给定一棵由N 个节点组成的树，任务是将给定的树转换为其 Sum Tree(包括其自身的权重)，并找到该 sum 树的任意两个节点的权重之间的最小差异。
注意：给定树的 N 个节点以从上到下的形式给出，N-1 行，其中每行描述两个连接的节点。
例子：
Input:
Output: 1
Explanation:
total weight of node 1: 3 (own weight) + (10 + 6 + 5 + 8 + 2 + 7 + 11) (sub-tree node’s weight) = 52
total weight of node 2: 5 (own weight) + (2 + 7 + 11) (sub-tree node’s weight) = 25
total weight of node 3: 8 (own weight) + (0) (sub-tree node’s weight) = 8
total weight of node 4: 10 (own weight) + (0) (sub-tree node’s weight) = 10
total weight of node 5: 2 (own weight) + (0) (sub-tree node’s weight) = 2
total weight of node 6: 6 (own weight) + (5 + 8 + 2 + 7 + 11) (sub-tree node’s weight) = 39
total weight of node 7: 7 (own weight) + (0) (sub-tree node’s weight) = 7
total weight of node 8: 11 (own weight) + (0) (sub-tree node’s weight) = 11
By observing the total weight of each node, Node 4 and 8 have a minimum difference(11-10) = 1
Input:
Output: 0
方法：
- 我们将从下面遍历给定的树，并将该节点的权重加上其子树节点的权重存储在一个数组中，并将每个节点的索引标记为已访问。因此，在两者之间，如果我们重新访问该节点，则不必再次计算该节点的权重。
- 我们将对存储每个节点总权重的数组进行排序。
- 现在找到排序数组中的成对差异，无论哪一对给出最小差异，最后打印出最小差异。
下面是上述方法的实现：
C++
// C++ program for the above approach
#include
using namespace std;
// Function to find minimum
// difference between any two node
void MinimumDifference(int total_weight[],
int N)
{
int min_difference = INT_MAX;
for (int i = 1; i < N; i++) {
// Pairwise difference
if (total_weight[i]
- total_weight[i - 1]
< min_difference) {
min_difference
= total_weight[i]
- total_weight[i - 1];
}
}
cout << min_difference << endl;
}
// Function to find total weight
// of each individual node
void SumTree(vector > v,
int individual_weight[],
int N)
{
// Array to store total weight
// of each node from 1 to N
int total_weight[N] = { 0 };
// Array to keep track of node
// previously counted or not
int visited[N] = { 0 };
// To store node no. from
/// N-1 lines
int first, second;
// To traverse from (N-1)
// line to 1 line
for (int i = (N - 2); i >= 0; i--) {
first = v[i].first;
second = v[i].second;
// Node is note visited
if (visited[second - 1] == 0) {
total_weight[second - 1]
+= individual_weight[second - 1];
// Make node visited
visited[second - 1] = 1;
}
total_weight[first - 1]
+= total_weight[second - 1];
// Node is note visited
if (visited[first - 1] == 0) {
total_weight[first - 1]
+= individual_weight[first - 1];
// Make node visited
visited[first - 1] = 1;
}
}
// Sort the total weight of each node
sort(total_weight, total_weight + N);
// Call function to find minimum
// difference
MinimumDifference(total_weight, N);
}
// Driver code
int main()
{
// Total node of rooted tree
int N = 8;
vector > v;
// N-1 lines describing
// rooted tree from top
// to bottom
v.push_back(make_pair(1, 4));
v.push_back(make_pair(1, 6));
v.push_back(make_pair(6, 2));
v.push_back(make_pair(6, 3));
v.push_back(make_pair(2, 5));
v.push_back(make_pair(2, 7));
v.push_back(make_pair(2, 8));
// Array describing weight
// of each node from 1 to N
int individual_weight[N] = { 3, 5, 8, 10,
2, 6, 7, 11 };
SumTree(v, individual_weight, N);
return 0;
} Java
// Java program for the above approach
import java.util.*;
class GFG{
static class pair
{
int first, second;
public pair(int first, int second)
{
this.first = first;
this.second = second;
}
}
// Function to find minimum
// difference between any two node
static void MinimumDifference(int total_weight[],
int N)
{
int min_difference = Integer.MAX_VALUE;
for(int i = 1; i < N; i++)
{
// Pairwise difference
if (total_weight[i] -
total_weight[i - 1] <
min_difference)
{
min_difference = total_weight[i] -
total_weight[i - 1];
}
}
System.out.print(min_difference + "\n");
}
// Function to find total weight
// of each individual node
static void SumTree(Vector v,
int individual_weight[],
int N)
{
// Array to store total weight
// of each node from 1 to N
int total_weight[] = new int[N];
// Array to keep track of node
// previously counted or not
int visited[] = new int[N];
// To store node no. from
/// N-1 lines
int first, second;
// To traverse from (N-1)
// line to 1 line
for(int i = (N - 2); i >= 0; i--)
{
first = v.get(i).first;
second = v.get(i).second;
// Node is note visited
if (visited[second - 1] == 0)
{
total_weight[second - 1] +=
individual_weight[second - 1];
// Make node visited
visited[second - 1] = 1;
}
total_weight[first - 1] +=
total_weight[second - 1];
// Node is note visited
if (visited[first - 1] == 0)
{
total_weight[first - 1] +=
individual_weight[first - 1];
// Make node visited
visited[first - 1] = 1;
}
}
// Sort the total weight of each node
Arrays.sort(total_weight);
