通过来自两端的等距元素最大化可能的 XOR 子序列

给定一个大小为N的数组A[] ，找到最大异或子序列，使得A [ i ]和A [ N – i – 1 ] 都属于这个子序列，其中 i 的范围在[0, N – 1] 之间。

例子：

Input: N = 8, A [ ] = {1, 2, 3, 4, 5, 6, 7, 8}
Output: 13
Explanation:
Maximum Xor Subsequence is {1, 3, 4, 5, 6, 8}

Input: N = 5, A [ ] = {3, 2, 6, 5, 4}
Output: 6
Explanation:
Maximum Xor Subsequence is {3, 2, 6, 5, 4}

方法：
因为 A[i] 和 A[Ni-1] 都应该出现在同一个子序列中，所以将它们配对，然后找到最大的异或和。对于每个有效索引i ，计算 (A[i] ^ A[Ni-1]) 并将其存储在新数组X 中。这个新数组的大小将为 N/2。

天真的解决方案：
对于这个问题，创建数组X[]后最简单的方法是递归生成X 的所有子序列，并找到最大的异或子序列。

以下是 maxXorSubseq(X, N, i) 的递归定义：

maxXorSubseq ( X, N, i ) = MAX ( X[ i ] ^ maxXorSubseq ( X, N, i + 1 ), maxXorSubseq (X, N, i + 1) )
 

可能的组合总数为2 ^N 。
时间复杂度： O(2 ^N )

下面是上述方法的实现：

C++

// C++ implementation
// of the above approach
#include 
using namespace std;
 
// Returns maximum xor
int maxXorSubseq(vector &x, int n,
                                 int i)
{
    if(i == n)
       return 0;
 
    return max(x[i] ^ maxXorSubseq(x, n,
                                   i + 1),
                      maxXorSubseq(x, n,
                                   i + 1));
}
 
vector compute(vector a, int n)
{
    vector x;
 
    // Calculate a[i]^a[n-i-1]
    for(int i = 0; i < n / 2; i++)
       x.push_back(a[i] ^ a[n - i - 1]);
 
    // If n is odd
    if(n & 1)
       x.push_back(a[n / 2]);
 
    return x;
}
 
// Driver code
int main()
{
    int n = 8;
    vector a = { 1, 2, 3, 4,
                      5, 6, 7, 8 };
     
    // Getting new array x
    vector x = compute(a, n);
     
    int mxXor = maxXorSubseq(x, x.size(), 0);
     
    cout << (mxXor);
    return 0;
}
 
// This code is contributed by mohit kumar 29

Java

// Java implementation
// of the above approach
import java.util.*;
 
class GFG{
 
// Returns maximum xor
static int maxXorSubseq(List x, int n,
                                         int i)
{
    if(i == n)
    return 0;
 
    return Math.max(x.get(i) ^ maxXorSubseq(x, n,
                                            i + 1),
                               maxXorSubseq(x, n,
                                            i + 1));
}
 
static List compute(List a, int n)
{
    List x = new ArrayList();
 
    // Calculate a[i]^a[n-i-1]
    for(int i = 0; i < n / 2; i++)
        x.add(a.get(i) ^ a.get(n - i - 1));
 
    // If n is odd
    if((n & 1) == 1)
        x.add(a.get(n / 2));
 
    return x;
}
 
// Driver code
public static void main(String[] args)
{
    int n = 8;
    List a = Arrays.asList( 1, 2, 3, 4,
                                     5, 6, 7, 8 );
     
    // Getting new array x
    List x = compute(a, n);
     
    int mxXor = maxXorSubseq(x, x.size(), 0);
     
    System.out.println((mxXor));
}
}
 
// This code is contributed by offbeat

Python3

# Python3 implementation
# of the above approach
 
# Returns maximum xor
def maxXorSubseq(x, n, i):
     
    if(i == n):
        return 0
         
    return max(
        x[i]^maxXorSubseq(
        x, n, i + 1), maxXorSubseq(
        x, n, i + 1))
 
def compute(a, n):
     
    x = []
     
    # Calculate a[i]^a[n-i-1]
    for i in range(n//2):    
        x.append(a[i]^a[n-i-1])
     
    # If n is odd
    if(n&1):
        x.append(a[n//2])
         
    return x
 
# Driver code
if __name__ =="__main__":
     
    n = 8
    a = [1, 2, 3, 4, 5, 6, 7, 8]
     
    # Getting new array x
    x = compute(a, n)
     
    mxXor = maxXorSubseq(x, len(x), 0)
     
    print(mxXor)

C#

// C# implementation 
// of the above approach 
using System;
using System.Collections.Generic;
class GFG {
     
    // Returns maximum xor
    static int maxXorSubseq(List x, int n, int i)
    {
        if(i == n)
        return 0;
       
        return Math.Max(x[i] ^ maxXorSubseq(x, n, i + 1), maxXorSubseq(x, n, i + 1));
    }
       
    static List compute(List a, int n)
    {
        List x = new List();
       
