Finding the missing number in an array of integers from 1 to n is a common technical interview coding question that assesses a candidate’s problem-solving skills and understanding of time and space complexity analysis. This how-to guide will walk through an effective approach to solve this problem in Python, along with sample code snippets, complexity analysis, and tips for acing the interview.

## Introduction

The missing number problem gives the candidate an array of unique integers that range from 1 to some number n, but with one number missing. The task is to write a Python function that takes this array as input and returns the missing integer.

This may seem like a simple coding challenge, but the optimal solution requires knowledge of efficient algorithms and Big O notation for analyzing time and space complexity. The interviewer wants to assess the candidate’s ability to select the appropriate data structures and algorithms for the problem at hand.

This guide will cover the following topics in Python:

## Table of Contents

## Open Table of Contents

## Formal Problem Statement

Given an array `nums`

containing `n`

distinct integers in the range `[1, n]`

, return the one number that is missing from the array.

**Example 1:**

```
Input: nums = [2,3,4,5,6,7,8,9,10]
Output: 1
```

**Example 2:**

```
Input: nums = [1,2,3,4,6,7,8,9,10]
Output: 5
```

**Constraints:**

`n == nums.length`

`1 <= n <= 10^5`

- All the integers of
`nums`

are unique.

## Brute Force Approach

The most straightforward solution is to use a brute force approach checking each number from 1 to n to see if it is present in the array.

Here is an implementation in Python:

```
def findMissingNumber(nums):
n = len(nums) + 1
for number in range(1, n+1):
if number not in nums:
return number
nums = [1,2,3,4,6,7,8,9,10]
missing = findMissingNumber(nums)
print(missing) # 5
```

This uses a linear search through each number from 1 to n, checking if it is in the `nums`

array using the `in`

operator. If we do not find a number, we return it as the missing number.

### Complexity Analysis

**Time complexity:** O(n) linear time, where n is the length of the input array. We need to check each number from 1 to n in the worst case.

**Space complexity:** O(1) constant space. We only use a constant amount of extra memory.

The brute force solution is quite simple and easy to implement but is inefficient for large inputs. We can optimize this using mathematical concepts to achieve faster runtime.

## Optimal Approach

A clever math-based approach can solve this in O(n) time with O(1) space.

The key insight is that the sum of integers from 1 to n can be calculated using the formula:

```
sum = n * (n + 1) / 2
```

For example, the sum of numbers from 1 to 10 is:

```
sum = 10 * (10 + 1) / 2 = 55
```

So if we calculate the expected sum of numbers from 1 to n, and subtract the sum of the numbers in the input array, the result will be the missing number!

Here is the Python implementation:

```
def findMissingNumber(nums):
n = len(nums) + 1
expectedSum = n * (n + 1) // 2
actualSum = sum(nums)
return expectedSum - actualSum
nums = [1,2,3,4,6,7,8,9,10]
missing = findMissingNumber(nums)
print(missing) # 5
```

We calculate the expected sum if all numbers from 1 to n were present. Then we find the actual sum of the input array and subtract to find the missing number.

### Complexity Analysis

**Time Complexity:** O(n) linear time. We need to calculate the sum of the input array which takes linear time.

**Space Complexity:** O(1) constant space. We only use a few extra variables.

This optimized solution reduces both time and space complexity. By leveraging mathematical concepts, we avoided the brute force nested loop and improved efficiency.

## Tips for Coding the Solution

Here are some tips for effectively coding this algorithm during an interview:

Clarify questions and edge cases - Ask about constraints, data types, edge cases like empty array.

Use helper functions - Break the logic into smaller reusable functions for sum, expected sum, etc.

Name variables intuitively - Use names like

`expectedSum`

and`actualSum`

for readability.Comment complex logic - Add comments explaining the key mathematical insights for the interviewer.

Check edge cases - Handle invalid inputs and edge cases explicitly.

Use built-in functions - Leverage built-ins like

`sum()`

to write clean code.Analyze out loud - Verbalize your thought process and analysis to show understanding.

## Testing the Code

Testing is crucial to ensure the code handles all cases correctly. Here are some ways to test this algorithm:

Use a sample input and walk through the logic step-by-step.

Check small and large test cases including edge cases.

Try different missing numbers at different positions.

Test with duplicates and invalid inputs.

Time the runtime with large random inputs to verify efficiency.

Write unit tests for each function covering key test cases.

Automate testing using frameworks like

`unittest`

.

Discuss your testing strategies with the interviewer to demonstrate thoroughness.

## Key Takeaways

Use the formula for sum of integers from 1 to n to find the missing number in O(n) time and O(1) space.

Break the problem down into clear functions and steps.

Discuss the brute force approach first before optimizing.

Analyze time and space complexity of the solutions.

Test your code thoroughly with example inputs and edge cases.

Explain your thought process clearly and document the code well.

Solving the missing number problem efficiently demonstrates strong analytical skills and coding abilities for technical interviews. Keep practicing with different algorithm questions and analyze the complexity tradeoffs between solutions.

## Conclusion

This guide covered an optimal Python solution to find the missing number in an array from 1 to n. Calculating the expected sum versus the actual array sum provides an efficient O(n) time and O(1) space solution.

Analyzing time and space complexity for algorithm questions is key in technical interviews. Always consider both brute force and optimal solutions. Use helper functions, intuitive names, and comments to write clean, well-documented code.

Thoroughly testing edge cases and explaining your thought process are just as important. With practice on common algorithm challenges, you will master the skills needed to succeed in any coding interview.