给定一棵树，它有N 个节点值从1到N和(N – 1) 条边和一个数字K ，任务是从树中删除最少数量的节点，使得每个子树最多有K 个节点。删除节点将删除从该节点到所有其他连接节点的边。
例子：
Input: N = 10, K = 3, Below is the graph:
Output:
Number of nodes removed: 2
Removed Nodes: 2 1
Explanation:
After removing the nodes 1 and 2, here, no subtree or tree has more than 3 nodes. Below is the resulting graph:
Input: N = 6, K = 3, Below is the graph:
Output:
Number of nodes removed: 1
Removed Nodes: 1
Explanation:
After removing the nodes 1, here, no subtree or tree has more than 3 nodes. Below is the resulting graph:
方法：思想是观察一个节点X的子树中的节点数是其子节点的子树中的节点数与该节点本身的总和。以下是步骤：
- 用动态规划和DFS存储节点的计数，在每个节点的子树容易。
- 现在，为了没有节点的子树超过 K 个节点，想法是在其子树中的节点超过 K 个时删除该节点，并将 0 传递给其父节点。
- 在上面的步骤中，我们让每个节点在其子树中的节点不大于 K 并最小化节点删除的数量。
下面是上述方法的实现：
C++
// C++ program for the above approach
#include
using namespace std;
#define N 20
// Function to perform DFS Traversal
int dfs(vector &visited, int s,
int &K, int &removals,
vector &removed_nodes,
vector> &adj)
{
// Mark the node as true
visited[s] = true;
int nodes = 1;
// Traverse adjacency list
// of child node
for(int child : adj[s])
{
// If already visited then
// omit the node
if (visited[child])
continue;
// Add number of nodes
// in subtree
nodes += dfs(visited, child, K,
removals, removed_nodes,
adj);
}
if (nodes > K)
{
// Increment the count
removals++;
removed_nodes.push_back(s);
nodes = 0;
}
// Return the nodes
return nodes;
}
// Function to add edges in graph
void addEdge(vector> &adj,
int a, int b)
{
adj[a].push_back(b);
adj[b].push_back(a);
}
// Function that finds the number
// of nodes to be removed such that
// every subtree has size at most K
void findRemovedNodes(vector &visited, int K,
int &removals,
vector &removed_nodes,
vector> &adj)
{
// Function Call to find the
// number of nodes to remove
dfs(visited, 1, K, removals,
removed_nodes, adj);
// Print Removed Nodes
cout << "Number of nodes removed: "
<< removals << endl;
cout << "Removed Nodes: ";
for(int node : removed_nodes)
{
cout << node << " ";
}
}
// Driver Code
int main()
{
// Variables used to store data globally
vector visited(N);
int K;
int removals = 0;
vector removed_nodes;
// Adjacency list representation of tree
vector> adj(N);
// Insert of nodes in graph
addEdge(adj, 1, 2);
addEdge(adj, 1, 3);
addEdge(adj, 2, 4);
addEdge(adj, 2, 5);
addEdge(adj, 3, 6);
// Required subtree size
K = 3;
// Function Call
findRemovedNodes(visited, K, removals,
removed_nodes, adj);
return 0;
}
// This code is contributed by sanjeev2552 Java
// Java program for the above approach
import java.util.*;
class GFG {
// Variables used to store data globally
static final int N = 20;
static boolean visited[] = new boolean[N];
static int K;
static int removals = 0;
static ArrayList removed_nodes
= new ArrayList<>();
// Adjacency list representation of tree
static ArrayList > adj
= new ArrayList<>();
// Function to perform DFS Traversal
static int dfs(int s)
{
// Mark the node as true
visited[s] = true;
int nodes = 1;
// Traverse adjacency list
// of child node
for (Integer child : adj.get(s)) {
// If already visited then
// omit the node
if (visited[child])
continue;
// Add number of nodes
// in subtree
nodes += dfs(child);
}
if (nodes > K) {
// Increment the count
removals++;
removed_nodes.add(s);
nodes = 0;
}
// Return the nodes
return nodes;
}
// Function to add edges in graph
static void addEdge(int a, int b)
{
adj.get(a).add(b);
adj.get(b).add(a);
}
// Function that finds the number
// of nodes to be removed such that
// every subtree has size at most K
public static void findRemovedNodes(int K)
{
// Function Call to find the
// number of nodes to remove
dfs(1);
// Print Removed Nodes
