给定一个由N个整数组成的数组arr [] ，其中N是一个偶数。任务是将给定的N个整数分成两个相等的子集，以使两个子集的所有元素的按位XOR之和最大。
例子：
Input: N= 4, arr[] = {1, 2, 3, 4}
Output: 10
Explanation:
There are 3 ways possible:
(1, 2)(3, 4) = (1^2)+(3^4) = 10
(1, 3)(2, 4) = (1^3)+(2^3) = 8
(1, 4)(2, 3) = (1^4)+(2^3) = 6
Hence, the maximum sum = 10
Input: N= 6, arr[] = {4, 5, 3, 2, 5, 6}
Output: 17
天真的方法：这个想法是检查N / 2对的每个可能的分布。打印两个子集的所有元素的总和的按位XOR，这是最大的。
时间复杂度： O(N * N！)
辅助空间： O(1)
高效方法：为了优化上述方法，其思想是使用“使用位屏蔽的动态编程”。请按照以下步骤解决问题：
- 最初，位掩码为0，如果设置了该位，则说明该对已经被选中。
- 遍历所有可能的对，并检查是否有可能选择一对，即在掩码中未设置i和j的位：
- 如果可以接受该对，则找到当前对的按位XOR总和，然后递归检查下一个对。
- 否则检查下一对元素。
- 在上述步骤中，为每个递归调用继续更新最大XOR对和。
- 打印存储在dp [mask]中的所有可能对的最大值。
下面是上述方法的实现：
C++
// C++ program for the above approach
#include
using namespace std;
// Function that finds the maximum
// Bitwise XOR sum of the two subset
int xorSum(int a[], int n,
int mask, int dp[])
{
// Check if the current state is
// already computed
if (dp[mask] != -1)
{
return dp[mask];
}
// Initialize answer to minimum value
int max_value = 0;
// Iterate through all possible pairs
for (int i = 0; i < n; i++)
{
for (int j = i + 1; j < n; j++)
{
// Check whether ith bit and
// jth bit of mask is not
// set then pick the pair
if (i != j &&
(mask & (1 << i)) == 0 &&
(mask & (1 << j)) == 0)
{
// For all possible pairs
// find maximum value pick
// current a[i], a[j] and
// set i, j th bits in mask
max_value = max(max_value, (a[i] ^ a[j]) +
xorSum(a, n, (mask | (1 << i) |
(1 << j)), dp));
}
}
}
// Store the maximum value
// and return the answer
return dp[mask] = max_value;
}
// Driver Code
int main()
{
int n = 4;
// Given array arr[]
int arr[] = { 1, 2, 3, 4 };
// Declare Initialize the dp states
int dp[(1 << n) + 5];
memset(dp, -1, sizeof(dp));
// Function Call
cout << (xorSum(arr, n, 0, dp));
}
// This code is contributed by Rohit_ranjan Java
// Java program for the above approach
import java.util.*;
import java.io.*;
public class GFG {
// Function that finds the maximum
// Bitwise XOR sum of the two subset
public static int xorSum(int a[], int n,
int mask, int[] dp)
{
// Check if the current state is
// already computed
if (dp[mask] != -1) {
return dp[mask];
}
// Initialize answer to minimum value
int max_value = 0;
// Iterate through all possible pairs
for (int i = 0; i < n; i++) {
for (int j = i + 1; j < n; j++) {
// Check whether ith bit and
// jth bit of mask is not
// set then pick the pair
if (i != j
&& (mask & (1 << i)) == 0
&& (mask & (1 << j)) == 0) {
// For all possible pairs
// find maximum value pick
// current a[i], a[j] and
// set i, j th bits in mask
max_value = Math.max(
max_value,
(a[i] ^ a[j])
+ xorSum(a, n,
(mask | (1 << i)
| (1 << j)),
dp));
}
}
}
// Store the maximum value
// and return the answer
return dp[mask] = max_value;
}
// Driver Code
public static void main(String args[])
{
int n = 4;
// Given array arr[]
int arr[] = { 1, 2, 3, 4 };
// Declare Initialize the dp states
int dp[] = new int[(1 << n) + 5];
Arrays.fill(dp, -1);
// Function Call
System.out.println(xorSum(arr, n, 0, dp));
}
}Python3
# Python3 program to implement
# the above approach
# Function that finds the maximum
# Bitwise XOR sum of the two subset
def xorSum(a, n, mask, dp):
# Check if the current state is
# already computed
if(dp[mask] != -1):
return dp[mask]
# Initialize answer to minimum value
max_value = 0
# Iterate through all possible pairs
for i in range(n):
for j in range(i + 1, n):
# Check whether ith bit and
# jth bit of mask is not
# set then pick the pair
if(i != j and
(mask & (1 << i)) == 0 and
(mask & (1 << j)) == 0):
# For all possible pairs
# find maximum value pick
# current a[i], a[j] and
# set i, j th bits in mask
max_value = max(max_value,
(a[i] ^ a[j]) +
xorSum(a, n,
(mask | (1 << i) |
(1 << j)), dp))
# Store the maximum value
# and return the answer
dp[mask] = max_value
return dp[mask]
# Driver Code
n = 4
# Given array arr[]
arr = [ 1, 2, 3, 4 ]
# Declare Initialize the dp states
dp = [-1] * ((1 << n) + 5)
# Function call
print(xorSum(arr, n, 0, dp))
# This code is contributed by Shivam SinghC#
// C# program for the above approach
using System;
class GFG{
// Function that finds the maximum
// Bitwise XOR sum of the two subset
public static int xorSum(int []a, int n,
int mask, int[] dp)
{
// Check if the current state is
// already computed
if (dp[mask] != -1)
{
return dp[mask];
}
// Initialize answer to minimum value
int max_value = 0;
// Iterate through all possible pairs
for(int i = 0; i < n; i++)
{
for(int j = i + 1; j < n; j++)
{
// Check whether ith bit and
// jth bit of mask is not
// set then pick the pair
if (i != j &&
(mask & (1 << i)) == 0 &&
(mask & (1 << j)) == 0)
{
// For all possible pairs
// find maximum value pick
// current a[i], a[j] and
// set i, j th bits in mask
max_value = Math.Max(
max_value,
(a[i] ^ a[j]) +
xorSum(a, n, (mask |
(1 << i) | (1 << j)), dp));
}
}
}
// Store the maximum value
// and return the answer
return dp[mask] = max_value;
}
// Driver Code
public static void Main(String []args)
{
int n = 4;
// Given array []arr
int []arr = { 1, 2, 3, 4 };
// Declare Initialize the dp states
int []dp = new int[(1 << n) + 5];
for(int i = 0; i < dp.Length; i++)
dp[i] = -1;
// Function call
Console.WriteLine(xorSum(arr, n, 0, dp));
}
}
// This code is contributed by amal kumar choubey输出：
10
时间复杂度： O(N 2 * 2 N )，其中N是给定数组的大小
辅助空间： O(N)