Find the Maximum Sequence Value of Array

Find the Maximum Sequence Value of Array

What is this problem about?

The Find the Maximum Sequence Value of Array coding problem asks you to find the maximum possible XOR sum (or another bitwise operation) of two disjoint subsequences of length exactly $k$. You are given an array and an integer $k$. You need to pick $k$ elements from the left and $k$ elements from the right and maximize their combined "value."

Why is this asked in interviews?

Google asks this "Hard" problem to test your proficiency with Bit Manipulation and Dynamic Programming with Subsets. It evaluation your ability to efficiently track all possible bitwise outcomes for a fixed-size subsequence. This is a common pattern in signal processing and cryptography-related coding challenges.

Algorithmic pattern used

This problem uses Prefix/Suffix Dynamic Programming with Bitsets.

1. Precompute left[i][mask]: Is it possible to get a bitwise OR (or XOR) of mask using exactly $k$ elements from the first $i$ elements?
2. Precompute right[i][mask] similarly from the end of the array.
3. Since the number of possible masks is small (e.g., $2^7 = 128$), you can store these as a boolean array or a bitset.
4. Iterate through all possible split points $i$.
5. For each split, combine every possible mask from the left with every possible mask from the right to find the maximum resulting value.

Example explanation

nums = [1, 2, 3, 4], $k=1$.

1. Left possible ORs up to index 1: {1, 2}.
2. Right possible ORs from index 2: {3, 4}.
3. Split at index 1:
   • (1 XOR 3) = 2.
   • (1 XOR 4) = 5.
   • (2 XOR 3) = 1.
   • (2 XOR 4) = 6. Max value: 6.

Common mistakes candidates make

• $O(2^N)$ Brute Force: Trying to generate all subsequences, which is impossible for $N=400$.
• Ignoring the fixed size: Forgetting that each subsequence must have exactly $k$ elements.
• State Management: Not correctly clearing or updating the DP masks as you move the split point.

Interview preparation tip

For bitwise subsequence problems, the state is usually dp[index][count][mask]. If mask is small, you can use bitsets to speed up the transitions significantly. This is a high-level optimization that shows strong computer architecture awareness.