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Find the Maximum Sequence Value of Array

Difficulty: Hard

Problem Description

You are given an integer array nums and a positive integer k. The value of a sequence seq of size 2 * x is defined as: (seq[0] OR seq[1] OR ... OR seq[x - 1]) XOR (seq[x] OR seq[x + 1] OR ... OR seq[2 * x - 1]). Return the maximum value of any subsequence of nums having size 2 * k.

Key Insights

  • The problem requires us to find a subsequence of size 2 * k that maximizes the XOR value of two sections of the subsequence.
  • Each section of the subsequence contributes to the overall result through bitwise OR operations followed by a bitwise XOR.
  • The constraints allow us to explore combinations of numbers, but the size of the array makes brute force infeasible for large k.
  • Efficiently managing the selections and computations is key to solving the problem within time limits.

Space and Time Complexity

Time Complexity: O(n^2 * k)
Space Complexity: O(n)

Solution

The solution involves iterating through the array to explore valid subsequences of size 2 * k. For each valid subsequence, we split it into two parts of size k, compute the OR for each part, and then calculate the XOR of these two results. The maximum found during these iterations is returned.

The approach can utilize dynamic programming or bit manipulation techniques to optimize the search for maximum OR values efficiently.

Code Solutions

def findMaxValue(nums, k):
    n = len(nums)
    max_value = 0
    
    for i in range(n):
        for j in range(i + 1, n):
            if (j - i + 1) >= 2 * k:  # Ensure the subsequence is at least 2*k
                first_part = 0
                second_part = 0
                
                # Calculate OR for the first k elements
                for x in range(k):
                    first_part |= nums[i + x]
                
                # Calculate OR for the second k elements
                for x in range(k):
                    second_part |= nums[j - k + 1 + x]
                
                max_value = max(max_value, first_part ^ second_part)
    
    return max_value
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