Problem Description

The problem presents a scenario where you are given a single integer num. Your task is to find two integers whose product is either num + 1 or num + 2. These two integers should be as close to each other as possible in terms of their absolute difference – meaning the difference between the two numbers without considering whether it is positive or negative should be minimal. The final solution does not require the integers to be in any specific order, thus either of the integers can come first.

Intuition

In solving this problem, the concept of factors of a number is key. For any given number, its factors are the numbers that divide it without leaving a remainder. In this case, our target numbers are num + 1 and num + 2. We look for the pair of factors, for each of these numbers, that are closest to each other. The intuition here is that the pair of factors that are closest to each other will have the smallest absolute difference.

To efficiently find such a pair for a given target number, we can start checking from the square root of the target number and move downwards. The square root gives us a good starting point since it is the largest number that can multiply by itself to not exceed the target. Therefore, any factor larger than the square root would result in a product larger than our target number, disqualifying it from being a correct answer.

Once we have the factors for both num + 1 and num + 2, we compare their absolute differences. The pair with the smaller absolute difference is the closest pair – this represents our final answer. The reason for checking both num + 1 and num + 2 is to fulfill the task's requirement of finding the closest integers in absolute difference.

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Solution Approach

The solution uses a straightforward approach by defining a helper function f(x) which takes a number x (which will either be num + 1 or num + 2) and finds the closest divisors of x.

Here's how the f(x) function works in detail:

1. Start by calculating the integer square root of x using int(sqrt(x)). This is the starting point for finding our factors.
2. Iterate downwards from this square root to 1 (inclusive), as any factor greater than the square root of x would result in a product larger than x when multiplied with another integer.
3. In each iteration, check if x % i == 0 which is the condition to confirm if i is a factor of x.
4. Once a factor is found, calculate the pair by dividing x by the factor i. The pair will be [i, x // i].
5. Return the pair of factors found.

The main function closestDivisors(num: int) -> List[int] calls this helper function twice: once with num + 1 and once with num + 2.

1. The function stores the pairs returned by f(num + 1) and f(num + 2) in variables a and b respectively.
2. It then compares the absolute differences of the two integers in each pair.
3. The pair with the smaller absolute difference is chosen as the result.
4. The result pair is returned.

Algorithmically, this is an efficient approach because it only searches up to the square root of the target numbers rather than iterating through all possible divisors, which significantly reduces the number of operations, especially for very large numbers.

To summarize, the solution algorithm involves:

• Finding pairs of factors (divisors) for two numbers num + 1 and num + 2.
• Starting the search from the square root of these numbers and iterating downwards to find these pairs.
• Returning the pair with the smallest absolute difference between the two factors.

Data structures used in this solution are basic and include primarily lists to store the pairs of factors. The pattern utilized here is an optimization over brute force where only necessary divisors are considered, which is made possible by the mathematical property that a number's divisors are symmetrical around the square root.

Example Walkthrough

Let's assume num = 8. We want to find two integers whose product is num + 1 or num + 2, which here would be 9 and 10. These two integers should have the smallest possible absolute difference. Let's apply our solution approach:

1. We first calculate the square root of num + 1 which is sqrt(9) = 3.

2. Then, we check if there's any number from 3 down to 1 that evenly divides 9:

   • 3 is a divisor of 9 since 9 % 3 == 0.
   • The pair is formed by 3 (the divisor) and 9 // 3 = 3. So the factor pair for num + 1 is [3, 3].

3. Next, we calculate the square root of num + 2, which is slightly over sqrt(10) but we use int(sqrt(10)) = 3 as our starting point.

4. We check divisors from 3 to 1 for the number 10:

   • 3 is not a divisor of 10 since 10 % 3 != 0.
   • 2 is a divisor since 10 % 2 == 0.
   • The pair for num + 2 is [2, 10 // 2], which is [2, 5].

5. Now we have two pairs: [3, 3] for num + 1 and [2, 5] for num + 2. We compare their absolute differences:

   • The absolute difference for [3, 3] is 3 - 3 = 0.
   • The absolute difference for [2, 5] is 5 - 2 = 3.

6. We choose the pair with the smallest absolute difference, which is [3, 3], with an absolute difference of 0.

7. The final result is [3, 3], indicating that these are the two integers (which in this case happen to be identical) that, when multiplied together, yield num + 1 (which is 9) and have the smallest absolute difference.

This example walk-through demonstrates the solution approach. By checking only the divisors from the square root and below, we quickly identify the closest pair of numbers satisfying the condition without unnecessary computation.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the function primarily depends on the for loop within the nested function f(x), which iterates over numbers starting from int(sqrt(x)) to 1.

Since the square root function essentially reduces the number of iterations to the square root of x, the time complexity for finding the divisors would be O(sqrt(x)).

Given that the function f(x) is called twice—once for num + 1 and once for num + 2—the overall time complexity is then O(sqrt(num + 1) + sqrt(num + 2)), which simplifies to O(sqrt(num)), as the higher order term dominates and adding 1 or 2 to num has a negligible effect for large numbers.

Space Complexity

The space complexity of this algorithm is O(1), as the space used does not grow with the input size num. The only extra space used is for a handful of variables that store the divisor pairs and their differences.

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