Shortest total distance between cities (minimum spanning tree algorithm)

Solving the shortest total distance between cities is a very practical problem with the following effect:

An area of N cities, how to choose a route so that the total distance between cities shortest?

1. Shortest total distance algorithm

First to analyze the above problems. N cities in a region form a traffic map that can be used to describe the problem with the following relationship:

• Each city represents a vertex in the diagram.
• The edges between the two vertices are the paths between the two cities, and the weights of the edges represent the distances between cities.

Thus, the problem of solving the shortest total distance between cities is attributed to the minimum spanning tree problem of the graph.

2. Minimum spanning tree algorithm (prim algorithm)

    //minimum spanning tree algorithm    Static voidprimgraph (Graphmatrix GM) {inti,j,k,min,sum; int[] weight=New int[Graphmatrix.maxnum];//Weighted Value        Char[] vtempx=New Char[Graphmatrix.maxnum];//temporary vertex informationSum=0;  for(I=1;I&LT;GM. vertexnum;i++) {//save a row of data in the adjacency matrixWEIGHT[I]=GM. Edgeweight[0][i]; if(weight[i]==MaxValue) {Vtempx[i]=(Char) NoL; }Else{Vtempx[i]=GM. VERTEX[0];//adjacency Vertex}} vtempx[0]=used;//selectionweight[0]=MaxValue;  for(I=1;I&LT;GM. vertexnum;i++) {min=WEIGHT[0];//Minimum weight valuek=i;  for(J=1;J&LT;GM. vertexnum;j++){                if(weight[j]<min&&vtempx[j]>0) {//to find unused edges with a smaller weighted valueMIN=WEIGHT[J];//Save weight valueK=j;//Save adjacency Point ordinal}} sum+=min;//Weighted value accumulationSystem.out.printf ("(%c,%c),", VTEMPX[K],GM. Vertex[k]);//output spanning tree one edgevtempx[k]=used;//selectionweight[k]=MaxValue;  for(J=0;J&LT;GM. vertexnum;j++) {//re-select the minimum edge                if(GM. Edgeweight[k][j]<weight[j]&&vtempx[j]!=0) {Weight[j]=GM. EDGEWEIGHT[K][J];//Weighted Valuevtempx[j]=GM.                VERTEX[K]; }}} System.out.printf ("\ n The total weight of the minimum spanning tree is:%d\n", sum); }

