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Formatted question description: https://leetcode.ca/all/2464.html
2464. Minimum Subarrays in a Valid Split
Description
You are given an integer array nums
.
Splitting of an integer array nums
into subarrays is valid if:
- the greatest common divisor of the first and last elements of each subarray is greater than
1
, and - each element of
nums
belongs to exactly one subarray.
Return the minimum number of subarrays in a valid subarray splitting of nums
. If a valid subarray splitting is not possible, return -1
.
Note that:
- The greatest common divisor of two numbers is the largest positive integer that evenly divides both numbers.
- A subarray is a contiguous non-empty part of an array.
Example 1:
Input: nums = [2,6,3,4,3] Output: 2 Explanation: We can create a valid split in the following way: [2,6] | [3,4,3]. - The starting element of the 1st subarray is 2 and the ending is 6. Their greatest common divisor is 2, which is greater than 1. - The starting element of the 2nd subarray is 3 and the ending is 3. Their greatest common divisor is 3, which is greater than 1. It can be proved that 2 is the minimum number of subarrays that we can obtain in a valid split.
Example 2:
Input: nums = [3,5] Output: 2 Explanation: We can create a valid split in the following way: [3] | [5]. - The starting element of the 1st subarray is 3 and the ending is 3. Their greatest common divisor is 3, which is greater than 1. - The starting element of the 2nd subarray is 5 and the ending is 5. Their greatest common divisor is 5, which is greater than 1. It can be proved that 2 is the minimum number of subarrays that we can obtain in a valid split.
Example 3:
Input: nums = [1,2,1] Output: -1 Explanation: It is impossible to create valid split.
Constraints:
1 <= nums.length <= 1000
1 <= nums[i] <= 105
Solutions
Solution 1: Memoization Search
We design a function $dfs(i)$ to represent the minimum number of partitions starting from index $i$. For index $i$, we can enumerate all partition points $j$, i.e., $i \leq j < n$, where $n$ is the length of the array. For each partition point $j$, we need to determine whether the greatest common divisor of $nums[i]$ and $nums[j]$ is greater than $1$. If it is greater than $1$, we can partition, and the number of partitions is $1 + dfs(j + 1)$; otherwise, the number of partitions is $+\infty$. Finally, we take the minimum of all partition numbers.
The time complexity is $O(n^2)$, and the space complexity is $O(n)$. Here, $n$ is the length of the array.
-
class Solution { private int n; private int[] f; private int[] nums; private int inf = 0x3f3f3f3f; public int validSubarraySplit(int[] nums) { n = nums.length; f = new int[n]; this.nums = nums; int ans = dfs(0); return ans < inf ? ans : -1; } private int dfs(int i) { if (i >= n) { return 0; } if (f[i] > 0) { return f[i]; } int ans = inf; for (int j = i; j < n; ++j) { if (gcd(nums[i], nums[j]) > 1) { ans = Math.min(ans, 1 + dfs(j + 1)); } } f[i] = ans; return ans; } private int gcd(int a, int b) { return b == 0 ? a : gcd(b, a % b); } }
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class Solution { public: const int inf = 0x3f3f3f3f; int validSubarraySplit(vector<int>& nums) { int n = nums.size(); vector<int> f(n); function<int(int)> dfs = [&](int i) -> int { if (i >= n) return 0; if (f[i]) return f[i]; int ans = inf; for (int j = i; j < n; ++j) { if (__gcd(nums[i], nums[j]) > 1) { ans = min(ans, 1 + dfs(j + 1)); } } f[i] = ans; return ans; }; int ans = dfs(0); return ans < inf ? ans : -1; } };
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class Solution: def validSubarraySplit(self, nums: List[int]) -> int: @cache def dfs(i): if i >= n: return 0 ans = inf for j in range(i, n): if gcd(nums[i], nums[j]) > 1: ans = min(ans, 1 + dfs(j + 1)) return ans n = len(nums) ans = dfs(0) dfs.cache_clear() return ans if ans < inf else -1
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func validSubarraySplit(nums []int) int { n := len(nums) f := make([]int, n) var dfs func(int) int const inf int = 0x3f3f3f3f dfs = func(i int) int { if i >= n { return 0 } if f[i] > 0 { return f[i] } ans := inf for j := i; j < n; j++ { if gcd(nums[i], nums[j]) > 1 { ans = min(ans, 1+dfs(j+1)) } } f[i] = ans return ans } ans := dfs(0) if ans < inf { return ans } return -1 } func min(a, b int) int { if a < b { return a } return b } func gcd(a, b int) int { if b == 0 { return a } return gcd(b, a%b) }