有n个城市，其中一些城市之间有道路。不知何故，所有的道路都同时受损。我们必须修复道路以重新连接城市。修理一条特定的道路是有固定成本的。找出通过修复道路连接所有城市的最低成本。输入为矩阵（城市）形式，如果 city[i][j] = 0 那么城市 i 和城市 j 之间没有任何道路，如果 city[i][j] = a > 0 那么重建城市的成本城市 i 和城市 j 之间的路径是 a。打印出连接所有城市的最低成本。
可以肯定的是，在道路受损之前，所有城市都已连通。
例子：
Input : {{0, 1, 2, 3, 4},
{1, 0, 5, 0, 7},
{2, 5, 0, 6, 0},
{3, 0, 6, 0, 0},
{4, 7, 0, 0, 0}};
Output : 10
Input : {{0, 1, 1, 100, 0, 0},
{1, 0, 1, 0, 0, 0},
{1, 1, 0, 0, 0, 0},
{100, 0, 0, 0, 2, 2},
{0, 0, 0, 2, 0, 2},
{0, 0, 0, 2, 2, 0}};
Output : 106方法：在这里，我们必须通过成本最低的路径连接所有城市。这样做的方法是找出城市地图的最小生成树（MST）（即每个城市是图的一个节点，城市之间所有损坏的道路都是边）。总成本是最小生成树中路径边值的相加。
先决条件： MST Prim 算法
C++
// C++ code to find out minimum cost
// path to connect all the cities
#include
using namespace std;
// Function to find out minimum valued node
// among the nodes which are not yet included in MST
int minnode(int n, int keyval[], bool mstset[]) {
int mini = numeric_limits::max();
int mini_index;
// Loop through all the values of the nodes
// which are not yet included in MST and find
// the minimum valued one.
for (int i = 0; i < n; i++) {
if (mstset[i] == false && keyval[i] < mini) {
mini = keyval[i], mini_index = i;
}
}
return mini_index;
}
// Function to find out the MST and
// the cost of the MST.
void findcost(int n, vector> city) {
// Array to store the parent node of a
// particular node.
int parent[n];
// Array to store key value of each node.
int keyval[n];
// Boolean Array to hold bool values whether
// a node is included in MST or not.
bool mstset[n];
// Set all the key values to infinite and
// none of the nodes is included in MST.
for (int i = 0; i < n; i++) {
keyval[i] = numeric_limits::max();
mstset[i] = false;
}
// Start to find the MST from node 0.
// Parent of node 0 is none so set -1.
// key value or minimum cost to reach
// 0th node from 0th node is 0.
parent[0] = -1;
keyval[0] = 0;
// Find the rest n-1 nodes of MST.
for (int i = 0; i < n - 1; i++) {
// First find out the minimum node
// among the nodes which are not yet
// included in MST.
int u = minnode(n, keyval, mstset);
// Now the uth node is included in MST.
mstset[u] = true;
// Update the values of neighbor
// nodes of u which are not yet
// included in MST.
for (int v = 0; v < n; v++) {
if (city[u][v] && mstset[v] == false &&
city[u][v] < keyval[v]) {
keyval[v] = city[u][v];
parent[v] = u;
}
}
}
// Find out the cost by adding
// the edge values of MST.
int cost = 0;
for (int i = 1; i < n; i++)
cost += city[parent[i]][i];
cout << cost << endl;
}
// Utility Program:
int main() {
// Input 1
int n1 = 5;
vector> city1 = {{0, 1, 2, 3, 4},
{1, 0, 5, 0, 7},
{2, 5, 0, 6, 0},
{3, 0, 6, 0, 0},
{4, 7, 0, 0, 0}};
findcost(n1, city1);
// Input 2
int n2 = 6;
vector> city2 = {{0, 1, 1, 100, 0, 0},
{1, 0, 1, 0, 0, 0},
{1, 1, 0, 0, 0, 0},
{100, 0, 0, 0, 2, 2},
{0, 0, 0, 2, 0, 2},
{0, 0, 0, 2, 2, 0}};
findcost(n2, city2);
return 0;
} Java
// Java code to find out minimum cost
// path to connect all the cities
import java.util.*;
class GFG{
// Function to find out minimum valued node
// among the nodes which are not yet included
// in MST
static int minnode(int n, int keyval[],
boolean mstset[])
{
int mini = Integer.MAX_VALUE;
int mini_index = 0;
// Loop through all the values of the nodes
// which are not yet included in MST and find
// the minimum valued one.
