Problem Description

In this problem, we are given a four-digit positive integer num. Our task is to split this integer into exactly two new integers, new1 and new2. To create these two new integers, we have to distribute the digits of num between them. Some important points to note about this task are:

• A digit can be placed in new1 or new2 without restriction, except that all digits from num must be used.
• Leading zeros are allowed in new1 and new2.
• We want to minimize the sum of new1 and new2.

As an example, consider num = 2932. We can create several pairs from these digits, such as [22, 93], [23, 92], [223, 9], and [2, 329]. However, our goal is to find the pair where the sum of new1 and new2 is the smallest. This requires a strategic approach to pairing the digits into two numbers where their combined value is minimized.

Intuition

The intuition behind solving this problem lies in understanding that to minimize the sum of two numbers created from the digits of num, we need to consider digit positions carefully. The smallest sum is obtained when the least significant digits (the ones and tens place) are as small as possible in both new numbers.

One effective strategy is to:

1. Sort the digits of the original number in ascending order so that the smallest digits are placed in positions that contribute the least to the sum.
2. Assign digits to new1 and new2 strategically to ensure that the least significant digits of both numbers are as small as possible. This is because adding a smaller digit in the tens place will reduce the sum more significantly than adding it in the ones place due to the positional value in decimal notation.

Here's how we can do it:

• After sorting the digits, the smallest digits will be at the front of the array. Manage these small digits so that they contribute to tens place of both the numbers.
• Specifically, we take the two smallest digits and place them into the tens places of new1 and new2. This is done by multiplying the sum of these two smallest digits by 10.
• Then we add the remaining two digits to the ones places.

Thus, the approach to arriving at the solution relies on sorting the four digits and then constructing two new numbers such that their sum is minimized by careful placement of the least significant digits.

Learn more about Greedy, Math and Sorting patterns.

Solution Approach

The solution is implemented in Python and uses fundamental programming constructs such as a list and sorting algorithm. Below is the step-by-step approach taken in the provided code:

1. First, we initialize an empty list, nums, which will hold the individual digits of the given num.

2. We then extract each digit from num using a while loop that continues until num becomes zero. We use the modulus operator % to get the least significant digit and append it to the list nums. Then, we use floor division // by 10 to remove the least significant digit from num.

   while num:
       nums.append(num % 10)
       num //= 10

3. The resulting nums list contains the individual digits in reverse order. We sort this list to arrange the digits in ascending order. Sorting is crucial to place the smallest digits in the tens places of new1 and new2 to minimize the sum.

4. The sorted nums list now has the smallest digit at index 0 and the second smallest at index 1. By adding the smallest two digits and multiplying by 10, we achieve two numbers with the smallest digits in the tens place:

   return 10 * (nums[0] + nums[1]) + nums[2] + nums[3]

5. To complete the numbers new1 and new2, we add the third smallest digit (nums[2]) and the largest digit (nums[3]) to the sum (which will occupy the ones places).

6. The final result returned is the minimum possible sum of new1 and new2 constructed from the digits of the original number, num.

The algorithm used is simple yet effective because it directly leverages the property of positional notation in decimal numbers. The choice of the list as the data structure enables easy manipulation of digits and the use of Python's built-in sort method streamlines digit organization.

Example Walkthrough

Let's walk through the solution approach with a small example. Suppose the given number is num = 4723.

Following the approach described, we perform these steps:

1. We start by initializing an empty list to store the digits. We call this list nums.

2. We then extract each digit from num and append it to nums. Processing the number 4723 gives us:

   nums = [3, 2, 7, 4]  # Note that digits are appended in reverse order

3. Next, we sort the nums list in ascending order:

   nums.sort()  # After sorting, nums = [2, 3, 4, 7]

4. Once the list is sorted, we identify the two smallest digits, which now are at indices 0 and 1. In our case, they are 2 and 3. According to our strategy, they will go into the tens places of new1 and new2. So, by adding them together and multiplying by 10, we obtain the tens place for both:

   Tens place sum = 10 * (2 + 3) = 50

5. Now, we take the remaining digits, which are 4 and 7, to fill in the ones places. Adding them to our tens place sum will give us the final value:

   Total sum = 50 + 4 + 7 = 61

6. Therefore, the smallest possible summed pair of numbers created from 4723 is $new1 = 23 and $new2 = 47, with a sum of 61.

This example demonstrates that by sorting the digits first and then strategically placing the smaller digits in the more significant tens positions across new1 and new2, we achieve the minimum sum as per the problem's requirement.

Solution Implementation

Time and Space Complexity

Time Complexity:

The provided function consists of a while loop that iterates through the digits of the input number (num). This loop will run a number of times equal to the number of digits in num. For a 32-bit integer, the maximum number of digits is 10 (2147483647 is the largest 32-bit integer). Therefore, the while loop has a constant runtime with respect to the size of the input and does not depend on the value of num. Sorting the nums list which contains at most 4 digits using a basic sorting algorithm (like Timsort, which is used by Python's sort() method) has a constant time complexity because the number of digits is fixed and small. The final return statement is a constant time operation. Hence, overall, the time complexity is O(1), a constant time complexity.

Space Complexity:

Space is allocated for the nums list which will always have at most 4 elements since the maximum number of digits for an integer is 4 in this problem context. Hence, the space complexity of the code is O(1), a constant space complexity, since it does not grow with the size of the input number.

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