#2221

Medium

LeetCode Problem Workspace

Find Triangular Sum of an Array

The problem asks for calculating the triangular sum of an array through repeated pairwise summation.

Problem Statement

You are given a 0-indexed integer array nums, where each element is between 0 and 9 (inclusive). The triangular sum of the array is the value that remains after performing the following operation repeatedly:

In each step, the array is updated by summing each adjacent pair of elements, and replacing the current array with the results of these sums. This process continues until only one element remains in the array. Your task is to return this final element, the triangular sum of nums.

Examples

Example 1

Input: nums = [1,2,3,4,5]

Output: 8

The above diagram depicts the process from which we obtain the triangular sum of the array.

Example 2

Input: nums = [5]

Output: 5

Since there is only one element in nums, the triangular sum is the value of that element itself.

Constraints

1 <= nums.length <= 1000
0 <= nums[i] <= 9

Solution Approach

Simulate the Process Iteratively

You can simulate the process by iterating through the array, summing each adjacent pair of elements, and updating the array. Repeat this process until a single element remains.

Optimize Space Complexity

Instead of modifying the array in-place, you could use a new array for the sums to avoid modifying the array in a way that affects the ongoing calculations.

Minimize Time Complexity with Direct Calculation

For larger arrays, it's essential to focus on minimizing time complexity by avoiding redundant operations or recalculations. An efficient approach ensures the operation is done in O(n) time.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity of the solution depends on the approach. Simulating the process iteratively will take O(n) time for each step, and since there are n steps, the overall complexity is O(n^2). Space complexity can be reduced by reusing the array, making the space complexity O(1).

What Interviewers Usually Probe

Look for understanding of iterative processes in arrays.
Check for awareness of space optimization techniques in array manipulation.
Evaluate familiarity with time complexity reductions and efficient problem-solving.

Common Pitfalls or Variants

Common pitfalls

Failing to correctly simulate the process by not updating the array properly in each step.
Not optimizing for space by using an additional array when the process can be done in-place.
Overcomplicating the solution by attempting to handle the problem in a more complex way when a simple iterative approach works.

Follow-up variants

Different array sizes could be tested, from the smallest case with one element to the largest with 1000 elements.
Modifying the problem to use elements larger than 9 to test how the solution handles different ranges of numbers.
Introducing more complex operations between elements (such as subtraction or multiplication) could test the adaptability of the algorithm.

FAQ

What is the triangular sum of an array?

The triangular sum of an array is the final element obtained by repeatedly summing adjacent pairs in the array until only one element remains.

How does the iterative process work in the "Find Triangular Sum of an Array" problem?

You repeatedly sum adjacent pairs of elements, updating the array with these sums, until you have only one element remaining.

What are the time and space complexities of solving this problem?

Time complexity is O(n^2) for the simulation, and space complexity is O(1) if done in-place, or O(n) if using an additional array.

How can I optimize my approach to solving the triangular sum problem?

Focus on minimizing the number of steps by efficiently handling array updates, and consider in-place array modifications to save space.

What is the best approach for large arrays in the "Find Triangular Sum of an Array" problem?

For large arrays, make sure to minimize time complexity by avoiding redundant operations, and reduce space complexity by reusing the array for updates.

terminal

Solution

Solution 1: Simulation

We can directly simulate the operations described in the problem. Perform $n - 1$ rounds of operations on the array $\textit{nums}$, updating the array $\textit{nums}$ according to the rules described in the problem for each round. Finally, return the only remaining element in the array $\textit{nums}$.

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class Solution:
    def triangularSum(self, nums: List[int]) -> int:
        for k in range(len(nums) - 1, 0, -1):
            for i in range(k):
                nums[i] = (nums[i] + nums[i + 1]) % 10
        return nums[0]

Continue Practicing

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