转载请注明出处：勿在浮沙筑高台http://blog.csdn.net/luoshixian099/article/details/51908175css
关于图的几个概念定义：node
- 连通图：在无向图中，若任意两个顶点vi” role=”presentation”>vivi都有路径相通，则称该无向图为连通图。
- 强连通图：在有向图中，若任意两个顶点vi” role=”presentation”>vivi都有路径相通，则称该有向图为强连通图。
- 连通网：在连通图中，若图的边具备必定的意义，每一条边都对应着一个数，称为权；权表明着链接连个顶点的代价，称这种连通图叫作连通网。
- 生成树：一个连通图的生成树是指一个连通子图，它含有图中所有n个顶点，但只有足以构成一棵树的n-1条边。一颗有n个顶点的生成树有且仅有n-1条边，若是生成树中再添加一条边，则一定成环。
- 最小生成树：在连通网的全部生成树中，全部边的代价和最小的生成树，称为最小生成树。
下面介绍两种求最小生成树算法ios
1.Kruskal算法
此算法能够称为“加边法”，初始最小生成树边数为0，每迭代一次就选择一条知足条件的最小代价边，加入到最小生成树的边集合里。
1. 把图中的全部边按代价从小到大排序；
2. 把图中的n个顶点当作独立的n棵树组成的森林；
3. 按权值从小到大选择边，所选的边链接的两个顶点ui,vi” role=”presentation”>ui,viui,vi,应属于两颗不一样的树，则成为最小生成树的一条边，并将这两颗树合并做为一颗树。
4. 重复(3),直到全部顶点都在一颗树内或者有n-1条边为止。web
2.Prim算法
此算法能够称为“加点法”，每次迭代选择代价最小的边对应的点，加入到最小生成树中。算法从某一个顶点s开始，逐渐长大覆盖整个连通网的全部顶点。算法
- 图的全部顶点集合为V” role=”presentation”>VV;
- 在两个集合u,v” role=”presentation”>u,vu,v并入到集合u中。
- 重复上述步骤，直到最小生成树有n-1条边或者n个顶点为止。
因为不断向集合u中加点，因此最小代价边必须同步更新；须要创建一个辅助数组closedge,用来维护集合v中每一个顶点与集合u中最小代价边信息，：数组
struct
{
char vertexData
UINT lowestcost
}closedge[vexCounts]
3.完整代码
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#include <iostream>
#include <vector>
#include <queue>
#include <algorithm>
using namespace std;
#define INFINITE 0xFFFFFFFF
#define VertexData unsigned int
#define UINT unsigned int
#define vexCounts 6
char vextex[] = {
'A',
'B',
'C',
'D',
'E',
'F' };
struct node
{
VertexData data;
unsigned int lowestcost;
}closedge[vexCounts];
typedef struct
{
VertexData u;
VertexData v;
unsigned int cost;
}Arc;
void AdjMatrix(
unsigned int adjMat[][vexCounts])
{
for (
int i =
0; i < vexCounts; i++)
for (
int j =
0; j < vexCounts; j++)
{
adjMat[i][j] = INFINITE;
}
adjMat[
0][
1] =
6; adjMat[
0][
2] =
1; adjMat[
0][
3] =
5;
adjMat[
1][
0] =
6; adjMat[
1][
2] =
5; adjMat[
1][
4] =
3;
adjMat[
2][
0] =
1; adjMat[
2][
1] =
5; adjMat[
2][
3] =
5; adjMat[
2][
4] =
6; adjMat[
2][
5] =
4;
adjMat[
3][
0] =
5; adjMat[
3][
2] =
5; adjMat[
3][
5] =
2;
adjMat[
4][
1] =
3; adjMat[
4][
2] =
6; adjMat[
4][
5] =
6;
adjMat[
5][
2] =
4; adjMat[
5][
3] =
2; adjMat[
5][
4] =
6;
}
int Minmum(
struct node * closedge)
{
unsigned int min = INFINITE;
int index = -
1;
for (
int i =
0; i < vexCounts;i++)
{
if (closedge[i].lowestcost < min && closedge[i].lowestcost !=
0)
{
min = closedge[i].lowestcost;
index = i;
}
}
return index;
}
void MiniSpanTree_Prim(
unsigned int adjMat[][vexCounts], VertexData s)
{
for (
int i =
0; i < vexCounts;i++)
{
closedge[i].lowestcost = INFINITE;
}
closedge[s].data = s;
closedge[s].lowestcost =
0;
for (
int i =
0; i < vexCounts;i++)
{
if (i != s)
{
closedge[i].data = s;
closedge[i].lowestcost = adjMat[s][i];
}
}
for (
int e =
1; e <= vexCounts -
1; e++)
{
int k = Minmum(closedge);
cout << vextex[closedge[k].data] <<
"--" << vextex[k] << endl;
closedge[k].lowestcost =
0;
for (
int i =
0; i < vexCounts;i++)
{
if ( adjMat[k][i] < closedge[i].lowestcost)
{
closedge[i].data = k;
closedge[i].lowestcost = adjMat[k][i];
}
}
}
}
void ReadArc(
unsigned int adjMat[][vexCounts],
vector<Arc> &vertexArc)
{
Arc * temp = NULL;
for (
unsigned int i =
0; i < vexCounts;i++)
{
for (
unsigned int j =
0; j < i; j++)
{
if (adjMat[i][j]!=INFINITE)
{
temp =
new Arc;
temp->u = i;
temp->v = j;
temp->cost = adjMat[i][j];
vertexArc.push_back(*temp);
}
}
}
}
bool compare(Arc A, Arc B)
{
return A.cost < B.cost ?
true :
false;
}
bool FindTree(VertexData u, VertexData v,
vector<vector<VertexData> > &Tree)
{
unsigned int index_u = INFINITE;
unsigned int index_v = INFINITE;
for (
unsigned int i =
0; i < Tree.size();i++)
{
if (find(Tree[i].begin(), Tree[i].end(), u) != Tree[i].end())
index_u = i;
if (find(Tree[i].begin(), Tree[i].end(), v) != Tree[i].end())
index_v = i;
}
if (index_u != index_v)
{
for (
unsigned int i =
0; i < Tree[index_v].size();i++)
{
Tree[index_u].push_back(Tree[index_v][i]);
}
Tree[index_v].clear();
return true;
}
return false;
}
void MiniSpanTree_Kruskal(
unsigned int adjMat[][vexCounts])
{
vector<Arc> vertexArc;
ReadArc(adjMat, vertexArc);
sort(vertexArc.begin(), vertexArc.end(), compare);
vector<vector<VertexData> > Tree(vexCounts);
for (
unsigned int i =
0; i < vexCounts; i++)
{
Tree[i].push_back(i);
}
for (
unsigned int i =
0; i < vertexArc.size(); i++)
{
VertexData u = vertexArc[i].u;
VertexData v = vertexArc[i].v;
if (FindTree(u, v, Tree))
{
cout << vextex[u] <<
"---" << vextex[v] << endl;
}
}
}
int main()
{
unsigned int adjMat[vexCounts][vexCounts] = {
0 };
AdjMatrix(adjMat);
cout <<
"Prim :" << endl;
MiniSpanTree_Prim(adjMat,
0);
cout <<
"-------------" << endl <<
"Kruskal:" << endl;
MiniSpanTree_Kruskal(adjMat);
return 0;
}
Reference:
数据结构–耿国华
算法导论–第三版markdown
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