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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 prime p strictly less than nums[i], then subtract p from nums[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

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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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

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class Solution {
    public boolean primeSubOperation(int[] nums) {
        List<Integer> p = new ArrayList<>();
        for (int i = 2; i <= 1000; ++i) {
            boolean ok = true;
            for (int j : p) {
                if (i % j == 0) {
                    ok = false;
                    break;
                }
            }
            if (ok) {
                p.add(i);
            }
        }
        int n = nums.length;
        for (int i = n - 2; i >= 0; --i) {
            if (nums[i] < nums[i + 1]) {
                continue;
            }
            int j = search(p, nums[i] - nums[i + 1]);
            if (j == p.size() || p.get(j) >= nums[i]) {
                return false;
            }
            nums[i] -= p.get(j);
        }
        return true;
    }

    private int search(List<Integer> nums, int x) {
        int l = 0, r = nums.size();
        while (l < r) {
            int mid = (l + r) >> 1;
            if (nums.get(mid) > x) {
                r = mid;
            } else {
                l = mid + 1;
            }
        }
        return l;
    }
}

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class Solution {
public:
    bool primeSubOperation(vector<int>& nums) {
        vector<int> p;
        for (int i = 2; i <= 1000; ++i) {
            bool ok = true;
            for (int j : p) {
                if (i % j == 0) {
                    ok = false;
                    break;
                }
            }
            if (ok) {
                p.push_back(i);
            }
        }
        int n = nums.size();
        for (int i = n - 2; i >= 0; --i) {
            if (nums[i] < nums[i + 1]) {
                continue;
            }
            int j = upper_bound(p.begin(), p.end(), nums[i] - nums[i + 1]) - p.begin();
            if (j == p.size() || p[j] >= nums[i]) {
                return false;
            }
            nums[i] -= p[j];
        }
        return true;
    }
};

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func primeSubOperation(nums []int) bool {
    p := []int{}
    for i := 2; i <= 1000; i++ {
        ok := true
        for _, j := range p {
            if i%j == 0 {
                ok = false
                break
            }
        }
        if ok {
            p = append(p, i)
        }
    }
    for i := len(nums) - 2; i >= 0; i-- {
        if nums[i] < nums[i+1] {
            continue
        }
        j := sort.SearchInts(p, nums[i]-nums[i+1]+1)
        if j == len(p) || p[j] >= nums[i] {
            return false
        }
        nums[i] -= p[j]
    }
    return true
}

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function primeSubOperation(nums: number[]): boolean {
    const p: number[] = [];
    for (let i = 2; i <= 1000; ++i) {
        let ok = true;
        for (const j of p) {
            if (i % j === 0) {
                ok = false;
                break;
            }
        }
        if (ok) {
            p.push(i);
        }
    }
    const search = (x: number): number => {
        let l = 0;
        let r = p.length;
        while (l < r) {
            const mid = (l + r) >> 1;
            if (p[mid] > x) {
                r = mid;
            } else {
                l = mid + 1;
            }
        }
        return l;
    };
    const n = nums.length;
    for (let i = n - 2; i >= 0; --i) {
        if (nums[i] < nums[i + 1]) {
            continue;
        }
        const j = search(nums[i] - nums[i + 1]);
        if (j === p.length || p[j] >= nums[i]) {
            return false;
        }
        nums[i] -= p[j];
    }
    return true;
}

Solution 2: Preprocessing prime numbers

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function primeSubOperation(nums: number[]): boolean {
    const p: number[] = [];
    const max = Math.max(...nums);

    for (let i = 2; i < max; i++) {
        let isPrime = true;

        for (const x of p) {
            if (i % x === 0) {
                isPrime = false;
                break;
            }
        }

        while (isPrime && p.length <= i) {
            p.push(i);
        }
    }

    for (let i = nums.length - 2; i >= 0; i--) {
        if (nums[i] < nums[i + 1]) continue;

        const [x, next] = [nums[i], nums[i + 1]];
        const prime = p[x - next + 1];

        if (!prime || prime >= x) return false;
        nums[i] -= prime;
    }

    return true;
}

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function primeSubOperation(nums) {
    const p = [];
    const max = Math.max(...nums);

    for (let i = 2; i < max; i++) {
        let isPrime = true;

        for (const x of p) {
            if (i % x === 0) {
                isPrime = false;
                break;
            }
        }

        while (isPrime && p.length <= i) {
            p.push(i);
        }
    }

    for (let i = nums.length - 2; i >= 0; i--) {
        if (nums[i] < nums[i + 1]) continue;

        const [x, next] = [nums[i], nums[i + 1]];
        const prime = p[x - next + 1];

        if (!prime || prime >= x) return false;
        nums[i] -= prime;
    }

    return true;
}

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