        // Calculate a[i]^a[n-i-1]
        for(int i = 0; i < n / 2; i++)
            x.Add(a[i] ^ a[n - i - 1]);
       
        // If n is odd
        if((n & 1) == 1)
            x.Add(a[n / 2]);
       
        return x;
    }
 
  static void Main() {
        int n = 8;
        List a = new List{ 1, 2, 3, 4, 5, 6, 7, 8 };
           
        // Getting new array x
        List x = compute(a, n);
           
        int mxXor = maxXorSubseq(x, x.Count, 0);
           
        Console.WriteLine((mxXor));
  }
}
 
// This code is contributed by divyeshrabadiya07

Javascript

 
    // Javascript implementation
    // of the above approach
     
    // Returns maximum xor
    function maxXorSubseq(x, n, i)
    {
        if(i == n)
        return 0;
        
        return Math.max(x[i] ^ maxXorSubseq(x, n, i + 1),
        maxXorSubseq(x, n, i + 1));
    }
        
    function compute(a, n)
    {
        let x = [];
        
        // Calculate a[i]^a[n-i-1]
        for(let i = 0; i < parseInt(n / 2, 10); i++)
            x.push(a[i] ^ a[n - i - 1]);
        
        // If n is odd
        if((n & 1) == 1)
            x.push(a[parseInt(n / 2, 10)]);
        
        return x;
    }
     
    let n = 8;
    let a = [ 1, 2, 3, 4, 5, 6, 7, 8 ];
 
    // Getting new array x
    let x = compute(a, n);
 
    let mxXor = maxXorSubseq(x, x.length, 0);
 
    document.write((mxXor));
 

输出：

13

有效的方法：
考虑到上面的实现，下面是输入X = [9, 5, 1]的部分递归树

在上面的部分递归树中，maxXorSubseq({1}) 被求解了四次。在绘制完整的递归树时，可以观察到有许多子问题被一次又一次地解决。所以这个问题有重叠的子问题，使用Memoization可以避免相同子问题的重新计算。

为了记忆，创建一个数组dp [ ]数组并存储：

dp[i] = max(x[i] ^ maxXorSubseq(x, n, i + 1), maxXorSubseq(x, n, idx + 1)
 

这避免了对先前计算的索引的重新计算，从而优化了计算复杂度。

下面是上述方法的实现：

C++

// C++ implementation of the above approach
#include 
using namespace std;
vector dp;
  
// Returns maximum xor sum
int maxXorSubseq(vector x, int n, int idx)
{
    if (idx == n)
    {
        return 0;
    }
      
    // If already precomputed
    if (dp[idx] != -1)
    {
        return dp[idx];
    }
    int ans = 0;
    ans = max(ans, x[idx] ^ maxXorSubseq(x, n, idx + 1));
    ans = max(ans, maxXorSubseq(x, n, idx + 1));
                     
    // Store the maximum
    dp[idx] = ans;
    return ans;
}
  
vector compute(int a[],int n)
{
    vector x;
      
    // Calculate a[i]^a[n-i-1]
    for(int i = 0; i < n / 2; i++)
    {
        x.push_back(a[i] ^ a[n - i - 1]);
    }
      
    //  If n is odd
    if ((n & 1) != 0)
    {
        x.push_back(a[n / 2]);
    }
    return x;
}
 
// Driver code
int main()
{
    int n = 8;
    int a[] = { 1, 2, 3, 4, 5, 6, 7, 8 };
      
    // Getting new array x
    vector x = compute(a, n);
      
    // Initialize dp array
    for(int i = 0; i < x.size(); i++)
    {
        dp.push_back(-1);
    }
    int mxXor = maxXorSubseq(x, x.size(), 0);    
    cout << mxXor << endl;
    return 0;
}
 
// This code is contributed by divyesh072019.

Java

// Java implementation of the above approach
import java.io.*;
import java.util.*;
 
class GFG{
     
static Vector dp = new Vector();
 
// Returns maximum xor sum
static int maxXorSubseq(Vector x,
                        int n, int idx)
{
    if (idx == n)
    {
        return 0;
    }
     
    // If already precomputed
    if (dp.get(idx) != -1)
    {
        return dp.get(idx);
    }
    int ans = 0;
    ans = Math.max(ans, x.get(idx) ^
                   maxXorSubseq(x, n, idx + 1));
    ans = Math.max(ans,
                   maxXorSubseq(x, n, idx + 1));
                    
    // Store the maximum
    dp.set(idx,ans);
    return ans;
}
 
static Vector compute(int[] a,int n)
{
    Vector x = new Vector();
     
    // Calculate a[i]^a[n-i-1]
    for(int i = 0; i < n / 2; i++)
    {
        x.add(a[i] ^ a[n - i - 1]);
    }
     
