Find the Key of the Numbers

#1857 Most AskedEasy

Find the Key of the Numbers #1857 Most Asked

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You are given three positive integers num1, num2, and num3.

The key of num1, num2, and num3 is defined as a four-digit number such that:

Initially, if any number has less than four digits, it is padded with leading zeros.
The i^th digit (1 <= i <= 4) of the key is generated by taking the smallest digit among the i^th digits of num1, num2, and num3.

Return the key of the three numbers without leading zeros (if any).

Example 1:

Input: num1 = 1, num2 = 10, num3 = 1000

Output: 0

Explanation:

On padding, num1 becomes "0001", num2 becomes "0010", and num3 remains "1000".

The 1^st digit of the key is min(0, 0, 1).
The 2^nd digit of the key is min(0, 0, 0).
The 3^rd digit of the key is min(0, 1, 0).
The 4^th digit of the key is min(1, 0, 0).

Hence, the key is "0000", i.e. 0.

Example 2:

Input: num1 = 987, num2 = 879, num3 = 798

Output: 777

Example 3:

Input: num1 = 1, num2 = 2, num3 = 3

Output: 1

Constraints:

1 <= num1, num2, num3 <= 9999

Try coding it in LeetCode

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What is the expected data type of the numbers in the input? Are they integers, floats, or something else?
Can the input contain negative numbers, zero, or null values?
Are there any constraints on the range of the numbers within the input?
If no 'key' exists based on the defined criteria, what should the function return?
Are the numbers in the input guaranteed to be unique?

Brute Force Solution

Approach

The brute force approach to finding the key involves trying every single possible number as the key. We then check if that number unlocks the lock by applying it to the numbers given. If it unlocks the lock, we found our answer.

Here's how the algorithm would work step-by-step:

Start with the first possible number for the key (usually 1 or 0, depending on the problem).
Use this number as the key and apply it to the numbers you have (according to whatever rules the problem defines for how the key works).
See if the result of applying the key to the numbers matches the desired outcome (like a specific target number or condition).
If it matches, then you've found the key!
If it doesn't match, then move on to the next possible number and repeat the process.
Keep doing this, trying every possible number until you find one that works.

Code Implementation

def find_key_brute_force(numbers, target_result, operation):    possible_key = 0
    while True:
        # Try each number as a possible key until we find the right one

        result = []
        for number in numbers:
            result.append(operation(number, possible_key))

        # Check if this key produces the target result when applied to all numbers
        if result == target_result:

            return possible_key

        # Increase possible_key for the next try
        possible_key += 1

Big(O) Analysis

Time Complexity

O(k*n) – The brute force approach iterates through every possible key in the range of possible keys, which we can denote as 'k'. For each potential key, we iterate through the 'n' given numbers to check if applying that key unlocks the lock. Therefore, for each of the 'k' possible keys, we perform 'n' operations. This results in approximately k * n operations, thus the time complexity is O(k*n).

Space Complexity

O(1) – The brute force approach, as described, primarily involves iteratively testing candidate keys. The algorithm doesn't mention the creation of any auxiliary data structures like lists, hash maps, or significant temporary variables to store intermediate results. The space required is limited to storing the current candidate key and potentially a flag to indicate whether a valid key has been found. Thus, the auxiliary space used remains constant regardless of the input size (N), resulting in O(1) space complexity.

Optimal Solution

Approach

The core idea is to use the information available to us to make the problem simpler to handle. The goal is to find a number (the key) which, when combined with the numbers we're given, always gives us values within a certain range. We can greatly reduce the search space for the key by focusing on the extremes.

Here's how the algorithm would work step-by-step:

First, figure out the smallest and largest numbers you have.
Next, consider what happens when you combine the smallest number with the key. What's the lowest possible range limit for that result?
Also consider combining the largest number with the key. What is the largest possible range limit for that result?
Now, look at these calculated limits. The key must ensure that all combined values fall inside these limits.
This creates a narrowed-down set of possible key values. Explore those potential key values only.
For each possible key, check if when combined with all of the original numbers, are results within the range. If the values are not within the range, this key is not valid and we can try another key.
If a valid key is found, then the process is complete.

Code Implementation

def find_key_of_numbers(numbers, lower_limit, upper_limit):
    smallest_number = min(numbers)
    largest_number = max(numbers)

    #Determine the possible range for the key based on lower limit
    minimum_possible_key = lower_limit - smallest_number
    maximum_possible_key = upper_limit - largest_number

    for possible_key in range(minimum_possible_key, maximum_possible_key + 1):

        is_valid_key = True

        #Ensure each number produces the proper range
        for number in numbers:
            combined_value = number + possible_key
            if combined_value < lower_limit or combined_value > upper_limit:
                is_valid_key = False
                break

        if is_valid_key:
            return possible_key

    return None

Big(O) Analysis

Time Complexity

O(n * (maxRange - minRange)) – The algorithm first finds the minimum and maximum numbers in the input array of size n, which takes O(n) time. Then, it calculates the possible range of the key based on the given range limits and the minimum and maximum values of the input array. The algorithm iterates through the narrowed-down set of possible key values, which is bounded by (maxRange - minRange). For each possible key, it iterates through the input array of size n to check if combining the key with all numbers falls within the specified range. Therefore, the dominant operation is the nested loop, resulting in a time complexity of O(n * (maxRange - minRange)), where (maxRange - minRange) is the size of the possible key space. Since finding the min/max initially is only O(n), it's less significant and ignored in the final complexity.

Space Complexity

O(1) – The algorithm's space complexity is primarily determined by the variables used to store the smallest and largest numbers, and a single variable for the potential key value. The number of variables does not scale with the input size, N, which represents the number of elements. No auxiliary data structures like arrays or hash maps are created to store intermediate results or track visited elements, leading to constant auxiliary space usage. Therefore, the space complexity is O(1).

Edge Cases

Case	How to Handle
Empty input array	Return an empty list as there are no elements to process.
Array with only one element	Return an empty list since a pair requires at least two elements.
Array with duplicate numbers where only one number satisfies the condition	Ensure the algorithm correctly identifies and uses the correct pair despite duplicates.
Array containing negative numbers	The algorithm should correctly handle negative numbers without causing errors.
Large input array approaching memory limits	Consider the space complexity of the chosen data structures and algorithm, potentially streaming or divide-and-conquer.
Integer overflow when calculating the key	Use appropriate data types or modular arithmetic to prevent integer overflow.
No valid key exists in the array	Return an empty list to indicate that no solution was found.
Array contains zero, potentially leading to division by zero if the calculation involves it	Implement explicit checks to avoid division by zero or handle zero values appropriately based on the key calculation logic.

Empty input array

How to Handle:

Return an empty list as there are no elements to process.

Array with only one element

How to Handle:

Return an empty list since a pair requires at least two elements.

Array with duplicate numbers where only one number satisfies the condition

How to Handle:

Ensure the algorithm correctly identifies and uses the correct pair despite duplicates.

Array containing negative numbers

How to Handle:

The algorithm should correctly handle negative numbers without causing errors.

Large input array approaching memory limits

How to Handle:

Consider the space complexity of the chosen data structures and algorithm, potentially streaming or divide-and-conquer.

Integer overflow when calculating the key

How to Handle:

Use appropriate data types or modular arithmetic to prevent integer overflow.

No valid key exists in the array

How to Handle:

Return an empty list to indicate that no solution was found.

Array contains zero, potentially leading to division by zero if the calculation involves it

How to Handle:

Implement explicit checks to avoid division by zero or handle zero values appropriately based on the key calculation logic.