Problem Description

The problem presents us with an integer array nums with each element being a digit from 0 to 9. We need to compute what is called the "triangular sum" of this array.

To find the triangular sum, we repeatedly perform the following process:

1. Check if the array nums has only one element. If yes, the process ends, and this element is our triangular sum.
2. If the array has more than one element, we create a new array newNums that has one less element than nums.
3. We then populate newNums such that each element at index i is equal to (nums[i] + nums[i+1]) % 10. This is the sum of consecutive elements in the original array modulo 10.
4. We replace nums with newNums and repeat the process from step 1.

Our goal is to keep repeating this process until there is only one element left in nums, which will be our answer.

Intuition

The solution is based on iterative reduction. Notice that in each iteration, the array size is reduced by 1. This process continues until only one element remains. Our task is to simulate this process programmatically.

The approach avoids creating multiple arrays for each iteration by continually updating the original array, which is both space-efficient and direct. It leverages in-place updates for nums, each time updating the element at index j based on the element at j and j + 1.

We initiate this process from the first element (index 0) to the second last element (n - 2) of the current iteration's array size, and afterward, we reduce our array size (n) by 1. This process is repeated until we reach the array size of 1, and this final single element is the triangular sum of the original array.

Learn more about Math and Combinatorics patterns.

Solution Approach

The reference solution code provided demonstrates the implementation of the intuitive approach we discussed earlier. It utlizes a nested loop to successively reduce the array and calculate the new values.

Here's a step-by-step breakdown of the algorithm applied in the solution code:

1. Determine the length of the nums array and store it in the variable n.
2. Start an outer loop to iteratively reduce the size of the nums array. This loop runs from the current size n down to 1.
3. Inside the outer loop, there's an inner loop which is where the actual update takes place. This loop runs from 0 to i - 1, where i is the current size of the nums array for that iteration.
4. Within the inner loop, calculate the new value for nums[j] as (nums[j] + nums[j + 1]) % 10. This formula is applied to every pair of adjacent elements, effectively shifting the entire nums array by one position to the left.
5. When the outer loop finishes (which is when the array size n reaches 1), we have our triangular sum stored in nums[0], and we return this value as the triangular sum of the array.

The data structure used here is just the initial integer array nums, and we are manipulating it in place to maintain space efficiency. No auxiliary data structures are used.

The pattern followed is iterative reduction combined with in-place updates, which is a common way to deal with problems where the size of the data structure changes in each iteration based on specific calculation rules.

Here is the core part of the solution code that aligns with the description above:

1n = len(nums)
2for i in range(n, 0, -1):
3    for j in range(i - 1):
4        nums[j] = (nums[j] + nums[j + 1]) % 10
5return nums[0]

As seen, the outer loop runs until the array is reduced to a single element, and the inner loop makes the required modifications to the nums array based on the transition rule (nums[i] + nums[i+1]) % 10.

This approach is both clean and efficient, allowing for easy understanding and avoiding any unnecessary space complexity.

Example Walkthrough

Let's take a small example to illustrate the solution approach. Suppose nums is [1, 2, 3, 4, 5]. We need to apply the solution approach to this input to find the triangular sum.

1. n = len(nums) will set n to be 5 since there are five elements in the input array.

2. We then enter the outer loop, which will continue until n is reduced to 1.

3. The first iteration of the outer loop will not change n, but will initiate the inner loop which runs n-1 times (4 times in this case).

4. In the inner loop for this iteration, we update nums as follows:

   • nums[0] becomes (1 + 2) % 10 = 3
   • nums[1] becomes (2 + 3) % 10 = 5
   • nums[2] becomes (3 + 4) % 10 = 7
   • nums[3] becomes (4 + 5) % 10 = 9 Now, nums looks like [3, 5, 7, 9].

5. For the next iteration of the outer loop, we decrement n to 4 and run the inner loop again to update nums:

   • nums[0] becomes (3 + 5) % 10 = 8
   • nums[1] becomes (5 + 7) % 10 = 2
   • nums[2] becomes (7 + 9) % 10 = 6 Now, nums looks like [8, 2, 6].

6. With n now at 3, the outer loop iterates and the inner loop continues updates:

   • nums[0] becomes (8 + 2) % 10 = 0
   • nums[1] becomes (2 + 6) % 10 = 8 Now, nums looks like [0, 8].

7. Lastly, with n at 2, we run the inner loop one more time:

   • nums[0] becomes (0 + 8) % 10 = 8 Now nums is simply [8].

8. When we exit the outer loop, we return nums[0], which is 8. This is the triangular sum of the array [1, 2, 3, 4, 5].

Following the problem's strategy, we successfully computed the triangular sum by iteratively reducing the array and updating it in place without any extra space. The final answer is 8.

Solution Implementation

Time and Space Complexity

The time complexity of the provided code can be analyzed as follows:

We have a nested loop where the outer loop runs n times (where n is the length of nums) and the inner loop performs i - 1 iterations, where i decreases from n to 1. This creates a triangular number of operations (hence the name "triangularSum"). The total number of operations can be calculated by the sum of the first n integers which is (n * (n + 1)) / 2. Consequently, removing the non-dominant terms and constants, we can describe the time complexity as O(n^2).

The space complexity of the code can be determined as follows:

The solution modifies the input list nums in-place and does not use any additional space that grows with the input size (other than a constant amount of space for the loop indices and the variable n). Hence, the space complexity is O(1).

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