#1799

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Maximize Score After N Operations

Maximize the score after n operations by selecting pairs from the array and using their GCD with dynamic programming or backtracking techniques.

array math dynamic programming backtracking bit manipulation

Problem Statement

You are given an array nums of size 2 * n. Your task is to perform n operations on this array, where in the ith operation, you pick two numbers, compute their GCD, and multiply it by i. The total score is the sum of these values across all operations.

The challenge lies in maximizing the score by selecting the optimal pairs to compute the GCD and considering the multiplication factor in each operation. Your task is to return the maximum score you can achieve after n operations.

Examples

Example 1

Input: nums = [1,2]

Output: 1

The optimal choice of operations is: (1 * gcd(1, 2)) = 1

Example 2

Input: nums = [3,4,6,8]

Output: 11

The optimal choice of operations is: (1 * gcd(3, 6)) + (2 * gcd(4, 8)) = 3 + 8 = 11

Example 3

Input: nums = [1,2,3,4,5,6]

Output: 14

The optimal choice of operations is: (1 * gcd(1, 5)) + (2 * gcd(2, 4)) + (3 * gcd(3, 6)) = 1 + 4 + 9 = 14

Constraints

1 <= n <= 7
nums.length == 2 * n
1 <= nums[i] <= 106

Solution Approach

State Transition Dynamic Programming

The problem can be solved efficiently using state transition dynamic programming. We can represent the state of the array and recursively calculate the maximum score by picking pairs of elements. By keeping track of the pairs already chosen, we avoid redundant calculations.

Backtracking with Memoization

Backtracking can be used to explore all possible pairings of elements in the array. With memoization, we store intermediate results to avoid redundant calculations, significantly reducing time complexity compared to a pure brute force solution.

Bitmasking

A bitmasking approach allows us to represent which elements of the array have been paired, making it easier to explore all valid pairings. This technique can be used to generate all possible pair combinations and calculate the maximum score.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity depends on the approach. A dynamic programming solution with bitmasking can achieve a time complexity of O(n * 2^n), where n is the size of the array. The space complexity is O(n * 2^n) due to the storage of intermediate states.

What Interviewers Usually Probe

Check for the candidate's ability to apply dynamic programming in combinatorial problems.
Assess how efficiently the candidate handles memoization and backtracking to reduce computational overhead.
Look for understanding of bitmasking techniques and how it can be applied to problems involving subsets.

Common Pitfalls or Variants

Common pitfalls

Failing to efficiently memoize subproblems, leading to redundant recalculations.
Not correctly handling the state transitions in dynamic programming, which can cause incorrect results or excessive time complexity.
Overcomplicating the problem with unnecessary optimizations or not fully exploring all pairings.

Follow-up variants

Consider the problem with different constraints, such as larger arrays or additional scoring rules.
Allow multiple solutions for pairing, where the goal could be to find the second-highest score instead of the maximum.
Modify the problem to return all possible score configurations and have the candidate find the one with the maximum score.

FAQ

What is the optimal approach for maximizing score in the "Maximize Score After N Operations" problem?

The optimal approach is state transition dynamic programming, where you explore all possible pairings of elements in the array while maximizing the score based on their GCD values.

Can backtracking be used to solve this problem?

Yes, backtracking can be used with memoization to efficiently explore all possible pairings while avoiding redundant calculations.

How does bitmasking help in solving the "Maximize Score After N Operations" problem?

Bitmasking allows you to represent which elements have been paired, making it easier to explore all valid pair combinations and calculate the maximum score.

What is the time complexity of the optimal solution?

The time complexity of the optimal solution using dynamic programming with bitmasking is O(n * 2^n), where n is the size of the array.

How can I avoid redundant calculations in this problem?

Memoization and backtracking are key to avoiding redundant calculations by storing intermediate results and only recalculating when necessary.

terminal

Solution

Solution 1: State Compression + Dynamic Programming

We can preprocess to get the greatest common divisor of any two numbers in the array `nums`, stored in the two-dimensional array $g$, where $g[i][j]$ represents the greatest common divisor of $nums[i]$ and $nums[j]$.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

class Solution:
    def maxScore(self, nums: List[int]) -> int:
        m = len(nums)
        f = [0] * (1 << m)
        g = [[0] * m for _ in range(m)]
        for i in range(m):
            for j in range(i + 1, m):
                g[i][j] = gcd(nums[i], nums[j])
        for k in range(1 << m):
            if (cnt := k.bit_count()) % 2 == 0:
                for i in range(m):
                    if k >> i & 1:
                        for j in range(i + 1, m):
                            if k >> j & 1:
                                f[k] = max(
                                    f[k],
                                    f[k ^ (1 << i) ^ (1 << j)] + cnt // 2 * g[i][j],
                                )
        return f[-1]

Continue Practicing

#1994 the number of good subsets #996 number of squareful arrays #1255 maximum score words formed by letters