2909. Minimum Sum of Mountain Triplets II

Problem Description

You are tasked with finding a specific type of triplet in an integer array nums, where the array indices start at 0. This triplet needs to follow the characteristics of a mountain, which means for indices i, j, k satisfying i < j < k, the elements at these indices must adhere to the pattern nums[i] < nums[j] and nums[k] < nums[j]. The goal is to determine the smallest possible sum of such a mountain triplet. If there are no triplets that form a mountain, you should return -1.

Intuition

The problem calls for an efficient way to find the smallest sum of a "mountain" within an array. To approach this, you would need to check each possible middle element (the peak of the mountain) to see if there's a smaller number before it (uphill) and a smaller number after it (downhill). A brute-force solution would be to try every possible triplet to find the minimum sum, but that would result in a high time complexity.

Instead, we can use pre-processing to optimize this procedure. By iterating through the array once, we can store the minimum value found to the right of each element in a new array called right. This pre-processing helps us quickly find the smallest element that can be the downhill part of the mountain for any middle element we're considering.

As we traverse the array to find the middle element of potential mountain triplets, we keep track of the smallest value encountered so far to serve as the uphill part of the mountain. This value is stored in a variable called left. At any point, if our current element is greater than both left and the corresponding value in right, we have found a valid mountain. We then check if the sum of left, the current element, and the corresponding right is less than the best answer we have found so far, updating our answer accordingly.

If after traversing the array no valid mountain is found, the answer remains at its initial infinite value and we return -1. Otherwise, we return the smallest sum found.

Solution Approach

The implementation of the solution is based on the Reference Solution Approach provided, which cleverly reduces the problem's complexity by pre-processing the array and using two pointers to track the minimum values required to identify a potential mountain.

Here's how the solution approach is implemented step by step:

1. Pre-processing: We create an array right which holds the minimum value to the right of each index in the nums array. We initialize every element in right to be inf (inf is a placeholder for infinity), which ensures that any number compared to it will be smaller. Starting from the second last element in nums and moving backwards, we populate right[i] with the minimum between right[i + 1] and nums[i].

2. One pass enumeration with two pointers: As we iterate through the nums array using the variable i, we maintain two pointers/values:

   • left: The minimum value in the subarray to the left of the current index i (including nums[i]).
   • ans: The result variable which keeps track of the minimum sum of a valid mountain that has been found so far.

3. Evaluating potential mountains: For each element nums[i], which we are considering to be the peak of the mountain (j), we check if we have a smaller number on its left (uphill) and on its right (downhill):

   • The check left < nums[i] ensures that the current peak is higher than the minimum value on the left.
   • The check right[i + 1] < nums[i] ensures that we have a valid downhill part for the mountain.
   • If both conditions are met, nums[i] can be the peak of a mountain, and we update ans with the sum of left, nums[i], and right[i + 1] if it's smaller than the current ans.

4. Updating the minimum left value: After each iteration, we update left to be the minimum value between the current left and nums[i].

5. Returning the result: Once we've iterated through the entire array, we check whether ans is still infinity. If it is, no valid mountain was found, and we return -1. Otherwise, we return the minimum sum stored in ans.

By following this approach, we avoid the need for a triple nested loop to check every possible triplet, which significantly optimizes the solution. The algorithm achieves a time complexity of O(n) since it only requires two passes through the nums array (one for pre-processing and one for the main iteration).

Example Walkthrough

To illustrate the solution approach, let's consider a small example of the nums array:

1nums = [2, 1, 4, 7, 3, 2, 5]

Step 1: Pre-processing

We begin by creating a right array that will contain the minimum value to the right of each index, initializing all elements to infinity:

1right = [inf, inf, inf, inf, inf, inf, inf]

We then update the right array by traversing from right to left:

1Starting with the second to last index, right[5] is min(inf, 2) = 2
2right = [inf, inf, inf, inf, inf, 2, inf]
3
4right[4] is min(2, 3) = 2
5right = [inf, inf, inf, inf, 2, 2, inf]
6
7right[3] is min(2, 7) = 2
8right = [inf, inf, inf, 2, 2, 2, inf]
9
10right[2] is min(2, 4) = 2
11right = [inf, inf, 2, 2, 2, 2, inf]
12
13right[1] is min(2, 1) = 1
14right = [inf, 1, 2, 2, 2, 2, inf]
15
16right[0] will not be used since it must be the leftmost element of a triplet.

Final right array after pre-processing:

1right = [inf, 1, 2, 2, 2, 2, inf]

Step 2: One pass enumeration with two pointers

We initialize two variables - left starting with infinity and ans also with infinity.

As we start iterating over the array, our pointers will update as follows:

1left = inf
2ans = inf

Step 3: Evaluating potential mountains

• When i is 0 (nums[i] is 2), i is at the leftmost edge, so left does not update because no valid mountain can start here.
• When i is 1 (nums[i] is 1), the left array so far [2] has a minimum of 2, and our right is 1. No valid mountain can have a peak at index 1 since left >= nums[i].
• When i is 2 (nums[i] is 4), we have left = 1 and right[3] is 2. This satisfies the mountain property (left < nums[i] > right[3]). We sum 1 + 4 + 2 to get the total sum of 7. Since ans was infinity, it is now updated to 7.
• When i is 3 (nums[i] is 7), a sum is possible here, but it won't be minimum because right[4] = 2 and the minimum sum would be 1 + 7 + 2 which is 10, larger than our current ans.
• For i 4, 5, and 6, while a valley exists to the right (right[i + 1] < nums[i]), no smaller value exists to the left (left >= nums[i]), so no valid mountain can exist at these points.

Step 4: Updating the minimum left value

After each iteration, left is updated to the minimum value it has seen so far (if smaller than nums[i]), which helps efficiently find the uphill part for the next iterations.

Step 5: Returning the result

After iterating through the entire nums array, we find that ans is 7. Since it's no longer infinity, we have found at least one valid mountain, and 7 is the smallest sum of such a mountain triplet.

Therefore, the smallest possible sum of a mountain triplet in the nums array is 7.

Solution Implementation

Time and Space Complexity

The time complexity of the provided code is indeed O(n). This is because the code consists of two loops over the nums list, each running for n iterations, where n is the length of the input list nums. The first loop is a reverse iteration used to build the right list, and the second loop goes through the nums list to compute the ans variable. Since both loops are independent and execute sequentially, their combined time complexity remains linear with the size of the input list.

The space complexity of the code is O(n) as well due to the allocation of the right list, which stores the minimum value encountered from the right side of the nums list. This right list has a length equal to n + 1, where n is the length of the nums list. The other variables used (ans, left, and a few others) do not depend on n and thus contribute a constant factor, which is ignorable when assessing space complexity.

Learn more about how to find time and space complexity quickly using problem constraints.