算法系列15天速成——第十五天 图【下】（大结局）

今天是大结局，说下“图”的最后一点东西，“最小生成树“和”最短路径“。

一： 最小生成树

1. 概念

首先看如下图，不知道大家能总结点什么。

对于一个连通图G，如果其全部顶点和一部分边构成一个子图G1，当G1满足：

① 刚好将图中所有顶点连通。②顶点不存在回路。则称G1就是G的“生成树”。

其实一句话总结就是：生成树是将原图的全部顶点以最小的边连通的子图，这不，如下的连通图可以得到下面的两个生成树。

② 对于一个带权的连通图，当生成的树不同，各边上的权值总和也不同，如果某个生成树的权值最小，则它就是“最小生成树”。

View Code

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;

namespace MatrixGraph
{
public class Program
{
static void Main(string[] args)
{
MatrixGraphManager manager = new MatrixGraphManager();

//创建图
MatrixGraph graph = manager.CreateMatrixGraph();

manager.OutMatrix(graph);

int sum = 0;

manager.Prim(graph, out sum);

Console.WriteLine("\n最小生成树的权值为：" + sum);

manager.Dijkstra(graph);

//Console.Write("广度递归:\t");

//manager.BFSTraverse(graph);

//Console.Write("\n深度递归:\t");

//manager.DFSTraverse(graph);

Console.ReadLine();

}
}

#region 邻接矩阵的结构图
/// <summary>
/// 邻接矩阵的结构图
/// </summary>
public class MatrixGraph
{
//保存顶点信息
public string[] vertex;

//保存边信息
public int[,] edges;

//深搜和广搜的遍历标志
public bool[] isTrav;

//顶点数量
public int vertexNum;

//边数量
public int edgeNum;

//图类型
public int graphType;

/// <summary>
/// 存储容量的初始化
/// </summary>
/// <param name="vertexNum"></param>
/// <param name="edgeNum"></param>
/// <param name="graphType"></param>
public MatrixGraph(int vertexNum, int edgeNum, int graphType)
{
this.vertexNum = vertexNum;
this.edgeNum = edgeNum;
this.graphType = graphType;

vertex = new string[vertexNum];
edges = new int[vertexNum, vertexNum];
isTrav = new bool[vertexNum];
}

}
#endregion

/// <summary>
/// 图的操作类
/// </summary>
public class MatrixGraphManager
{
#region 图的创建
/// <summary>
/// 图的创建
/// </summary>
/// <param name="g"></param>
public MatrixGraph CreateMatrixGraph()
{
Console.WriteLine("请输入创建图的顶点个数，边个数，是否为无向图(0,1来表示)，已逗号隔开。");

var initData = Console.ReadLine().Split(',').Select(i => int.Parse(i)).ToList();

MatrixGraph graph = new MatrixGraph(initData[0], initData[1], initData[2]);

//我们默认“正无穷大为没有边”
for (int i = 0; i < graph.vertexNum; i++)
{
for (int j = 0; j < graph.vertexNum; j++)
{
graph.edges[i, j] = short.MaxValue;
}
}

Console.WriteLine("请输入各顶点信息：");

for (int i = 0; i < graph.vertexNum; i++)
{
Console.Write("\n第" + (i + 1) + "个顶点为:");

var single = Console.ReadLine();

//顶点信息加入集合中
graph.vertex[i] = single;
}

Console.WriteLine("\n请输入构成两个顶点的边和权值，以逗号隔开。\n");

for (int i = 0; i < graph.edgeNum; i++)
{
Console.Write("第" + (i + 1) + "条边:\t");

initData = Console.ReadLine().Split(',').Select(j => int.Parse(j)).ToList();

int start = initData[0];
int end = initData[1];
int weight = initData[2];

//给矩阵指定坐标位置赋值
graph.edges[start - 1, end - 1] = weight;

//如果是无向图，则数据呈“二，四”象限对称
if (graph.graphType == 1)
{
graph.edges[end - 1, start - 1] = weight;
}
}

return graph;
}
#endregion

#region 输出矩阵数据
/// <summary>
/// 输出矩阵数据
/// </summary>
/// <param name="graph"></param>
public void OutMatrix(MatrixGraph graph)
{
for (int i = 0; i < graph.vertexNum; i++)
{
for (int j = 0; j < graph.vertexNum; j++)
{
if (graph.edges[i, j] == short.MaxValue)
Console.Write("∽\t");
else
Console.Write(graph.edges[i, j] + "\t");
}
//换行
Console.WriteLine();
}
}
#endregion

#region 广度优先
/// <summary>
/// 广度优先
/// </summary>
/// <param name="graph"></param>
public void BFSTraverse(MatrixGraph graph)
{
//访问标记默认初始化
for (int i = 0; i < graph.vertexNum; i++)
{
graph.isTrav[i] = false;
}

//遍历每个顶点
for (int i = 0; i < graph.vertexNum; i++)
{
//广度遍历未访问过的顶点
if (!graph.isTrav[i])
{
BFSM(ref graph, i);
}
}
}

/// <summary>
/// 广度遍历具体算法
/// </summary>
/// <param name="graph"></param>
public void BFSM(ref MatrixGraph graph, int vertex)
{
//这里就用系统的队列
Queue<int> queue = new Queue<int>();

//先把顶点入队
queue.Enqueue(vertex);

//标记此顶点已经被访问
graph.isTrav[vertex] = true;

//输出顶点
Console.Write(" ->" + graph.vertex[vertex]);

//广度遍历顶点的邻接点
while (queue.Count != 0)
{
var temp = queue.Dequeue();

//遍历矩阵的横坐标
for (int i = 0; i < graph.vertexNum; i++)
{
if (!graph.isTrav[i] && graph.edges[temp, i] != 0)
{
graph.isTrav[i] = true;

queue.Enqueue(i);

