#2601

Medium

auto_awesomeBinary search over the valid answer space

LeetCode Problem Workspace

Prime Subtraction Operation

Determine if it's possible to make the array strictly increasing using prime subtractions.

array math binary search greedy number theory

Problem Statement

You are given a 0-indexed integer array nums of length n. You can perform the following operation as many times as you want: for any i (0 <= i < n), you may pick a prime p such that nums[i] - p is still a valid number. The goal is to determine if it's possible to use these operations to make the array strictly increasing.

Return true if you can make nums a strictly increasing array using the above operation, and false otherwise. You should focus on using binary search over the valid primes that can be subtracted to achieve the increasing order.

Examples

Example 1

Input: nums = [4,9,6,10]

Output: true

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

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

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

Solution Approach

Prime Selection via Binary Search

A binary search can be used to efficiently find the largest prime number smaller than or equal to each element in nums. This allows for optimal prime subtraction at each step, improving the solution's efficiency.

Greedy Subtraction Strategy

By using a greedy approach, always subtract the largest valid prime at each index in nums to keep the array increasing. This ensures that the solution is both optimal and correct.

Mathematical Insights into Primes

Understanding prime numbers and their properties is essential. Leveraging prime tables and efficient prime checking algorithms is crucial to solving this problem within time constraints.

Complexity Analysis

Metric	Value
Time	O(n + m \log \log (m))
Space	O(m)

The time complexity is O(n + m \log \log (m)) where n is the length of the array and m is the range of values in nums. The space complexity is O(m) due to the prime number storage required for efficient subtraction.

What Interviewers Usually Probe

Assess the candidate's understanding of prime number properties.
Look for knowledge in binary search optimization.
Check if the candidate is comfortable using greedy algorithms.

Common Pitfalls or Variants

Common pitfalls

Forgetting to subtract the correct prime at each index.
Not applying binary search efficiently, leading to unnecessary computation.
Confusing prime subtraction with other array manipulation techniques.

Follow-up variants

Adjust the problem by limiting the available primes to smaller sets.
Alter the array length to test scalability and performance.
Change the problem to require the array to be non-decreasing instead of strictly increasing.

FAQ

What is the optimal approach to solving the Prime Subtraction Operation problem?

The optimal approach involves using binary search to efficiently select the largest valid prime to subtract from each element in the array.

How does binary search improve the solution for the Prime Subtraction Operation problem?

Binary search allows you to quickly find the largest prime smaller than or equal to each element, minimizing unnecessary operations and ensuring efficiency.

Can this problem be solved with a greedy algorithm?

Yes, a greedy algorithm is a key component of the solution, as you always want to subtract the largest possible prime at each step to make the array strictly increasing.

What is the time complexity of the Prime Subtraction Operation problem?

The time complexity is O(n + m \log \log (m)) where n is the length of the array and m is the range of values in nums.

Are there any common mistakes to avoid when solving the Prime Subtraction Operation problem?

Common mistakes include failing to select the correct prime at each index, inefficient prime searching, and misunderstanding the problem's greedy nature.

terminal

Solution

Solution 1: Preprocessing prime numbers + binary search

We first preprocess all the primes within $1000$ and record them in the array $p$.

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class Solution:
    def primeSubOperation(self, nums: List[int]) -> bool:
        p = []
        for i in range(2, max(nums)):
            for j in p:
                if i % j == 0:
                    break
            else:
                p.append(i)

        n = len(nums)
        for i in range(n - 2, -1, -1):
            if nums[i] < nums[i + 1]:
                continue
            j = bisect_right(p, nums[i] - nums[i + 1])
            if j == len(p) or p[j] >= nums[i]:
                return False
            nums[i] -= p[j]
        return True

Solution 2: Preprocessing prime numbers

#### TypeScript

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class Solution:
    def primeSubOperation(self, nums: List[int]) -> bool:
        p = []
        for i in range(2, max(nums)):
            for j in p:
                if i % j == 0:
                    break
            else:
                p.append(i)

        n = len(nums)
        for i in range(n - 2, -1, -1):
            if nums[i] < nums[i + 1]:
                continue
            j = bisect_right(p, nums[i] - nums[i + 1])
            if j == len(p) or p[j] >= nums[i]:
                return False
            nums[i] -= p[j]
        return True

Continue Practicing

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