How do you solve Maximum AND Sum of Array on LeetCode?

Maximum AND Sum of Array (LeetCode #2172) can be solved using Brute Force or Optimal Solution. The optimal approach is Optimal Solution with O(n * 2^m) time complexity and O(2^m) space complexity. Key insight: Using bitmasking allows us to efficiently track the state of the slots.

What is the brute force solution for Maximum AND Sum of Array?

The brute force approach for Maximum AND Sum of Array is Brute Force, with O(n!) time complexity and O(n) space complexity. The brute-force approach involves generating all possible combinations of placing numbers into the slots. This method ensures that we explore every possible arrangement to find the maximum AND sum. The optimal Optimal Solution approach improves this to O(n * 2^m).

What algorithm or data structure is used to solve Maximum AND Sum of Array?

Maximum AND Sum of Array uses the following concepts: Array, Dynamic Programming, Bit Manipulation, Bitmask. The recommended patterns to study are: Dynamic Programming, Bitmasking, Backtracking.

What are the interview tips for Maximum AND Sum of Array?

Clearly explain your thought process and how you arrived at the solution. Be prepared to discuss the trade-offs between brute-force and optimal solutions. Practice implementing bitmasking and dynamic programming concepts in advance.

What are common mistakes when solving Maximum AND Sum of Array?

Failing to properly manage the state of the slots when using bitmasking. Not considering edge cases where slots might remain empty.

#2172

Maximum AND Sum of Array

Hard

ArrayDynamic ProgrammingBit ManipulationBitmaskDynamic ProgrammingBitmaskingBacktracking

LeetCode ↗

Approaches

Brute ForceOptimal

Complexity Comparison

	Brute Force	Optimal Solution★
Time	O(n!)	O(n * 2^m)
Space	O(n)	O(2^m)

💡

Intuition

Time O(n * 2^m)Space O(2^m)

The optimal solution uses dynamic programming with bitmasking to efficiently track the state of the slots and calculate the maximum AND sum. This approach significantly reduces the number of combinations we need to evaluate.

⚙️

Algorithm

3 steps

1Step 1: Use a bitmask to represent the state of the slots, where each bit indicates whether a slot is filled and how many numbers it contains.
2Step 2: Use a recursive function with memoization to explore all possible placements of numbers into the slots while calculating the AND sum.
3Step 3: Return the maximum AND sum found across all valid placements.

solution.py14 lines

1def max_and_sum_optimal(nums, numSlots):
2    from functools import lru_cache
3    n = len(nums)
4    @lru_cache(None)
5    def dp(mask, count):
6        if count == n:
7            return 0
8        max_sum = 0
9        for i in range(numSlots):
10            if (mask >> i) & 3 < 2:
11                new_mask = mask | (1 << (i * 2)) | (1 << (i * 2 + 1))
12                max_sum = max(max_sum, (nums[count] & (i + 1)) + dp(new_mask, count + 1))
13        return max_sum
14    return dp(0, 0)

ℹ

Complexity note: This complexity arises from the number of states represented by the bitmask (2^m) and the number of recursive calls (n) made for each state.

1Using bitmasking allows us to efficiently track the state of the slots.
2Dynamic programming helps avoid redundant calculations by storing previously computed results.

Solutions and explanations are original Tejav content. Problem titles © LeetCode — use the LeetCode button above for the full problem statement.