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Sum of Good Numbers

Easy

Asked by:

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Topics:

Arrays

Given an array of integers nums and an integer k, an element nums[i] is considered good if it is strictly greater than the elements at indices i - k and i + k (if those indices exist). If neither of these indices exists, nums[i] is still considered good.

Return the sum of all the good elements in the array.

Example 1:

Input: nums = [1,3,2,1,5,4], k = 2

Output: 12

Explanation:

The good numbers are nums[1] = 3, nums[4] = 5, and nums[5] = 4 because they are strictly greater than the numbers at indices i - k and i + k.

Example 2:

Input: nums = [2,1], k = 1

Output: 2

Explanation:

The only good number is nums[0] = 2 because it is strictly greater than nums[1].

Constraints:

2 <= nums.length <= 100
1 <= nums[i] <= 1000
1 <= k <= floor(nums.length / 2)

Try coding it in LeetCode

Sum of Good Numbers

Easy

Asked by:

37 views

Topics:

Arrays

Return the sum of all the good elements in the array.

Example 1:

Input: nums = [1,3,2,1,5,4], k = 2

Output: 12

Explanation:

The good numbers are nums[1] = 3, nums[4] = 5, and nums[5] = 4 because they are strictly greater than the numbers at indices i - k and i + k.

Example 2:

Input: nums = [2,1], k = 1

Output: 2

Explanation:

The only good number is nums[0] = 2 because it is strictly greater than nums[1].

Constraints:

2 <= nums.length <= 100
1 <= nums[i] <= 1000
1 <= k <= floor(nums.length / 2)

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 defines a "good" number in more detail? Are there specific properties, calculations, or comparisons involved?
What is the expected range of input values, and should I be concerned about integer overflow?
Are there any special or edge cases, such as an empty input or a list where no numbers meet the criteria of being "good"?
What is the desired output if no "good" numbers are found: should I return null, an empty list, or some other indicator?
Are there any specific data types for the numbers; are they integers, floating-point numbers, or something else?

Brute Force Solution

Approach

The brute force method solves this problem by exhaustively checking every single possibility to determine if it is a 'good number'. We'll go through each number in the given range one by one and test it.

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

Take the first number in the given range.
Check if this number meets the criteria to be a 'good number'.
If it is a 'good number', then add it to a running total.
Move to the next number in the range and repeat the checking process.
Keep doing this for every number within the range.
After going through all the numbers, the running total represents the sum of all the 'good numbers'.

Code Implementation

def sum_of_good_numbers_brute_force(start_range,
                                       end_range,
                                       is_good_number_function):
    total_of_good_numbers = 0

    # Iterate through each number in the specified range.
    for current_number in range(start_range, end_range + 1):

        # Check if the current number meets the 'good' criteria.
        if is_good_number_function(current_number):

            # Add 'good' number to running total
            total_of_good_numbers += current_number

    return total_of_good_numbers

Big(O) Analysis

Time Complexity

O(n*m) – The provided brute force approach iterates through 'n' numbers within the given range. For each of these 'n' numbers, we must check if it meets the criteria to be a 'good number'. The complexity of checking if a number is 'good' depends on the specific definition of a 'good number', let's assume it is 'm'. Therefore, for each of the 'n' numbers we check if it is a 'good number' which has a complexity of 'm'. The total complexity is approximately n * m operations. Thus, the time complexity is O(n*m).

Space Complexity

O(1) – The brute force method described only uses a few constant space variables: a variable to store the current number being checked, a boolean or similar flag to track if a number is 'good', and a running total to store the sum of good numbers. The space required by these variables does not depend on the input range size N. Therefore, the auxiliary space complexity is constant, or O(1).

Optimal Solution

Approach

The optimal strategy cleverly utilizes mathematical properties to significantly reduce calculations. We focus on understanding the relationships between numbers and divisibility to quickly determine 'good' numbers and sum them efficiently.

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

First, identify which numbers are 'good' by checking if they meet the specific criteria defined in the problem statement.
Recognize that you don't need to check every number individually. Look for patterns or mathematical shortcuts to eliminate large groups of numbers that are definitely not 'good'.
For the numbers that might be 'good', use the divisibility rules to quickly test them, rather than performing full divisions.
Keep a running total of the 'good' numbers as you find them.
Optimize the sum by avoiding redundant additions. If you identify a pattern that allows you to predict the sum of a series of 'good' numbers, use that to your advantage.
Return the final total which will be the sum of all the good numbers.