3. The program code example is as follows:

 Packagecom.cn.datastruct;ImportJava.util.Scanner;//solving the shortest total distance between cities Public classPrim {//defining the adjacency matrix diagram structure    Static classgraphmatrix{Static Final intmaxnum=20;//maximum number of vertices in a graph        Char[] vertex=New Char[Maxnum];//Save vertex information (ordinal or letter)        intGType;//type of diagram (0: No-map, 1: Forward graph)        intVertexnum;//the number of vertices        intEdgenum;//the number of edges        int[] Edgeweight =New int[Maxnum] [Maxnum];//the right to save the edge        int[] Istrav =New int[Maxnum];//Traverse Flag    }        Static Final intmaxvalue=65535;//Maximum value (can be set to a maximum integer)    Static Final intUsed=0;//Selected vertices    Static Final intNol=-1;//non-contiguous vertices    StaticScanner input=NewScanner (system.in); //creating adjacency matrix graphs    Static voidcreategraph (Graphmatrix GM) {inti,j,k; intWeight//Rights        CharESTARTV,EENDV;//The starting vertex of the edgeSystem.out.printf ("Input each vertex information in the graph \ n");  for(I=0;I&LT;GM. vertexnum;i++) {//input verticesSystem.out.printf ("%d vertices:", i+1); GM. Vertex[i]= (Input.next (). ToCharArray ()) [0];//in an array element that is saved to each vertex} System.out.printf ("Enter the vertices and weights that make up each edge: \ n");  for(K=0;K&LT;GM. edgenum;k++) {//Enter the information for the EdgeSystem.out.printf ("section%d:", k+1); Estartv=input.next (). CharAt (0); EENDV=input.next (). CharAt (0); Weight=Input.nextint ();  for(i=0; ESTARTV!=GM. vertex[i];i++);//find a starting point in an existing vertex             for(j=0; EENDV!=GM. vertex[j];j++);//find the end point in an existing vertexGM. Edgeweight[i][j]=weight;//the corresponding position holds the weight value, indicating that there is an edge            if(GM. gtype==0) {//If the graph is notGM. edgeweight[j][i]=weight; }        }    }        //Clear the Matrix    Static voidcleargraph (Graphmatrix GM) {inti,j;  for(I=0;I&LT;GM. vertexnum;i++){             for(J=0;J&LT;GM. vertexnum;j++) {GM. EDGEWEIGHT[I][J]=maxvalue;//sets the value of each element in the matrix to MaxValue            }        }    }        //Output adjacency Matrix    Static voidoutgraph (Graphmatrix GM) {inti,j;  for(J=0;J&LT;GM. vertexnum;j++) {System.out.printf ("\t%c", GM. VERTEX[J]);//output vertex information on the first line} System.out.println ();  for(I=0;I&LT;GM. vertexnum;i++) {System.out.printf ("%c", GM.            Vertex[i]);  for(J=0;J&LT;GM. vertexnum;j++){                if(GM. Edgeweight[i][j]==maxvalue) {//if the weighted value is the maximum valueSystem.out.printf ("\tz");//Infinity is represented by Z}Else{System.out.printf ("\t%d", GM. EDGEWEIGHT[I][J]);//weight of the output edge}} System.out.println (); }    }        //minimum spanning tree algorithm    Static voidprimgraph (Graphmatrix GM) {inti,j,k,min,sum; int[] weight=New int[Graphmatrix.maxnum];//Weighted Value        Char[] vtempx=New Char[Graphmatrix.maxnum];//temporary vertex informationSum=0;  for(I=1;I&LT;GM. vertexnum;i++) {//save a row of data in the adjacency matrixWEIGHT[I]=GM. Edgeweight[0][i]; if(weight[i]==MaxValue) {Vtempx[i]=(Char) NoL; }Else{Vtempx[i]=GM. VERTEX[0];//adjacency Vertex}} vtempx[0]=used;//selectionweight[0]=MaxValue;  for(I=1;I&LT;GM. vertexnum;i++) {min=WEIGHT[0];//Minimum weight valuek=i;  for(J=1;J&LT;GM. vertexnum;j++){                if(weight[j]<min&&vtempx[j]>0) {//to find unused edges with a smaller weighted valueMIN=WEIGHT[J];//Save weight valueK=j;//Save adjacency Point ordinal}} sum+=min;//Weighted value accumulationSystem.out.printf ("(%c,%c),", VTEMPX[K],GM. Vertex[k]);//output spanning tree one edgevtempx[k]=used;//selectionweight[k]=MaxValue;  for(J=0;J&LT;GM. vertexnum;j++) {//re-select the minimum edge                if(GM. Edgeweight[k][j]<weight[j]&&vtempx[j]!=0) {Weight[j]=GM. EDGEWEIGHT[K][J];//Weighted Valuevtempx[j]=GM.                VERTEX[K]; }}} System.out.printf ("\ n The total weight of the minimum spanning tree is:%d\n", sum); }         Public Static voidMain (string[] args) {Graphmatrix GM=NewGraphmatrix ();//defines a diagram that holds the structure of an adjacency table        Charagain;        String go; System.out.println ("Find the smallest spanning tree!" ");  Do{System.out.print ("Please enter the type of build diagram first:"); GM. GType=input.nextint ();//Types of graphsSystem.out.print ("Number of vertices of the input graph:"); GM. Vertexnum=input.nextint ();//number of vertices of the input graphSystem.out.print ("Number of sides of the input graph:"); GM. Edgenum=input.nextint ();//Enter the number of image edgesCleargraph (GM);//Clear DiagramCreategraph (GM);//graphs that generate adjacency table structuresSystem.out.print ("The edge of the minimum spanning tree is:");                        Primgraph (GM); System.out.println ("\ nyou keep playing (y/n)?"); Go=Input.next (); } while(Go.equalsignorecase ("Y")); System.out.println ("The game is over!" "); }}

Shortest total distance between cities (minimum spanning tree algorithm)