#2818

Hard

auto_awesomeStack-based state management

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Apply Operations to Maximize Score

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Maximize the score by applying operations on a subarray at most k times, utilizing stack-based state management.

Problem Statement

You are given an array nums of n positive integers and an integer k. Initially, you start with a score of 1. Your goal is to maximize your score by applying the following operation at most k times: select a subarray nums[l, ..., r] of any length, then multiply the score by the smallest element in the subarray. The goal is to choose subarrays and apply the operation to maximize the score within the given limits.

The value of the subarray is determined by the smallest element, and the score multiplies by this element. This operation can be repeated up to k times. Your task is to calculate the highest score possible. Efficiently managing the subarrays and understanding their prime scores will be key to solving the problem.

Examples

Example 1

Input: nums = [8,3,9,3,8], k = 2

Output: 81

To get a score of 81, we can apply the following operations:

Choose subarray nums[2, ..., 2]. nums[2] is the only element in this subarray. Hence, we multiply the score by nums[2]. The score becomes 1 * 9 = 9.
Choose subarray nums[2, ..., 3]. Both nums[2] and nums[3] have a prime score of 1, but nums[2] has the smaller index. Hence, we multiply the score by nums[2]. The score becomes 9 * 9 = 81. It can be proven that 81 is the highest score one can obtain.

Example 2

Input: nums = [19,12,14,6,10,18], k = 3

Output: 4788

To get a score of 4788, we can apply the following operations:

Choose subarray nums[0, ..., 0]. nums[0] is the only element in this subarray. Hence, we multiply the score by nums[0]. The score becomes 1 * 19 = 19.
Choose subarray nums[5, ..., 5]. nums[5] is the only element in this subarray. Hence, we multiply the score by nums[5]. The score becomes 19 * 18 = 342.
Choose subarray nums[2, ..., 3]. Both nums[2] and nums[3] have a prime score of 2, but nums[2] has the smaller index. Hence, we multipy the score by nums[2]. The score becomes 342 * 14 = 4788. It can be proven that 4788 is the highest score one can obtain.

Constraints

1 <= nums.length == n <= 105
1 <= nums[i] <= 105
1 <= k <= min(n * (n + 1) / 2, 109)

Solution Approach

Prime Factorization

To calculate the prime score for each element in the array, you will need to find its prime factorization. Use an optimized approach, such as O(sqrt(nums[i])) time, to quickly calculate the prime score for each number.

Stack-based State Management

Utilize a stack to manage the state of the subarrays. The stack will help in choosing the smallest element and maximizing the score by carefully selecting which subarrays to apply the operation on, considering the score optimization.

Greedy Approach for Subarray Selection

Employ a greedy approach to select the subarrays that will maximize the score. Evaluate each potential subarray and apply the operation when it yields the highest score, taking advantage of the stack to efficiently manage the subarrays and operations.

Complexity Analysis

Metric	Value
Time	O\left(n \times \left(\log{n} + \frac{\sqrt{m}}{\log{m}} + \log{k}\right) + m \log{\log{m}}\right)
Space	O(m + n)

The time complexity is dominated by the factorization of each number and the use of the stack to manage subarrays. The overall time complexity is O(n × (log n + √m / log m + log k) + m log log m), where n is the array length and m is the largest number in nums. The space complexity is O(m + n) due to the storage required for prime factors and stack management.

What Interviewers Usually Probe

Evaluates understanding of efficient prime factorization and number theory.
Assesses ability to use stack-based data structures for state management.
Tests knowledge of greedy algorithms and subarray optimization techniques.

Common Pitfalls or Variants

Common pitfalls

Overlooking the importance of optimizing prime factorization time, which can lead to inefficient solutions.
Incorrect subarray selection, which can result in suboptimal scores.
Failing to effectively manage the stack, leading to incorrect subarray evaluations.

Follow-up variants

Change the scoring operation to sum instead of multiply.
Consider allowing a different number of operations, increasing complexity.
Increase the size of the array or range of k to test scalability of the solution.

FAQ

How does prime factorization help in solving Apply Operations to Maximize Score?

Prime factorization is used to calculate the prime score for each element in the array, which is crucial for deciding which subarrays maximize the score.

What is the role of stack-based state management in this problem?

Stack-based state management helps in efficiently selecting and managing subarrays while optimizing the score with each operation.

What are the common mistakes when solving Apply Operations to Maximize Score?

Common mistakes include inefficient prime factorization, incorrect subarray selection, and improper use of stack data structures.

How do greedy algorithms apply to Apply Operations to Maximize Score?

A greedy approach is used to select the optimal subarrays that will maximize the score, based on the smallest element in each subarray.

What is the primary challenge in applying operations to maximize score?

The primary challenge is selecting the best subarrays and applying operations efficiently within the constraints of k operations and the array length.

terminal

Solution

Solution 1: Monotonic Stack + Greedy

It is not difficult to see that the number of subarrays with the highest prime score of an element $nums[i]$ is $cnt = (i - l) \times (r - i)$, where $l$ is the leftmost index such that $primeScore(nums[l]) \ge primeScore(nums[i])$, and $r$ is the rightmost index such that $primeScore(nums[r]) \ge primeScore(nums[i])$.

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def primeFactors(n):
    i = 2
    ans = set()
    while i * i <= n:
        while n % i == 0:
            ans.add(i)
            n //= i
        i += 1
    if n > 1:
        ans.add(n)
    return len(ans)


class Solution:
    def maximumScore(self, nums: List[int], k: int) -> int:
        mod = 10**9 + 7
        arr = [(i, primeFactors(x), x) for i, x in enumerate(nums)]
        n = len(nums)

        left = [-1] * n
        right = [n] * n
        stk = []
        for i, f, x in arr:
            while stk and stk[-1][0] < f:
                stk.pop()
            if stk:
                left[i] = stk[-1][1]
            stk.append((f, i))

        stk = []
        for i, f, x in arr[::-1]:
            while stk and stk[-1][0] <= f:
                stk.pop()
            if stk:
                right[i] = stk[-1][1]
            stk.append((f, i))

        arr.sort(key=lambda x: -x[2])
        ans = 1
        for i, f, x in arr:
            l, r = left[i], right[i]
            cnt = (i - l) * (r - i)
            if cnt <= k:
                ans = ans * pow(x, cnt, mod) % mod
                k -= cnt
            else:
                ans = ans * pow(x, k, mod) % mod
                break
        return ans

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