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962. Maximum Width Ramp

Description

A ramp in an integer array nums is a pair (i, j) for which i < j and nums[i] <= nums[j]. The width of such a ramp is j - i.

Given an integer array nums, return the maximum width of a ramp in nums. If there is no ramp in nums, return 0.

Β 

Example 1:

Input: nums = [6,0,8,2,1,5]
Output: 4
Explanation: The maximum width ramp is achieved at (i, j) = (1, 5): nums[1] = 0 and nums[5] = 5.

Example 2:

Input: nums = [9,8,1,0,1,9,4,0,4,1]
Output: 7
Explanation: The maximum width ramp is achieved at (i, j) = (2, 9): nums[2] = 1 and nums[9] = 1.

Β 

Constraints:

  • 2 <= nums.length <= 5 * 104
  • 0 <= nums[i] <= 5 * 104

Solutions

Solution 1: Monotonic Stack

According to the problem, we can find that the subsequence formed by all possible \(\textit{nums}[i]\) must be monotonically decreasing. Why is that? Let's prove it by contradiction.

Suppose there exist \(i_1<i_2\) and \(\textit{nums}[i_1]\leq\textit{nums}[i_2]\), then actually \(\textit{nums}[i_2]\) cannot be a candidate value, because \(\textit{nums}[i_1]\) is more to the left and would be a better value. Therefore, the subsequence formed by \(\textit{nums}[i]\) must be monotonically decreasing, and \(i\) must start from 0.

We use a monotonically decreasing stack \(\textit{stk}\) (from bottom to top) to store all possible \(\textit{nums}[i]\). Then we traverse \(j\) starting from the right boundary. If we encounter \(\textit{nums}[\textit{stk.top()}]\leq\textit{nums}[j]\), it means a ramp is formed at this point. We cyclically pop the top elements from the stack and update \(\textit{ans}\).

The time complexity is \(O(n)\) and the space complexity is \(O(n)\), where \(n\) represents the length of \(\textit{nums}\).

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class Solution:
    def maxWidthRamp(self, nums: List[int]) -> int:
        stk = []
        for i, v in enumerate(nums):
            if not stk or nums[stk[-1]] > v:
                stk.append(i)
        ans = 0
        for i in range(len(nums) - 1, -1, -1):
            while stk and nums[stk[-1]] <= nums[i]:
                ans = max(ans, i - stk.pop())
            if not stk:
                break
        return ans

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class Solution {
    public int maxWidthRamp(int[] nums) {
        int n = nums.length;
        Deque<Integer> stk = new ArrayDeque<>();
        for (int i = 0; i < n; ++i) {
            if (stk.isEmpty() || nums[stk.peek()] > nums[i]) {
                stk.push(i);
            }
        }
        int ans = 0;
        for (int i = n - 1; i >= 0; --i) {
            while (!stk.isEmpty() && nums[stk.peek()] <= nums[i]) {
                ans = Math.max(ans, i - stk.pop());
            }
            if (stk.isEmpty()) {
                break;
            }
        }
        return ans;
    }
}

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class Solution {
public:
    int maxWidthRamp(vector<int>& nums) {
        int n = nums.size();
        stack<int> stk;
        for (int i = 0; i < n; ++i) {
            if (stk.empty() || nums[stk.top()] > nums[i]) stk.push(i);
        }
        int ans = 0;
        for (int i = n - 1; i; --i) {
            while (!stk.empty() && nums[stk.top()] <= nums[i]) {
                ans = max(ans, i - stk.top());
                stk.pop();
            }
            if (stk.empty()) break;
        }
        return ans;
    }
};

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func maxWidthRamp(nums []int) int {
    n := len(nums)
    stk := []int{}
    for i, v := range nums {
        if len(stk) == 0 || nums[stk[len(stk)-1]] > v {
            stk = append(stk, i)
        }
    }
    ans := 0
    for i := n - 1; i >= 0; i-- {
        for len(stk) > 0 && nums[stk[len(stk)-1]] <= nums[i] {
            ans = max(ans, i-stk[len(stk)-1])
            stk = stk[:len(stk)-1]
        }
        if len(stk) == 0 {
            break
        }
    }
    return ans
}

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function maxWidthRamp(nums: number[]): number {
    let [ans, n] = [0, nums.length];
    const stk: number[] = [];

    for (let i = 0; i < n - 1; i++) {
        if (stk.length === 0 || nums[stk.at(-1)!] > nums[i]) {
            stk.push(i);
        }
    }

    for (let i = n - 1; i >= 0; i--) {
        while (stk.length && nums[stk.at(-1)!] <= nums[i]) {
            ans = Math.max(ans, i - stk.pop()!);
        }
        if (stk.length === 0) break;
    }

    return ans;
}

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function maxWidthRamp(nums) {
    let [ans, n] = [0, nums.length];
    const stk = [];

    for (let i = 0; i < n - 1; i++) {
        if (stk.length === 0 || nums[stk.at(-1)] > nums[i]) {
            stk.push(i);
        }
    }

    for (let i = n - 1; i >= 0; i--) {
        while (stk.length && nums[stk.at(-1)] <= nums[i]) {
            ans = Math.max(ans, i - stk.pop());
        }
        if (stk.length === 0) break;
    }

    return ans;
}

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