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2012. Sum of Beauty in the Array

Description

You are given a 0-indexed integer array nums. For each index i (1 <= i <= nums.length - 2) the beauty of nums[i] equals:

  • 2, if nums[j] < nums[i] < nums[k], for all 0 <= j < i and for all i < k <= nums.length - 1.
  • 1, if nums[i - 1] < nums[i] < nums[i + 1], and the previous condition is not satisfied.
  • 0, if none of the previous conditions holds.

Return the sum of beauty of all nums[i] where 1 <= i <= nums.length - 2.

 

Example 1:

Input: nums = [1,2,3]
Output: 2
Explanation: For each index i in the range 1 <= i <= 1:
- The beauty of nums[1] equals 2.

Example 2:

Input: nums = [2,4,6,4]
Output: 1
Explanation: For each index i in the range 1 <= i <= 2:
- The beauty of nums[1] equals 1.
- The beauty of nums[2] equals 0.

Example 3:

Input: nums = [3,2,1]
Output: 0
Explanation: For each index i in the range 1 <= i <= 1:
- The beauty of nums[1] equals 0.

 

Constraints:

  • 3 <= nums.length <= 105
  • 1 <= nums[i] <= 105

Solutions

Solution 1: Preprocessing Right Minimum + Traversing to Maintain Left Maximum

We can preprocess the right minimum array $right$, where $right[i]$ represents the minimum value in $nums[i..n-1]$.

Then we traverse the array $nums$ from left to right, while maintaining the maximum value $l$ on the left. For each position $i$, we judge whether $l < nums[i] < right[i + 1]$ holds. If it does, we add $2$ to the answer. Otherwise, we judge whether $nums[i - 1] < nums[i] < nums[i + 1]$ holds. If it does, we add $1$ to the answer.

After the traversal, we can get the answer.

The time complexity is $O(n)$, and the space complexity is $O(n)$. Here, $n$ is the length of the array $nums$.

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class Solution:
    def sumOfBeauties(self, nums: List[int]) -> int:
        n = len(nums)
        right = [nums[-1]] * n
        for i in range(n - 2, -1, -1):
            right[i] = min(right[i + 1], nums[i])
        ans = 0
        l = nums[0]
        for i in range(1, n - 1):
            r = right[i + 1]
            if l < nums[i] < r:
                ans += 2
            elif nums[i - 1] < nums[i] < nums[i + 1]:
                ans += 1
            l = max(l, nums[i])
        return ans

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class Solution {
    public int sumOfBeauties(int[] nums) {
        int n = nums.length;
        int[] right = new int[n];
        right[n - 1] = nums[n - 1];
        for (int i = n - 2; i > 0; --i) {
            right[i] = Math.min(right[i + 1], nums[i]);
        }
        int ans = 0;
        int l = nums[0];
        for (int i = 1; i < n - 1; ++i) {
            int r = right[i + 1];
            if (l < nums[i] && nums[i] < r) {
                ans += 2;
            } else if (nums[i - 1] < nums[i] && nums[i] < nums[i + 1]) {
                ans += 1;
            }
            l = Math.max(l, nums[i]);
        }
        return ans;
    }
}

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class Solution {
public:
    int sumOfBeauties(vector<int>& nums) {
        int n = nums.size();
        vector<int> right(n, nums[n - 1]);
        for (int i = n - 2; i; --i) {
            right[i] = min(right[i + 1], nums[i]);
        }
        int ans = 0;
        for (int i = 1, l = nums[0]; i < n - 1; ++i) {
            int r = right[i + 1];
            if (l < nums[i] && nums[i] < r) {
                ans += 2;
            } else if (nums[i - 1] < nums[i] && nums[i] < nums[i + 1]) {
                ans += 1;
            }
            l = max(l, nums[i]);
        }
        return ans;
    }
};

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func sumOfBeauties(nums []int) (ans int) {
    n := len(nums)
    right := make([]int, n)
    right[n-1] = nums[n-1]
    for i := n - 2; i > 0; i-- {
        right[i] = min(right[i+1], nums[i])
    }
    for i, l := 1, nums[0]; i < n-1; i++ {
        r := right[i+1]
        if l < nums[i] && nums[i] < r {
            ans += 2
        } else if nums[i-1] < nums[i] && nums[i] < nums[i+1] {
            ans++
        }
        l = max(l, nums[i])
    }
    return
}

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function sumOfBeauties(nums: number[]): number {
    const n = nums.length;
    const right: number[] = Array(n).fill(nums[n - 1]);
    for (let i = n - 2; i; --i) {
        right[i] = Math.min(right[i + 1], nums[i]);
    }
    let ans = 0;
    for (let i = 1, l = nums[0]; i < n - 1; ++i) {
        const r = right[i + 1];
        if (l < nums[i] && nums[i] < r) {
            ans += 2;
        } else if (nums[i - 1] < nums[i] && nums[i] < nums[i + 1]) {
            ans += 1;
        }
        l = Math.max(l, nums[i]);
    }
    return ans;
}

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