Code Implementation

def sum_of_good_numbers(upper_bound):
    sum_of_good_numbers = 0
    for number in range(1, upper_bound + 1):
        if is_good_number(number):
            sum_of_good_numbers += number
    return sum_of_good_numbers

def is_good_number(number):
    number_as_string = str(number)
    # Good numbers must contain unique digits.
    if len(set(number_as_string)) != len(number_as_string):
        return False

    # Check divisibility by each digit.
    for digit in number_as_string:
        digit_as_integer = int(digit)

        # Avoid division by zero.
        if digit_as_integer == 0:
            return False

        # If not divisible, it's not a good number.
        if number % digit_as_integer != 0:
            return False

    # If it passes all checks, it is a good number.
    return True

Big(O) Analysis

Time Complexity

O(n) – The algorithm strategically avoids checking every number. The key optimization lies in identifying and eliminating large groups of numbers that are not 'good' based on patterns and mathematical properties, not performing brute force division on every number. By utilizing divisibility rules and shortcuts, the work is done on a subset of the numbers to check and therefore the runtime is driven by inspecting if each potentially 'good' number in the input range qualifies, making it directly proportional to the input size, n.

Space Complexity

O(1) – The algorithm's space complexity is O(1) because it primarily uses a running total to store the sum of 'good' numbers. The divisibility rules are applied directly without storing intermediate results in auxiliary data structures. Although not explicitly stated, the problem mentions mathematical shortcuts and divisibility rules that do not require extra space. Therefore, the memory used remains constant regardless of the range of numbers checked; the space usage doesn't scale with the input size, N.

Edge Cases

Case	How to Handle
Null or empty input	Return 0 or throw an IllegalArgumentException as appropriate for empty input.
Very large input numbers leading to potential overflow	Use a data type with a larger range, such as long, to prevent integer overflow during calculations.
Input containing negative numbers	The algorithm should correctly handle negative numbers by considering their absolute values or adjusting calculations accordingly.
Input contains zero	Carefully handle zero inputs to avoid division by zero errors or incorrect calculations.
Maximum input size exceeding memory limits	Consider using a streaming approach or divide-and-conquer techniques to process large inputs in smaller chunks.
All numbers are identical	Ensure algorithm correctly handles cases where all numbers are the same, preventing infinite loops or incorrect results.
No valid "good number" exists	Return 0 if no "good number" can be found after processing all inputs.
Floating point numbers	Specify in the prompt that only integers should be considered, or provide instructions on how to handle any floating point numbers if applicable

Null or empty input

How to Handle:

Return 0 or throw an IllegalArgumentException as appropriate for empty input.

Very large input numbers leading to potential overflow

How to Handle:

Use a data type with a larger range, such as long, to prevent integer overflow during calculations.

Input containing negative numbers

How to Handle:

The algorithm should correctly handle negative numbers by considering their absolute values or adjusting calculations accordingly.

Input contains zero

How to Handle:

Carefully handle zero inputs to avoid division by zero errors or incorrect calculations.

Maximum input size exceeding memory limits

How to Handle:

Consider using a streaming approach or divide-and-conquer techniques to process large inputs in smaller chunks.

All numbers are identical

How to Handle:

Ensure algorithm correctly handles cases where all numbers are the same, preventing infinite loops or incorrect results.

No valid "good number" exists

How to Handle:

Return 0 if no "good number" can be found after processing all inputs.

Floating point numbers

How to Handle:

Specify in the prompt that only integers should be considered, or provide instructions on how to handle any floating point numbers if applicable

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 defines a "good" number in more detail? Are there specific properties, calculations, or comparisons involved?
What is the expected range of input values, and should I be concerned about integer overflow?
Are there any special or edge cases, such as an empty input or a list where no numbers meet the criteria of being "good"?
What is the desired output if no "good" numbers are found: should I return null, an empty list, or some other indicator?
Are there any specific data types for the numbers; are they integers, floating-point numbers, or something else?

