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Apply Operations to Maximize Score

Difficulty: Hard

Problem Description

You are given an array nums of n positive integers and an integer k. Initially, you start with a score of 1. You have to maximize your score by applying a specific operation at most k times. The operation involves selecting a non-empty subarray, choosing an element with the highest prime score from that subarray, and multiplying your score by that element. The prime score of an integer is the number of distinct prime factors of that integer. The goal is to return the maximum possible score after applying at most k operations, modulo 10^9 + 7.

Key Insights

  • The prime score is determined by the distinct prime factors of each number.
  • The operations can be performed on any non-empty subarray, allowing flexible selection of elements.
  • The optimal strategy involves selecting elements with the highest prime scores while considering their indices.
  • The constraints allow for a large number of elements and operations, necessitating efficient algorithms for factorization and score calculation.

Space and Time Complexity

Time Complexity: O(n log(max(nums))) for calculating prime scores and O(k log(k)) for selecting top k scores. Space Complexity: O(n) for storing prime scores and results.

Solution

To solve the problem, we first need to calculate the prime score for each element in the nums array. This can be efficiently achieved using a sieve method to determine the number of distinct prime factors for each integer up to the maximum value in nums. After calculating the prime scores, we can maintain a list of tuples containing each number and its prime score. We then sort this list based on prime scores in descending order, and in case of ties, by the index of the numbers. Finally, we multiply the top k scores together to maximize the final score, returning it modulo 10^9 + 7.

Code Solutions

def maxScore(nums, k):
    MOD = 10**9 + 7

    def prime_score(n):
        count = 0
        for i in range(2, int(n**0.5) + 1):
            if n % i == 0:
                count += 1
                while n % i == 0:
                    n //= i
        if n > 1:
            count += 1
        return count

    scores = [(num, prime_score(num)) for num in nums]
    scores.sort(key=lambda x: (-x[1], nums.index(x[0])))

    result = 1
    for i in range(min(k, len(scores))):
        result = (result * scores[i][0]) % MOD

    return result
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