for(int i = 0; i < n; i++)
{
if (mstset[i] == false &&
keyval[i] < mini)
{
mini = keyval[i];
mini_index = i;
}
}
return mini_index;
}
// Function to find out the MST and
// the cost of the MST.
static void findcost(int n, int city[][])
{
// Array to store the parent node of a
// particular node.
int parent[] = new int[n];
// Array to store key value of each node.
int keyval[] = new int[n];
// Boolean Array to hold bool values whether
// a node is included in MST or not.
boolean mstset[] = new boolean[n];
// Set all the key values to infinite and
// none of the nodes is included in MST.
for(int i = 0; i < n; i++)
{
keyval[i] = Integer.MAX_VALUE;
mstset[i] = false;
}
// Start to find the MST from node 0.
// Parent of node 0 is none so set -1.
// key value or minimum cost to reach
// 0th node from 0th node is 0.
parent[0] = -1;
keyval[0] = 0;
// Find the rest n-1 nodes of MST.
for(int i = 0; i < n - 1; i++)
{
// First find out the minimum node
// among the nodes which are not yet
// included in MST.
int u = minnode(n, keyval, mstset);
// Now the uth node is included in MST.
mstset[u] = true;
// Update the values of neighbor
// nodes of u which are not yet
// included in MST.
for(int v = 0; v < n; v++)
{
if (city[u][v] > 0 &&
mstset[v] == false &&
city[u][v] < keyval[v])
{
keyval[v] = city[u][v];
parent[v] = u;
}
}
}
// Find out the cost by adding
// the edge values of MST.
int cost = 0;
for(int i = 1; i < n; i++)
cost += city[parent[i]][i];
System.out.println(cost);
}
// Driver code
public static void main(String args[])
{
// Input 1
int n1 = 5;
int city1[][] = { { 0, 1, 2, 3, 4 },
{ 1, 0, 5, 0, 7 },
{ 2, 5, 0, 6, 0 },
{ 3, 0, 6, 0, 0 },
{ 4, 7, 0, 0, 0 } };
findcost(n1, city1);
// Input 2
int n2 = 6;
int city2[][] = { { 0, 1, 1, 100, 0, 0 },
{ 1, 0, 1, 0, 0, 0 },
{ 1, 1, 0, 0, 0, 0 },
{ 100, 0, 0, 0, 2, 2 },
{ 0, 0, 0, 2, 0, 2 },
{ 0, 0, 0, 2, 2, 0 } };
findcost(n2, city2);
}
}
// This code is contributed by adityapande88Python3
# Python3 code to find out minimum cost
# path to connect all the cities
# Function to find out minimum valued
# node among the nodes which are not
# yet included in MST
def minnode(n, keyval, mstset):
mini = 999999999999
mini_index = None
# Loop through all the values of
# the nodes which are not yet
# included in MST and find the
# minimum valued one.
for i in range(n):
if (mstset[i] == False and
keyval[i] < mini):
mini = keyval[i]
mini_index = i
return mini_index
# Function to find out the MST and
# the cost of the MST.
def findcost(n, city):
# Array to store the parent
# node of a particular node.
parent = [None] * n
# Array to store key value
# of each node.
keyval = [None] * n
# Boolean Array to hold bool
# values whether a node is
# included in MST or not.
mstset = [None] * n
# Set all the key values to infinite and
# none of the nodes is included in MST.
for i in range(n):
keyval[i] = 9999999999999
mstset[i] = False
# Start to find the MST from node 0.
# Parent of node 0 is none so set -1.
# key value or minimum cost to reach
# 0th node from 0th node is 0.
parent[0] = -1
keyval[0] = 0
# Find the rest n-1 nodes of MST.
for i in range(n - 1):
# First find out the minimum node
# among the nodes which are not yet
# included in MST.
u = minnode(n, keyval, mstset)
# Now the uth node is included in MST.
mstset[u] = True
# Update the values of neighbor
# nodes of u which are not yet
# included in MST.
for v in range(n):
if (city[u][v] and mstset[v] == False and
city[u][v] < keyval[v]):
keyval[v] = city[u][v]
parent[v] = u
# Find out the cost by adding
# the edge values of MST.
cost = 0
for i in range(1, n):
cost += city[parent[i]][i]
print(cost)
# Driver Code
if __name__ == '__main__':
# Input 1
n1 = 5
city1 = [[0, 1, 2, 3, 4],
[1, 0, 5, 0, 7],
[2, 5, 0, 6, 0],
[3, 0, 6, 0, 0],
[4, 7, 0, 0, 0]]
findcost(n1, city1)
# Input 2
n2 = 6
city2 = [[0, 1, 1, 100, 0, 0],
[1, 0, 1, 0, 0, 0],
[1, 1, 0, 0, 0, 0],
[100, 0, 0, 0, 2, 2],
[0, 0, 0, 2, 0, 2],
[0, 0, 0, 2, 2, 0]]
findcost(n2, city2)
# This code is contributed by PranchalKC#
// C# code to find out minimum cost
// path to connect all the cities
using System;
class GFG{
// Function to find out minimum valued node
// among the nodes which are not yet included
// in MST
static int minnode(int n, int[] keyval,
bool[] mstset)
{
int mini = Int32.MaxValue;
int mini_index = 0;
// Loop through all the values of the nodes
// which are not yet included in MST and find
// the minimum valued one.
for(int i = 0; i < n; i++)
{
if (mstset[i] == false && keyval[i] < mini)
{
mini = keyval[i];
mini_index = i;
}
}
return mini_index;
}
// Function to find out the MST and
// the cost of the MST.
static void findcost(int n, int[,] city)
{
// Array to store the parent node of a
// particular node.
int[] parent = new int[n];
// Array to store key value of each node.
int[] keyval = new int[n];
// Boolean Array to hold bool values whether
// a node is included in MST or not.
bool[] mstset = new bool[n];
// Set all the key values to infinite and
// none of the nodes is included in MST.
for(int i = 0; i < n; i++)
{
keyval[i] = Int32.MaxValue;
mstset[i] = false;
}
// Start to find the MST from node 0.
// Parent of node 0 is none so set -1.
// key value or minimum cost to reach
// 0th node from 0th node is 0.
parent[0] = -1;
keyval[0] = 0;
// Find the rest n-1 nodes of MST.
for(int i = 0; i < n - 1; i++)
{
// First find out the minimum node
// among the nodes which are not yet
// included in MST.
int u = minnode(n, keyval, mstset);
// Now the uth node is included in MST.
mstset[u] = true;
// Update the values of neighbor
// nodes of u which are not yet
// included in MST.
for(int v = 0; v < n; v++)
{
if (city[u, v] > 0 && mstset[v] == false &&
city[u, v] < keyval[v])
{
keyval[v] = city[u, v];
parent[v] = u;
}
}
}
// Find out the cost by adding
// the edge values of MST.
int cost = 0;
for(int i = 1; i < n; i++)
cost += city[parent[i], i];
Console.WriteLine(cost);
}
// Driver code
public static void Main(string[] args)
{
// Input 1
int n1 = 5;
int[,] city1 = { { 0, 1, 2, 3, 4 },
{ 1, 0, 5, 0, 7 },
{ 2, 5, 0, 6, 0 },
{ 3, 0, 6, 0, 0 },
{ 4, 7, 0, 0, 0 } };
findcost(n1, city1);
// Input 2
int n2 = 6;
int[,] city2 = { { 0, 1, 1, 100, 0, 0 },
{ 1, 0, 1, 0, 0, 0 },
{ 1, 1, 0, 0, 0, 0 },
{ 100, 0, 0, 0, 2, 2 },
{ 0, 0, 0, 2, 0, 2 },
{ 0, 0, 0, 2, 2, 0 } };
findcost(n2, city2);
}
}
// This code is contributed by ukaspJavascript
输出：
10
106复杂性：外循环（即向 MST 添加新节点的循环）运行 n 次，在循环的每次迭代中，需要 O(n) 时间来找到最小节点和 O(n) 时间来更新 u 的相邻节点-th 节点。因此整体复杂度为 O(n 2 )
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