2601. Prime Subtraction Operation
Description
You are given a 0-indexed integer array nums
of length n
.
You can perform the following operation as many times as you want:
- Pick an index
i
that you haven’t picked before, and pick a primep
strictly less thannums[i]
, then subtractp
fromnums[i]
.
Return true if you can make nums
a strictly increasing array using the above operation and false otherwise.
A strictly increasing array is an array whose each element is strictly greater than its preceding element.
Example 1:
Input: nums = [4,9,6,10] Output: true Explanation: In the first operation: Pick i = 0 and p = 3, and then subtract 3 from nums[0], so that nums becomes [1,9,6,10]. In the second operation: i = 1, p = 7, subtract 7 from nums[1], so nums becomes equal to [1,2,6,10]. After the second operation, nums is sorted in strictly increasing order, so the answer is true.
Example 2:
Input: nums = [6,8,11,12] Output: true Explanation: Initially nums is sorted in strictly increasing order, so we don't need to make any operations.
Example 3:
Input: nums = [5,8,3] Output: false Explanation: It can be proven that there is no way to perform operations to make nums sorted in strictly increasing order, so the answer is false.
Constraints:
1 <= nums.length <= 1000
1 <= nums[i] <= 1000
nums.length == n
Solutions
Solution 1: Preprocessing prime numbers + binary search
We first preprocess all the primes within $1000$ and record them in the array $p$.
For each element $nums[i]$ in the array $nums$, we need to find a prime $p[j]$ such that $p[j] \gt nums[i] - nums[i + 1]$ and $p[j]$ is as small as possible. If there is no such prime, it means that it cannot be strictly increased by subtraction operations, return false
. If there is such a prime, we will subtract $p[j]$ from $nums[i]$ and continue to process the next element.
If all the elements in $nums$ are processed, it means that it can be strictly increased by subtraction operations, return true
.
The time complexity is $O(n \log n)$ and the space complexity is $O(n)$. where $n$ is the length of the array $nums$.
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