC[i] = 0;
}
int currentEdge = 0; // the shortest edge not yet in the mst
int mstCnt = 0; // no of edges currently in the mst
int v, w;
// The MST can have at most min (g->E, g->V-1) edges
while ((currentEdge < g->E) && (mstCnt < g->V)) {
v = edgesSorted[currentEdge].v;
w = edgesSorted[currentEdge].w;
printf ("Looking at Edge ");
GRAPHEdgePrint (edgesSorted[currentEdge]);
if ((connectedComponent[v] == -1) ||
(connectedComponent[w] == -1)) {
GRAPHinsertE (mst, edgesSorted[currentEdge]);
mstCnt++;
if (connectedComponent[v] == connectedComponent[w]) {
connectedComponent[v] = mstCnt;
connectedComponent[w] = mstCnt;
sizeOfCC[mstCnt] = 2; // initialise a new CC
} else {
connectedComponent[v] = max (connectedComponent[w], connectedComponent[v]);
connectedComponent[w] = max (connectedComponent[w], connectedComponent[v]);
sizeOfCC[connectedComponent[w]]++;
}
printf (" is in MST\n");
} else if (connectedComponent[v] == connectedComponent[w]) {
printf (" is not in MST\n");
} else {
printf (" is in MST, connecting two msts\n");
GRAPHinsertE (mst, edgesSorted[currentEdge]);
mstCnt++;
// update the CC label of all the vertices in the smaller CC
// (size is only important for performance, not correctness)
if (sizeOfCC[connectedComponent[w]] > sizeOfCC[connectedComponent[v]]) {
updateConnectedComponent (mst, v, v, connectedComponent[w], connectedComponent);
sizeOfCC[connectedComponent[w]] += sizeOfCC[connectedComponent[v]];
} else {
updateConnectedComponent (mst, w, w, connectedComponent[v], connectedComponent);
sizeOfCC[connectedComponent[v]] += sizeOfCC[connectedComponent[w]];
}
}
currentEdge++;
}
free (edgesSorted);
free (connectedComponent);
free (sizeOfCC);
return mst;
}
edgeLink newEdgeList(void) {
return NULL;
}
edgeLink addEdgeList(edgeLink list, int node, link edge) {
printf("EdgeListStart");
edgeLink temp = list;
list = malloc(sizeof(edgeNode));
list->w = node;
list->v = edge->v;
list->weight = edge->weight;
list->next = temp;
printf("EdgeListEnd");
return list;
}
edgeLink findSmallest(edgeLink waitList, int nodeIsConnected[]) {
printf("SmallestSTart");
edgeLink smallest = waitList;
int small = 99999;
while(waitList != NULL) {
if((waitList->weight < small)&&(nodeIsConnected[waitList->v] == 0)) {
smallest = waitList;
small = smallest->weight;
} else {
printf("\n\n smallest already used %d", waitList->v);
}
waitList = waitList->next;
}
printf("SmallestEnd");
if(nodeIsConnected[smallest->v] == 0){
return smallest;
} else {
printf("Returning NULL");
return NULL;
}
}
link addList(edgeLink smallest, link list, int v) {
printf(":istsatt");
link temp = list;
list = malloc(sizeof(node));
list->v = v;
list->weight = smallest->weight;
list->next = temp;
printf("Listend");
return list;
}
Graph GRAPHmstPrim (Graph g) {
Graph mst = GRAPHinit (g->V); // graph to hold the MST
int i = 0;
int v = 0;
int w = 0;
int nodeIsConnected[g->V]; // array to hold whether a vertex has been added to MST
int loopStarted = 0;
edgeLink smallest = NULL;
// initially all nodes are not in the MST
for(i = 0; i < g->V; i++) {
nodeIsConnected[i] = 0;
}
while((smallest != NULL)||(loopStarted == 0)) {
printf("v is : %d", v);
// add the very first node to the MST
nodeIsConnected[v] = 1;
loopStarted = 1;
// push all of its links onto the list
link vertex = g->adj[v];
edgeLink waitList = newEdgeList();
while(vertex != NULL) {
waitList = addEdgeList(waitList, v, vertex);
vertex = vertex->next;
}
// find the smallest edge from the list
// which doesn't duplicate a connection
smallest = findSmallest(waitList, nodeIsConnected);
// no nodes don't duplicate a connection
// return the current MST
if(smallest == NULL){
return mst;
}
// otherwise add the attributes to the MST graph
w = smallest->w;
v = smallest->v;
mst->adj[v] = addList(smallest, mst->adj[v], w);
mst->adj[w] = addList(smallest, mst->adj[w], v);
}
return mst;
}
更改摘要： - 添加了 edgeList 以保存可能进入 MST 的边 - 数组 nodeIsConnected[] 以跟踪节点是否在 MST 中 - 选择最小节点的函数。如果没有不复制链接的节点，则返回 NULL