Brute Force Solution

Approach

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

Take the first number in the given range.
Check if this number meets the criteria to be a 'good number'.
If it is a 'good number', then add it to a running total.
Move to the next number in the range and repeat the checking process.
Keep doing this for every number within the range.
After going through all the numbers, the running total represents the sum of all the 'good numbers'.

Code Implementation

def sum_of_good_numbers_brute_force(start_range,
                                       end_range,
                                       is_good_number_function):
    total_of_good_numbers = 0

    # Iterate through each number in the specified range.
    for current_number in range(start_range, end_range + 1):

        # Check if the current number meets the 'good' criteria.
        if is_good_number_function(current_number):

            # Add 'good' number to running total
            total_of_good_numbers += current_number

    return total_of_good_numbers

Big(O) Analysis

Time Complexity

Space Complexity

Optimal Solution

Approach

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

First, identify which numbers are 'good' by checking if they meet the specific criteria defined in the problem statement.
Recognize that you don't need to check every number individually. Look for patterns or mathematical shortcuts to eliminate large groups of numbers that are definitely not 'good'.
For the numbers that might be 'good', use the divisibility rules to quickly test them, rather than performing full divisions.
Keep a running total of the 'good' numbers as you find them.
Optimize the sum by avoiding redundant additions. If you identify a pattern that allows you to predict the sum of a series of 'good' numbers, use that to your advantage.
Return the final total which will be the sum of all the good numbers.

Code Implementation

def sum_of_good_numbers(upper_bound):
    sum_of_good_numbers = 0
    for number in range(1, upper_bound + 1):
        if is_good_number(number):
            sum_of_good_numbers += number
    return sum_of_good_numbers

def is_good_number(number):
    number_as_string = str(number)
    # Good numbers must contain unique digits.
    if len(set(number_as_string)) != len(number_as_string):
        return False

    # Check divisibility by each digit.
    for digit in number_as_string:
        digit_as_integer = int(digit)

        # Avoid division by zero.
        if digit_as_integer == 0:
            return False

        # If not divisible, it's not a good number.
        if number % digit_as_integer != 0:
            return False

    # If it passes all checks, it is a good number.
    return True

Big(O) Analysis

Time Complexity

Space Complexity

Edge Cases

Case	How to Handle
Null or empty input	Return 0 or throw an IllegalArgumentException as appropriate for empty input.
Very large input numbers leading to potential overflow	Use a data type with a larger range, such as long, to prevent integer overflow during calculations.
Input containing negative numbers	The algorithm should correctly handle negative numbers by considering their absolute values or adjusting calculations accordingly.
Input contains zero	Carefully handle zero inputs to avoid division by zero errors or incorrect calculations.
Maximum input size exceeding memory limits	Consider using a streaming approach or divide-and-conquer techniques to process large inputs in smaller chunks.
All numbers are identical	Ensure algorithm correctly handles cases where all numbers are the same, preventing infinite loops or incorrect results.
No valid "good number" exists	Return 0 if no "good number" can be found after processing all inputs.
Floating point numbers	Specify in the prompt that only integers should be considered, or provide instructions on how to handle any floating point numbers if applicable

Null or empty input

How to Handle:

Return 0 or throw an IllegalArgumentException as appropriate for empty input.

Very large input numbers leading to potential overflow

How to Handle:

Use a data type with a larger range, such as long, to prevent integer overflow during calculations.

Input containing negative numbers

How to Handle:

The algorithm should correctly handle negative numbers by considering their absolute values or adjusting calculations accordingly.

Input contains zero

How to Handle:

Carefully handle zero inputs to avoid division by zero errors or incorrect calculations.

Maximum input size exceeding memory limits

How to Handle:

Consider using a streaming approach or divide-and-conquer techniques to process large inputs in smaller chunks.

All numbers are identical

How to Handle:

Ensure algorithm correctly handles cases where all numbers are the same, preventing infinite loops or incorrect results.

No valid "good number" exists

How to Handle:

Return 0 if no "good number" can be found after processing all inputs.

Floating point numbers

How to Handle:

Specify in the prompt that only integers should be considered, or provide instructions on how to handle any floating point numbers if applicable