Add to Array-Form of Integers

This can be solved by using a simple approach of arithmetic and array manipulation.

Iterate over the array from the end (i.e., least significant digit)
In each iteration, add the current digit to the remaining value of k. It’s crucial to add k to the least significant digits because k could have more digits than the num array.
Use the division and modulus operations to update k and replace the current digit. The updated digit becomes (current digit + k) % 10, and the remaining k becomes (current digit + k) / 10.
After going through all the digits of num, if k is not zero, it means we have remaining carry, or k had more digits than num. We need to add each digit of k to the beginning of the num array.
Finally, return the num array as the result.

Python solution:

1
2
3
4
5
6
7
8
class Solution:
    def addToArrayForm(self, num: List[int], k: int) -> List[int]:
        for i in range(len(num)-1, -1, -1):
            num[i], k = (num[i] + k) % 10, (num[i] + k) // 10
        while k > 0:
            num.insert(0, k % 10)
            k //= 10
        return num

This solution treats num and k as numbers and follows the standard algorithm for adding numbers: add corresponding digits and propagate the carry. This approach ensures that each digit of both numbers is taken into account. This solution will pass all test cases because it correctly implements the arithmetic of adding two numbers.

What are the building blocks of the solution?

The solution to this problem relies on three main building blocks:

Arithmetic operations: The solution uses the modulus and division operations to calculate the new digit and carry over (if any) when the current digit of num and k are added. Specifically, the new digit is (current digit + k) % 10, and the remaining k is (current digit + k) // 10.
Iteration: The solution involves iterating over the num array from the end (i.e., the least significant digit). This corresponds to how we usually perform addition manually - from right to left, adding corresponding digits of two numbers.
Array manipulation: If k is still greater than 0 after iterating through all the digits of num, we add each remaining digit of k to the beginning of the num array. This uses the insert operation of an array.

Together, these building blocks allow us to accurately perform the addition operation and transform the resulting number back into an array form.

I recognize Reduction, can you go up on the abstract level and identify more ?

At a higher level of abstraction, there are several important concepts at play in this problem:

Digit-by-Digit Calculation: This problem emphasizes the process of adding numbers in a digit-by-digit manner, akin to how we perform manual addition. This procedure is crucial in the context of larger arithmetic and numerical problems.
Carry Handling: The mechanism for dealing with the carry from adding two digits is integral to the solution. It not only affects the current digit but also the computation of the subsequent digit, demonstrating the concept of state propagation in computational problems.
Reverse Iteration: This problem involves iterating over the input in reverse, which is a common technique for problems involving digits or sequences where the last element affects the outcome.
Problem Transformation: The initial problem, which seems to be about arrays, is essentially reduced to a problem of numerical addition with carry handling. Recognizing this transformation is a critical part of devising the solution.

By identifying these higher-level concepts, we can better understand the nature of the problem and design an efficient algorithm to solve it.

The // operator in Python performs floor division. This means it divides the operand on its left by the operand on its right and rounds down the result to the nearest whole number. For example:

5 // 2 = 2
9 // 4 = 2

In the first example, 5 divided by 2 is 2.5. When we use //, Python rounds this down to 2. Similarly, in the second example, 9 divided by 4 is 2.25, which is rounded down to 2.

This operation is different from the standard division operator /, which performs floating point division and does not round down the result. For instance, 5 / 2 would give 2.5 in Python.

The // operator is used in this solution to help handle the carry in the addition operation.

Let’s take a closer look at this line: carry, curr = divmod(carry + k % 10, 10).

Here, divmod() is a built-in Python function that takes two arguments a and b, and returns a pair of numbers (a // b, a % b).

carry + k % 10 adds the carry from the previous step and the last digit of k. The result is then divided by 10. The quotient represents the next carry, and the remainder represents the current digit.

By using carry // 10, we are extracting the next carry value. This is important because when you’re adding two digits and the result is greater or equal to 10, the “tens place” digit becomes the carry for the addition of the next set of digits in the numbers being added.

In the context of this problem, // allows us to correctly propagate the carry value from right to left as we perform the addition of the numbers represented by num and k.

Coding Drills

Exercise 1: Understanding and using the divmod function

The divmod function is crucial to this solution. Understand how it works with different values and how it returns a pair of numbers - the quotient and the remainder of the division.

1
print(divmod(17, 5))  # Output: (3, 2)

This means that when 17 is divided by 5, the quotient is 3 and the remainder is 2.

Exercise 2: Reversing a list

The solution uses list reversal to align the digits of num and k from right to left. Understand how to reverse a list in Python.

1
2
list1 = [1, 2, 3, 4, 5]
print(list1[::-1])  # Output: [5, 4, 3, 2, 1]

Exercise 3: List comprehension and the enumerate function

The solution makes use of list comprehension to simplify the addition of corresponding digits from num and k. This technique is crucial to understanding how the solution works.

1
2
list1 = [1, 2, 3, 4, 5]
print([i * 2 for i in list1])  # Output: [2, 4, 6, 8, 10]

Also, the enumerate function is used in combination with list comprehension. Understand how enumerate works and how it can be used in list comprehension.

1
2
3
list1 = ['a', 'b', 'c']
for index, value in enumerate(list1):
    print(f"At index {index}, the value is {value}.")

Exercise 4: Combining it all together

Take the components from the above exercises and try to write a function that performs addition of a list of digits and an integer, similar to the problem description.

In each exercise, focus on the specific concept at hand, and by the end, try to put it all together to solve the original problem. This approach will help in breaking down complex problems into more manageable components.