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 the following operation at most k times:
- Choose any non-empty subarray nums[l, ..., r] that you haven't chosen previously.
- Choose an element x of nums[l, ..., r] with the highest prime score. If multiple such elements exist, choose the one with the smallest index.
- Multiply your score by x.
Here, nums[l, ..., r] denotes the subarray of nums starting at index l and ending at the index r, both ends being inclusive.
The prime score of an integer x is equal to the number of distinct prime factors of x. For example, the prime score of 300 is 3 since 300 = 2 * 2 * 3 * 5 * 5.
Return the maximum possible score after applying at most k operations.
Since the answer may be large, return it modulo 10^9 + 7.
Example 1:
Input: nums = [8,3,9,3,8], k = 2
Output: 81
Explanation: 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
Explanation: 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 <= 10^5
1 <= nums[i] <= 10^5
1 <= k <= min(n * (n + 1) / 2, 10^9)
Approach
1 Compute Prime Scores:
- Calculate the prime score for each integer in the array nums. Prime score represents the number of distinct prime factors of an integer.
- Initialize a boolean array prime of size upper, where upper is the maximum element in nums plus 1.
- Initialize an integer array primeScore of the same size.
- Set prime[0] and prime[1] to false.
- Iterate over integers from 2 to upper - 1, and update primeScore and prime based on their prime factors.
2 Compute Next Greater Elements:
- Initialize arrays nextGreaterElement and prevGreaterOrEqualElement of size n, where n is the length of nums.
- Use a monotonic stack to find the next greater element with a greater prime score for each element in nums.
- Iterate through nums and maintain a stack of indices.
- For each element, pop elements from the stack if their prime score is less than or equal to the current element's prime score.
- Record the index of the top of the stack as the nextGreaterElement if the stack is not empty, else set it to n.
- Repeat the above process in reverse to compute prevGreaterOrEqualElement.
3 Sort and Process Elements:
- Create an array of tuples (num, i) where num is the value of an element and i is its index in nums.
- Sort the tuples in descending order of the first element (num).
- Loop through the sorted tuples and perform the following steps:Compute the number of operations as the minimum of (i - prevGreaterOrEqualElement[i]) * (nextGreaterElement[i] - i) and k.Update res by multiplying it with pow(num, operations) modulo MOD using the helper function pow.Decrement k by the number of operations.If k becomes 0, return res.
4 Helper Function for Exponentiation:
- Implement the pow function to calculate exponentiation efficiently using modular arithmetic.
Complexity
- Time complexity: O(max(nums) * log(max(nums)) + n * log(n)). Accounting for computing prime scores, using the stack to compute next greater elements, and sorting the tuples.
- Space complexity: O(max(nums) + n). Considering the space required for arrays and the stack used for computation.
Code (Swift)
class Solution {
func maximumScore(_ nums: [Int], _ k: Int) -> Int {
let MOD = 1_000_000_007
var k = k // Make a mutable copy of k
let n = nums.count
var upper = nums.max()! + 1
var prime = [Bool](repeating: true, count: upper)
prime[0] = false
prime[1] = false
var primeScore = [Int](repeating: 0, count: upper)
for i in 2..<upper {
if prime[i] {
var j = i
while j < upper {
primeScore[j] += 1
prime[j] = false
j += i
}
}
}
var nextGreaterElement = [Int](repeating: n, count: n)
var s = [Int]()
for i in (0..<n).reversed() {
while !s.isEmpty && primeScore[nums[i]] >= primeScore[nums[s.last!]] {
s.popLast()
}
nextGreaterElement[i] = s.isEmpty ? n : s.last!
s.append(i)
}
var prevGreaterOrEqualElement = [Int](repeating: -1, count: n)
s.removeAll()
for i in 0..<n {
while !s.isEmpty && primeScore[nums[i]] > primeScore[nums[s.last!]] {
s.popLast()
}
prevGreaterOrEqualElement[i] = s.isEmpty ? -1 : s.last!
s.append(i)
}
var res = 1
var tuples = [(num: Int, index: Int)]()
for i in 0..<n {
tuples.append((nums[i], i))
}
tuples.sort { a, b in
a.num > b.num
}
for (num, i) in tuples {
let operations = min(
(i - prevGreaterOrEqualElement[i]) * (nextGreaterElement[i] - i), k)
res = (res * pow(num, operations, MOD)) % MOD
k -= operations
if k == 0 {
return res
}
}
return res
}
func pow(_ x: Int, _ n: Int, _ mod: Int) -> Int {
var res = 1
var x = x
var n = n
while n > 0 {
if n % 2 == 1 {
res = (res * x) % mod
}
x = (x * x) % mod
n /= 2
}
return res
}
}
Source: Github
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