// Call function to find minimum
// difference
MinimumDifference(total_weight, N);
}
// Driver code
public static void main(String[] args)
{
// Total node of rooted tree
int N = 8;
Vector v = new Vector<>();
// N-1 lines describing
// rooted tree from top
// to bottom
v.add(new pair(1, 4));
v.add(new pair(1, 6));
v.add(new pair(6, 2));
v.add(new pair(6, 3));
v.add(new pair(2, 5));
v.add(new pair(2, 7));
v.add(new pair(2, 8));
// Array describing weight
// of each node from 1 to N
int individual_weight[] = { 3, 5, 8, 10,
2, 6, 7, 11 };
SumTree(v, individual_weight, N);
}
}
// This code is contributed by Amit Katiyar Python3
# Python3 program for the above approach
import sys
# Function to find minimum difference
# between any two node
def minimum_difference(total_weight, n):
min_difference = sys.maxsize
for i in range(1, n):
# Pairwise difference
if (total_weight[i] -
total_weight[i - 1] <
min_difference):
min_difference = (total_weight[i] -
total_weight[i - 1])
print(min_difference)
# Function to find total weight
# of each individual node
def SumTree(v, individual_weight, N):
# Array to store total weight of
# each node from 1 to n
total_weight = [0 for i in range(N)]
# Array to keep track of node
# previously counted or not
visited = [0 for i in range(N)]
# To traverse from (n-1) line to 1 line
for i in range(N - 2, -1, -1):
first = v[i][0]
second = v[i][1]
if visited[second - 1] == 0:
total_weight[second - 1] += (
individual_weight[second - 1])
# Make node visited
visited[second - 1] = 1
total_weight[first - 1] += (
total_weight[second - 1])
# Node is note visited
if visited[first - 1] == 0:
total_weight[first - 1] += (
individual_weight[first - 1])
# Make node visited
visited[first - 1] = 1
# Sort the total weight of each node
total_weight.sort()
# Call function to find minimum difference
minimum_difference(total_weight, n)
# Driver Code
if __name__=='__main__':
# Total node of rooted tree
n = 8
v = []
# n-1 lines describing rooted
# tree from top to bottom
v.append([1, 4])
v.append([1, 6])
v.append([6, 2])
v.append([6, 3])
v.append([2, 5])
v.append([2, 7])
v.append([2, 8])
# Array describing weight of each
# node from 1 to n
individual_weight = [ 3, 5, 8, 10,
2, 6, 7, 11 ]
SumTree(v, individual_weight, n)
# This code is contributed by rutvik_56C#
// C# program for the
// above approach
using System;
using System.Collections.Generic;
class GFG{
class pair
{
public int first,
second;
public pair(int first,
int second)
{
this.first = first;
this.second = second;
}
}
// Function to find minimum
// difference between any two node
static void MinimumDifference(int []total_weight,
int N)
{
int min_difference = int.MaxValue;
for(int i = 1; i < N; i++)
{
// Pairwise difference
if (total_weight[i] -
total_weight[i - 1] <
min_difference)
{
min_difference = total_weight[i] -
total_weight[i - 1];
}
}
Console.Write(min_difference + "\n");
}
// Function to find total weight
// of each individual node
static void SumTree(List v,
int []individual_weight,
int N)
{
// Array to store total weight
// of each node from 1 to N
int []total_weight = new int[N];
// Array to keep track of node
// previously counted or not
int []visited = new int[N];
// To store node no. from
/// N-1 lines
int first, second;
// To traverse from (N-1)
// line to 1 line
for(int i = (N - 2); i >= 0; i--)
{
first = v[i].first;
second = v[i].second;
// Node is note visited
if (visited[second - 1] == 0)
{
total_weight[second - 1] +=
individual_weight[second - 1];
// Make node visited
visited[second - 1] = 1;
}
total_weight[first - 1] +=
total_weight[second - 1];
// Node is note visited
if (visited[first - 1] == 0)
{
total_weight[first - 1] +=
individual_weight[first - 1];
// Make node visited
visited[first - 1] = 1;
}
}
// Sort the total weight
// of each node
Array.Sort(total_weight);
// Call function to find minimum
// difference
MinimumDifference(total_weight, N);
}
// Driver code
public static void Main(String[] args)
{
// Total node of rooted tree
int N = 8;
List v = new List();
// N-1 lines describing
// rooted tree from top
// to bottom
v.Add(new pair(1, 4));
v.Add(new pair(1, 6));
v.Add(new pair(6, 2));
v.Add(new pair(6, 3));
v.Add(new pair(2, 5));
v.Add(new pair(2, 7));
v.Add(new pair(2, 8));
// Array describing weight
// of each node from 1 to N
int []individual_weight = {3, 5, 8, 10,
2, 6, 7, 11};
SumTree(v, individual_weight, N);
}
}
// This code is contributed by shikhasingrajput 输出：
1
时间复杂度： O(N * Log(N)) ，其中 N 是有根树中的总节点数。
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