    //  If n is odd
    if ((n & 1) != 0)
    {
        x.add(a[n / 2]);
    }
    return x;
}
 
// Driver code
public static void main(String[] args)
{
    int n = 8;
    int[] a = { 1, 2, 3, 4, 5, 6, 7, 8 };
     
    // Getting new array x
    Vector x = compute(a, n);
     
    // Initialize dp array
    for(int i = 0; i < x.size(); i++)
    {
        dp.add(-1);
    }
    int mxXor = maxXorSubseq(x, x.size(), 0);
     
    System.out.println(mxXor);
}
}
 
//  This code is contributed by avanitrachhadiya2155

蟒蛇3

# Python3 implementation
# of the above approach
 
# Returns maximum xor sum
def maxXorSubseq(x, n, idx):
     
    if(idx == n):
        return 0
         
    # If already precomputed
    if(dp[idx]!=-1):
        return dp[idx]
 
    ans = 0
     
    ans = max(
        ans, x[idx] ^ maxXorSubseq(
                    x, n, idx + 1))
     
    ans = max(ans, maxXorSubseq(
                    x, n, idx + 1))
     
    # Store the maximum
    dp[idx] = ans
     
    return ans
 
def compute(a, n):
     
    x = []
     
    # Calculate a[i]^a[n-i-1]
    for i in range(n//2):
        x.append(a[i]^a[n-i-1])
     
    # If n is odd
    if(n&1):
        x.append(a[n//2])
         
    return x
     
# Declared dp[] array globally
dp = []
 
# Driver code
if __name__ =="__main__":
     
    n = 8
    a =[1, 2, 3, 4, 5, 6, 7, 8]
     
    # Getting new array x
    x = compute(a, n)
 
    # Initialize dp array
    dp = [-1 for i in range(len(x))]
     
    mxXor = maxXorSubseq(x, len(x), 0)
     
    print(mxXor)
    

C＃

// C# implementation of the above approach
using System;
using System.Collections.Generic;
class GFG
{
    static List dp = new List();
   
    // Returns maximum xor sum
    static int maxXorSubseq(List x, int n, int idx)
    {
        if (idx == n)
        {
            return 0;
        }
       
        // If already precomputed
        if(dp[idx] != -1)
        {
            return dp[idx];
        }
        int ans = 0;
        ans = Math.Max(ans, x[idx]^maxXorSubseq(x, n, idx + 1));
        ans = Math.Max(ans, maxXorSubseq(x, n, idx + 1));
       
        // Store the maximum
        dp[idx] = ans;
        return ans;
    }
    static List compute(int[] a,int n)
    {
        List x = new List();
       
        // Calculate a[i]^a[n-i-1]
        for(int i = 0; i < n / 2; i++)
        {
            x.Add(a[i] ^ a[n - i - 1]);
        }
       
        //  If n is odd
        if((n & 1) != 0)
        {
            x.Add(a[n / 2]);
        }
        return x;
    }
   
    // Driver code
    static public void Main ()
    {
        int n = 8;
        int[] a = { 1, 2, 3, 4, 5, 6, 7, 8 };
       
        // Getting new array x
        List x = compute(a, n);
       
        // Initialize dp array
        for(int i = 0; i < x.Count; i++)
        {
            dp.Add(-1);
        }
        int mxXor = maxXorSubseq(x, x.Count, 0);
        Console.WriteLine(mxXor);
    }
}
 
// This code is contributed by rag2127

Javascript

    // Javascript implementation of the above approach
     
    let dp = [];
    
    // Returns maximum xor sum
    function maxXorSubseq(x, n, idx)
    {
        if (idx == n)
        {
            return 0;
        }
        
        // If already precomputed
        if(dp[idx] != -1)
        {
            return dp[idx];
        }
        let ans = 0;
        ans = Math.max(ans, x[idx]^maxXorSubseq(x, n, idx + 1));
        ans = Math.max(ans, maxXorSubseq(x, n, idx + 1));
        
        // Store the maximum
        dp[idx] = ans;
        return ans;
    }
     
    function compute(a, n)
    {
        let x = [];
        
        // Calculate a[i]^a[n-i-1]
        for(let i = 0; i < parseInt(n / 2, 10); i++)
        {
            x.push(a[i] ^ a[n - i - 1]);
        }
        
        //  If n is odd
        if((n & 1) != 0)
        {
            x.push(a[n / 2]);
        }
        return x;
    }
     
    let n = 8;
    let a = [ 1, 2, 3, 4, 5, 6, 7, 8 ];
 
    // Getting new array x
    let x = compute(a, n);
 
    // Initialize dp array
    for(let i = 0; i < x.length; i++)
    {
      dp.push(-1);
    }
    let mxXor = maxXorSubseq(x, x.length, 0);
    document.write(mxXor);
 
 
// This code is contributed by suresh07.

输出：

13

时间复杂度： O(N)

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