System.out.println("Number of nodes"
+ " removed: "
+ removals);
System.out.print("Removed Nodes: ");
for (int node : removed_nodes)
System.out.print(node + " ");
}
// Driver Code
public static void main(String[] args)
{
// Creating list for all nodes
for (int i = 0; i < N; i++)
adj.add(new ArrayList<>());
// Insert of nodes in graph
addEdge(1, 2);
addEdge(1, 3);
addEdge(2, 4);
addEdge(2, 5);
addEdge(3, 6);
// Required subtree size
K = 3;
// Function Call
findRemovedNodes(K);
}
} Python3
# Python3 program for the above approach
# Variables used to store data globally
N = 20
visited = [False for i in range(N)]
K = 0
removals = 0
removed_nodes = []
# Adjacency list representation of tree
adj = [[] for i in range(N)]
# Function to perform DFS Traversal
def dfs(s):
global removals
# Mark the node as true
visited[s] = True
nodes = 1
# Traverse adjacency list
# of child node
for child in adj[s]:
# If already visited then
# omit the node
if (visited[child]):
continue
# Add number of nodes
# in subtree
nodes += dfs(child)
if (nodes > K):
# Increment the count
removals += 1
removed_nodes.append(s)
nodes = 0
# Return the nodes
return nodes
# Function to add edges in graph
def addEdge(a, b):
adj[a].append(b)
adj[b].append(a)
# Function that finds the number
# of nodes to be removed such that
# every subtree has size at most K
def findRemovedNodes(K):
# Function Call to find the
# number of nodes to remove
dfs(1)
# Print Removed Nodes
print("Number of nodes removed: ", removals)
print("Removed Nodes: ", end = ' ')
for node in removed_nodes:
print(node, end = ' ')
# Driver Code
if __name__ == "__main__":
# Insert of nodes in graph
addEdge(1, 2)
addEdge(1, 3)
addEdge(2, 4)
addEdge(2, 5)
addEdge(3, 6)
# Required subtree size
K = 3
# Function Call
findRemovedNodes(K)
# This code is contributed by rutvik_56C#
// C# program for the above approach
using System;
using System.Collections.Generic;
class GFG{
// Variables used to store data globally
static readonly int N = 20;
static bool []visited = new bool[N];
static int K;
static int removals = 0;
static List removed_nodes
= new List();
// Adjacency list representation of tree
static List > adj
= new List>();
// Function to perform DFS Traversal
static int dfs(int s)
{
// Mark the node as true
visited[s] = true;
int nodes = 1;
// Traverse adjacency list
// of child node
foreach (int child in adj[s])
{
// If already visited then
// omit the node
if (visited[child])
continue;
// Add number of nodes
// in subtree
nodes += dfs(child);
}
if (nodes > K)
{
// Increment the count
removals++;
removed_nodes.Add(s);
nodes = 0;
}
// Return the nodes
return nodes;
}
// Function to add edges in graph
static void addEdge(int a, int b)
{
adj[a].Add(b);
adj[b].Add(a);
}
// Function that finds the number
// of nodes to be removed such that
// every subtree has size at most K
public static void findRemovedNodes(int K)
{
// Function Call to find the
// number of nodes to remove
dfs(1);
// Print Removed Nodes
Console.WriteLine("Number of nodes" +
" removed: " +
removals);
Console.Write("Removed Nodes: ");
foreach (int node in removed_nodes)
Console.Write(node + " ");
}
// Driver Code
public static void Main(String[] args)
{
// Creating list for all nodes
for (int i = 0; i < N; i++)
adj.Add(new List());
// Insert of nodes in graph
addEdge(1, 2);
addEdge(1, 3);
addEdge(2, 4);
addEdge(2, 5);
addEdge(3, 6);
// Required subtree size
K = 3;
// Function Call
findRemovedNodes(K);
}
}
// This code is contributed by Rohit_ranjan 输出
Number of nodes removed: 1
Removed Nodes: 1时间复杂度： O(N)
辅助空间： O(N)
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