寫在前面:
我也是一名java語言的愛好者,僅以此文作爲學習的記錄,對於文中出現的代碼規範,代碼格式,算法效率等問題,希望各路大神不吝賜教,在下感激不盡。同是學習的同學也同樣希望互相交流,取長補短。
大體上來說,一個無向圖G的最小生成樹就是由該圖的那些連接G的所有頂點的邊構成的樹,且其總價值最低
Prim算法
計算最小生成樹的一種方法是使其連續的一步步的長成,每一步都要把一個節點當作根並往上加邊。在算法的任意時刻,我們都可以看到一組已經添加到樹上的節點,而其餘頂點尚未加到這棵樹中。此時,算法在每一時刻都可以通過選擇邊(u,v)使得(u,v)的值是所有u在樹上但v不在樹上的邊的值中的最小值而找出一個新的頂點添加到這棵樹中。
算法設計:
Prim算法基本和求最短路徑的Dijsktra算法相同,設計數據類型時,對每個頂點保留dist和path以及一個known指標(表示頂點已知或未知)。這裏dist是連接到已知頂點最短邊的權,path是導致該頂點改變的最後頂點。
數據類型:
private static int MaxLen = 1000;
private static Vertex[] verArray = new Vertex[7];
class Vertex{
public List<Integer> adjacency;
public boolean known;
public int dist;
public Vertex path;
public int vname;
public Vertex(){
known = false;
dist = MaxLen;
path = null;
}
}
算法僞代碼
//Prim算法僞代碼:
/**
* Prim(Vertex s){
* s.dist = 0;
* while(there is an unknown vertex here){
* Vertex v = the smallest dist vertex;
* v.known = true;
* for each Vertex w adjacent to v if( !w.known ) {
* int dvw = the distance bewteen v and w;
* if(dvw<w.dist){
* w.dist = dvw;
* w.path = v;
* }
* }
* }
* }
*
*/
Java語言實現Prim算法:
import java.util.ArrayList;
import java.util.Arrays;
import java.util.List;
public class PrimA {
/**
* @param args
*/
private static int MaxLen = 1000;
private static Vertex[] verArray = new Vertex[7];
class Vertex{
public List<Integer> adjacency;
public boolean known;
public int dist;
public Vertex path;
public int vname;
public Vertex(){
known = false;
dist = MaxLen;
path = null;
}
}
//判斷是否還有unknown vertex存在
public static boolean Judge(){
for (Vertex v : verArray) {
if (v.known==false) {
return true;
}
}
return false;
}
//返回Dist最小的unknown vertex
public static Vertex SmallestVertex(){
int Vnum = 0;
int value = MaxLen;
for(int i=0;i<7;i++){
if((!verArray[i].known)&&(verArray[i].dist<value)){
Vnum = i;
value = verArray[i].dist;
}
}
return verArray[Vnum];
}
public static void Prim(Vertex s){
s.dist = 0;
while(Judge()){
Vertex v = SmallestVertex();
v.known = true;
for (Integer i : v.adjacency) {
if ((!verArray[v.adjacency.indexOf(i)].known)&&(i<verArray[v.adjacency.indexOf(i)].dist)) {
verArray[v.adjacency.indexOf(i)].dist = i;
verArray[v.adjacency.indexOf(i)].path = v;
}
}
}
}
public static void printTree(Vertex s){
System.out.println("最小生成樹爲:");
for (Vertex v : verArray) {
if (v!=s) {
System.out.println("V"+v.vname+"-----"+"V"+v.path.vname+" "+"邊權值爲"+v.adjacency.get(v.path.vname));
}
}
}
public static void main(String[] args) {
// TODO Auto-generated method stub
//初始化
verArray[0] = new PrimA().new Vertex();
verArray[1] = new PrimA().new Vertex();
verArray[2] = new PrimA().new Vertex();
verArray[3] = new PrimA().new Vertex();
verArray[4] = new PrimA().new Vertex();
verArray[5] = new PrimA().new Vertex();
verArray[6] = new PrimA().new Vertex();
verArray[0].adjacency = new ArrayList<Integer>(Arrays.asList(0,2,4,1,MaxLen,MaxLen,MaxLen));
verArray[0].vname = 0;
verArray[1].adjacency = new ArrayList<Integer>(Arrays.asList(2,0,MaxLen,3,10,MaxLen,MaxLen));
verArray[1].vname = 1;
verArray[2].adjacency = new ArrayList<Integer>(Arrays.asList(4,MaxLen,0,2,MaxLen,5,MaxLen));
verArray[2].vname = 2;
verArray[3].adjacency = new ArrayList<Integer>(Arrays.asList(1,3,2,0,7,8,4));
verArray[3].vname = 3;
verArray[4].adjacency = new ArrayList<Integer>(Arrays.asList(MaxLen,10,MaxLen,7,0,MaxLen,6));
verArray[4].vname = 4;
verArray[5].adjacency = new ArrayList<Integer>(Arrays.asList(MaxLen,MaxLen,5,8,MaxLen,0,1));
verArray[5].vname = 5;
verArray[6].adjacency = new ArrayList<Integer>(Arrays.asList(MaxLen,MaxLen,MaxLen,4,6,1,0));
verArray[6].vname = 6;
//起點從v0開始
Prim(verArray[0]);
printTree(verArray[0]);
}
}
運行結果: