给定两个整数V和E表示无向图G(V, E)的顶点和边数、边列表EdgeList和数组A[]表示为每个节点着色的成本,任务是找到使用以下操作为图形着色的最低成本:
When a node is colored, all the nodes that can be reached from it are colored without any additional cost.
例子:
Input: V = 6, E = 5, A[] = {12, 25, 8, 11, 10, 7}, EdgeList = {{1, 2}, {1, 3}, {3, 2}, {2, 5}, {4, 6}}
Output: 15
Explanation:
On coloring the vertex 3 for a cost of 8, the vertices {1, 2, 5} gets colored at no additional cost.
On coloring the vertex 6 for a cost of 7, the only remaining vertex {4} also gets colored.
Therefore, the minimum cost = 8 + 7 = 15.
Input: V =7, E = 6, A[] = {3, 5, 8, 6, 9, 11, 10}, EdgeList = {{1, 4}, {2, 1}, {2, 7}, {3, 4}, {3, 5}, {5, 6}}
Output: 5
方法:
请按照以下步骤解决问题:
- 所有节点都可以从一个给定的节点到达,形成一个连接组件。
- 因此,对于每个连通分量,使用深度优先搜索,在图中的连通分量中找到成本最低的节点。
下面是上述方法的实现:
C++
// C++ Program to find the minimum
// cost to color all vertices of an
// Undirected Graph
#include
using namespace std;
#define MAX 10
vector adj[MAX];
// Function to add edge in the
// given graph
void addEdge(int u, int v)
{
adj[u].push_back(v);
adj[v].push_back(u);
}
// Function to perform DFS traversal and
// find the node with minimum cost
void dfs(int v, int cost[], bool vis[],
int& min_cost_node)
{
vis[v] = true;
// Update the minimum cost
min_cost_node
= min(min_cost_node, cost[v - 1]);
for (int i = 0; i < adj[v].size(); i++) {
// Recur for all connected nodes
if (vis[adj[v][i]] == false) {
dfs(adj[v][i], cost, vis,
min_cost_node);
}
}
}
// Function to calculate and return
// the minimum cost of coloring all
// vertices of the Undirected Graph
int minimumCost(int V, int cost[])
{
// Marks if a vertex is
// visited or not
bool vis[V + 1];
// Initialize all vertices as unvisited
memset(vis, false, sizeof(vis));
int min_cost = 0;
// Perform DFS traversal
for (int i = 1; i <= V; i++) {
// If vertex is not visited
if (!vis[i]) {
int min_cost_node = INT_MAX;
dfs(i, cost, vis, min_cost_node);
// Update minimum cost
min_cost += min_cost_node;
}
}
// Return the final cost
return min_cost;
}
// Driver Code
int main()
{
int V = 6, E = 5;
int cost[] = { 12, 25, 8, 11, 10, 7 };
addEdge(1, 2);
addEdge(1, 3);
addEdge(3, 2);
addEdge(2, 5);
addEdge(4, 6);
int min_cost = minimumCost(V, cost);
cout << min_cost << endl;
return 0;
}
Java
// Java program to find the minimum
// cost to color all vertices of an
// Undirected Graph
import java.util.*;
class GFG{
static final int MAX = 10;
@SuppressWarnings("unchecked")
static Vector []adj = new Vector[MAX];
static int min_cost_node;
// Function to add edge in the
// given graph
static void addEdge(int u, int v)
{
adj[u].add(v);
adj[v].add(u);
}
// Function to perform DFS traversal and
// find the node with minimum cost
static void dfs(int v, int cost[], boolean vis[])
{
vis[v] = true;
// Update the minimum cost
min_cost_node = Math.min(min_cost_node,
cost[v - 1]);
for(int i = 0; i < adj[v].size(); i++)
{
// Recur for all connected nodes
if (vis[adj[v].get(i)] == false)
{
dfs(adj[v].get(i), cost, vis);
}
}
}
// Function to calculate and return
// the minimum cost of coloring all
// vertices of the Undirected Graph
static int minimumCost(int V, int cost[])
{
// Marks if a vertex is
// visited or not
boolean []vis = new boolean[V + 1];
// Initialize all vertices as unvisited
Arrays.fill(vis, false);
int min_cost = 0;
// Perform DFS traversal
for(int i = 1; i <= V; i++)
{
// If vertex is not visited
if (!vis[i])
{
min_cost_node = Integer.MAX_VALUE;
dfs(i, cost, vis);
// Update minimum cost
min_cost += min_cost_node;
}
}
// Return the final cost
return min_cost;
}
// Driver Code
public static void main(String[] args)
{
int V = 6, E = 5;
int cost[] = { 12, 25, 8, 11, 10, 7 };
for(int i = 0; i < adj.length; i++)
adj[i] = new Vector();
addEdge(1, 2);
addEdge(1, 3);
addEdge(3, 2);
addEdge(2, 5);
addEdge(4, 6);
int min_cost = minimumCost(V, cost);
System.out.print(min_cost + "\n");
}
}
// This code is contributed by 29AjayKumar
Python3
# Python3 program to find the minimum
# cost to color all vertices of an
# Undirected Graph
import sys
MAX = 10
adj = [[] for i in range(MAX)]
# Function to add edge in the
# given graph
def addEdge(u, v):
adj[u].append(v)
adj[v].append(u)
# Function to perform DFS traversal and
# find the node with minimum cost
def dfs(v, cost, vis, min_cost_node):
vis[v] = True
# Update the minimum cost
min_cost_node = min(min_cost_node,
cost[v - 1])
for i in range(len(adj[v])):
# Recur for all connected nodes
if (vis[adj[v][i]] == False):
min_cost_node = dfs(adj[v][i],
cost, vis,
min_cost_node)
return min_cost_node
# Function to calculate and return
# the minimum cost of coloring all
# vertices of the Undirected Graph
def minimumCost(V, cost):
# Marks if a vertex is
# visited or not
vis = [False for i in range(V + 1)]
min_cost = 0
# Perform DFS traversal
for i in range(1, V + 1):
# If vertex is not visited
if (not vis[i]):
min_cost_node = sys.maxsize
min_cost_node = dfs(i, cost, vis,
min_cost_node)
# Update minimum cost
min_cost += min_cost_node
# Return the final cost
return min_cost
# Driver Code
if __name__=="__main__":
V = 6
E = 5
cost = [ 12, 25, 8, 11, 10, 7 ]
addEdge(1, 2)
addEdge(1, 3)
addEdge(3, 2)
addEdge(2, 5)
addEdge(4, 6)
min_cost = minimumCost(V, cost)
print(min_cost)
# This code is contributed by rutvik_56
C#
// C# program to find the minimum
// cost to color all vertices of an
// Undirected Graph
using System;
using System.Collections.Generic;
class GFG{
static readonly int MAX = 10;
static List []adj = new List[MAX];
static int min_cost_node;
// Function to add edge in the
// given graph
static void addEdge(int u, int v)
{
adj[u].Add(v);
adj[v].Add(u);
}
// Function to perform DFS traversal and
// find the node with minimum cost
static void dfs(int v, int []cost, bool []vis)
{
vis[v] = true;
// Update the minimum cost
min_cost_node = Math.Min(min_cost_node,
cost[v - 1]);
for(int i = 0; i < adj[v].Count; i++)
{
// Recur for all connected nodes
if (vis[adj[v][i]] == false)
{
dfs(adj[v][i], cost, vis);
}
}
}
// Function to calculate and return
// the minimum cost of coloring all
// vertices of the Undirected Graph
static int minimumCost(int V, int []cost)
{
// Marks if a vertex is
// visited or not
bool []vis = new bool[V + 1];
int min_cost = 0;
// Perform DFS traversal
for(int i = 1; i <= V; i++)
{
// If vertex is not visited
if (!vis[i])
{
min_cost_node = int.MaxValue;
dfs(i, cost, vis);
// Update minimum cost
min_cost += min_cost_node;
}
}
// Return the readonly cost
return min_cost;
}
// Driver Code
public static void Main(String[] args)
{
int V = 6;
int []cost = { 12, 25, 8, 11, 10, 7 };
for(int i = 0; i < adj.Length; i++)
adj[i] = new List();
addEdge(1, 2);
addEdge(1, 3);
addEdge(3, 2);
addEdge(2, 5);
addEdge(4, 6);
int min_cost = minimumCost(V, cost);
Console.Write(min_cost + "\n");
}
}
// This code is contributed by Amit Katiyar
Javascript
输出:
15
时间复杂度: O(V+E)
辅助空间: O(V)
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