In this problem, you're tasked with maximizing your score by applying operations to a subarray. The challenge involves calculating prime scores for each number, optimizing the score using stack-based state management, and repeating the operation up to k times. Understanding the operations and efficiently applying them will lead to the highest score.
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.
How GhostInterview Helps
- Provides optimized techniques for calculating prime scores efficiently in O(sqrt(nums[i])) time.
- Helps practice stack-based state management strategies, a critical part of this problem's solution.
- Guides on using greedy methods for selecting optimal subarrays to maximize the score.
Topic Pages
Related GhostInterview Pages
- LeetCode Interview Copilot - Use GhostInterview as a live solver when you want direct help with LeetCode-style coding questions.
- Coding Interview Assistant - See how GhostInterview supports array, string, linked list, graph, and tree interview workflows.
- How GhostInterview Works - Review the screenshot, reasoning, and answer flow before using the solver in a live interview.
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.
Need direct help with Apply Operations to Maximize Score instead of spending more time grinding it?
Download GhostInterview when you want a LeetCode solver, not another long practice loop. Capture Apply Operations to Maximize Score from a screenshot, get the answer path and complexity, and use supported stealth workflows that stay outside captured layers.
Capture the prompt fast instead of rewriting the problem by hand.
Get the solution path, trade-offs, and complexity summary in one pass.
Stay outside captured layers on supported screen-share workflows.
Stay in the same pattern family
Identify all weak characters in a game by analyzing attack and defense values using a stack-based greedy sorting approach efficiently.
Open problem page#2454 Next Greater Element IVFind the second greater integer for each element in an array using binary search and monotonic stack techniques.
Open problem page#769 Max Chunks To Make SortedThe Max Chunks To Make Sorted problem requires you to split an array into the maximum number of chunks that can be sorted independently to form a sorted array.
Open problem page