//输出未被访问的顶点
Console.Write(" ->" + graph.vertex[i]);
}
}
}
}
#endregion

#region 深度优先
/// <summary>
/// 深度优先
/// </summary>
/// <param name="graph"></param>
public void DFSTraverse(MatrixGraph graph)
{
//访问标记默认初始化
for (int i = 0; i < graph.vertexNum; i++)
{
graph.isTrav[i] = false;
}

//遍历每个顶点
for (int i = 0; i < graph.vertexNum; i++)
{
//广度遍历未访问过的顶点
if (!graph.isTrav[i])
{
DFSM(ref graph, i);
}
}
}

#region 深度递归的具体算法
/// <summary>
/// 深度递归的具体算法
/// </summary>
/// <param name="graph"></param>
/// <param name="vertex"></param>
public void DFSM(ref MatrixGraph graph, int vertex)
{
Console.Write("->" + graph.vertex[vertex]);

//标记为已访问
graph.isTrav[vertex] = true;

//要遍历的六个点
for (int i = 0; i < graph.vertexNum; i++)
{
if (graph.isTrav[i] == false && graph.edges[vertex, i] != 0)
{
//深度递归
DFSM(ref graph, i);
}
}
}
#endregion
#endregion

#region prim算法获取最小生成树
/// <summary>
/// prim算法获取最小生成树
/// </summary>
/// <param name="graph"></param>
public void Prim(MatrixGraph graph, out int sum)
{
//已访问过的标志
int used = 0;

//非邻接顶点标志
int noadj = -1;

//定义一个输出总权值的变量
sum = 0;

//临时数组，用于保存邻接点的权值
int[] weight = new int[graph.vertexNum];

//临时数组，用于保存顶点信息
int[] tempvertex = new int[graph.vertexNum];

//取出邻接矩阵的第一行数据，也就是取出第一个顶点并将权和边信息保存于临时数据中
for (int i = 1; i < graph.vertexNum; i++)
{
//保存于邻接点之间的权值
weight[i] = graph.edges[0, i];

//等于0则说明V1与该邻接点没有边
if (weight[i] == short.MaxValue)
tempvertex[i] = noadj;
else
tempvertex[i] = int.Parse(graph.vertex[0]);
}

//从集合V中取出V1节点，只需要将此节点设置为已访问过，weight为0集合
var index = tempvertex[0] = used;
var min = weight[0] = short.MaxValue;

//在V的邻接点中找权值最小的节点
for (int i = 1; i < graph.vertexNum; i++)
{
index = i;
min = short.MaxValue;

for (int j = 1; j < graph.vertexNum; j++)
{
//用于找出当前节点的邻接点中权值最小的未访问点
if (weight[j] < min && tempvertex[j] != 0)
{
min = weight[j];
index = j;
}
}
//累加权值
sum += min;

Console.Write("({0},{1})  ", tempvertex[index], graph.vertex[index]);

//将取得的最小节点标识为已访问
weight[index] = short.MaxValue;
tempvertex[index] = 0;

//从最新的节点出发，将此节点的weight比较赋值
for (int j = 0; j < graph.vertexNum; j++)
{
//已当前节点为出发点，重新选择最小边
if (graph.edges[index, j] < weight[j] && tempvertex[j] != used)
{
weight[j] = graph.edges[index, j];

//这里做的目的将较短的边覆盖点上一个节点的邻接点中的较长的边
tempvertex[j] = int.Parse(graph.vertex[index]);
}
}
}
}
#endregion

#region dijkstra求出最短路径
/// <summary>
/// dijkstra求出最短路径
/// </summary>
/// <param name="g"></param>
public void Dijkstra(MatrixGraph g)
{
int[] weight = new int[g.vertexNum];

int[] path = new int[g.vertexNum];

int[] tempvertex = new int[g.vertexNum];

Console.WriteLine("\n请输入源点的编号：");

//让用户输入要遍历的起始点
int vertex = int.Parse(Console.ReadLine()) - 1;

for (int i = 0; i < g.vertexNum; i++)
{
//初始赋权值
weight[i] = g.edges[vertex, i];

if (weight[i] < short.MaxValue && weight[i] > 0)
path[i] = vertex;

tempvertex[i] = 0;
}

tempvertex[vertex] = 1;
weight[vertex] = 0;

for (int i = 0; i < g.vertexNum; i++)
{
int min = short.MaxValue;

int index = vertex;

for (int j = 0; j < g.vertexNum; j++)
{
//顶点的权值中找出最小的
if (tempvertex[j] == 0 && weight[j] < min)
{
min = weight[j];
index = j;
}
}

tempvertex[index] = 1;

//以当前的index作为中间点，找出最小的权值
for (int j = 0; j < g.vertexNum; j++)
{
if (tempvertex[j] == 0 && weight[index] + g.edges[index, j] < weight[j])
{
weight[j] = weight[index] + g.edges[index, j];
path[j] = index;
}
}
}

Console.WriteLine("\n顶点{0}到各顶点的最短路径为：（终点 < 源点） " + g.vertex[vertex]);

//最后输出
for (int i = 0; i < g.vertexNum; i++)
{
if (tempvertex[i] == 1)
{
var index = i;

while (index != vertex)
{
var j = index;
Console.Write("{0} < ", g.vertex[index]);
index = path[index];
}
Console.WriteLine("{0}\n", g.vertex[index]);
}
else
{
Console.WriteLine("{0} <- {1}: 无路径\n", g.vertex[i], g.vertex[vertex]);
}
}
}
#endregion
}
}

算法速成系列至此就全部结束了，公司给我们的算法培训也于上周五结束，呵呵，赶一下同步。最后希望大家能对算法重视起来，

学好算法，终身收益。