Set Mismatch

We are given numbers from 1 to n and some number x is here twice and another number y is missing. Let us calculate sum of all numbers and sum of squares of all numbes and given this information we can get our answer, using math! For this we need to use couple of facts:

1 + 2 + … + n = n*(n+1)//2. But even if you do not know this formula you can get it in O(n) time and O(1) space. 1^2 + 2^2 + … + n^2 = n*(n+1)(2n+1)//6. Again if you do not know this formula, you can get it in O(n) time and O(1) space.

Now, let us consider number A = -sum(nums) + n*(n+1)//2. It is equal to y - x, because each element not equal to x and y will be canceled out. In similar way if we define B = -sum(ii for i in nums) + n(n+1)(2n+1)//6, it will be equal to yy - xx. So, we have system of equations now:

A = y - x B = yy - xx.

Dividing one by another we have B/A = y + x, so x = (B/A - A)/2 and y = (B/A + A)/2, which we just return.

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class Solution:
    def findErrorNums(self, nums: List[int]) -> List[int]:
        n = len(nums)
        A = -sum(nums) + n*(n+1)//2
        B = -sum(i*i for i in nums) + n*(n+1)*(2*n+1)//6

        return [(B-A*A)//2//A, (B+A*A)//2//A]

Identifying Problem Isomorphism

“Set Mismatch” can be mapped to the problem “Find the Duplicate Number”.

Both problems involve an array of n+1 integers where each integer is between 1 and n inclusive, with some twist that violates this perfect 1-to-n matching. You are asked to find the elements causing this violation.

In “Set Mismatch”, you are given an array where exactly one of the numbers in 1 to n is missing, and another one is repeated twice. The task is to find both of these numbers.

In “Find the Duplicate Number”, you are given an array where exactly one number is duplicated one or more times, and you have to find that number.

Although the specifics of the problems are slightly different, the underlying concept of finding a duplicate in an array of integers is shared between the two. The techniques used to solve one of these problems can be applied to the other.

“Find the Duplicate Number” is simpler as it only asks for the duplicate number, whereas “Set Mismatch” also asks for the missing number.

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class Solution(object):
    def findErrorNums(self, nums):
        """
        :type nums: List[int]
        :rtype: List[int]
        """
        n = len(nums)
        s = n*(n+1)//2
        miss = s - sum(set(nums))
        duplicate = sum(nums) + miss - s
        return [duplicate, miss]

Problem Classification

Array manipulation - The problem revolves around handling and processing an array (integer array nums).

Mathematical calculations - The problem involves calculating a mathematical solution based on the provided integer array.

Error detection - The problem requires detecting an error (duplication and missing number) in a data set.

The “What” components of the problem statement are:

  1. An integer array ’nums’ that represents a set of integers from 1 to n, but with an error - one number has been duplicated, and another number is missing.

  2. The task to find the duplicated number and the missing number.

The problem can be further classified as a searching and detection problem where the goal is to find specific elements (the duplicate and missing numbers) within the given data set.

My categorizations are based on the need to manipulate data in an array format, apply mathematical calculations to deduce a solution, and perform an error detection process to find the numbers that are duplicated and missing. These require a good understanding of how to handle arrays and apply basic mathematical principles, as well as a strategic approach to problem-solving to detect and resolve the errors in the given data set.

Language Agnostic Coding Drills

  1. Dissect the Code into Distinct Concepts

Here are the distinct concepts contained in this code:

a. Defining Functions: This code defines a function findErrorNums. This is a fundamental concept in all programming languages.

b. Arithmetic Operations: The code includes basic arithmetic operations like addition, subtraction, multiplication, and division, including the use of the floor division operator (//).

c. Array Length Determination: The code uses the len function to determine the number of elements in an array.

d. Summation of Array Elements: The sum function is used to add up all elements of an array.

e. Set Conversion: The set function is used to convert the list to a set, which automatically removes duplicate elements.

f. List Construction and Return: The code creates a list with two elements and returns it.

g. Variable Assignment: The code uses variables to store intermediate and final results.

  1. Order of Increasing Difficulty

a. Defining Functions: This is a fundamental concept that most programmers learn early on.

b. Variable Assignment: This is another basic concept. A firm understanding of this is crucial for any kind of coding.

c. Arithmetic Operations: This involves simple math operations.

d. Array Length Determination: A more specific concept requiring understanding of arrays/lists.

e. Summation of Array Elements: This requires knowledge of inbuilt functions and handling of arrays/lists.

f. List Construction and Return: This requires an understanding of list manipulation.

g. Set Conversion: This concept requires knowledge about sets, which is a slightly more complex data structure compared to lists.

  1. Problem-Solving Approach

The approach here is to use mathematical principles and operations to find the duplicate and missing numbers. First, the expected sum of numbers from 1 to n is calculated (using the formula for the sum of an arithmetic series). Then, the sum of the unique elements in the array is subtracted from this expected sum to find the missing number.

The duplicate number is found by calculating the actual sum of all elements in the array, then adding the missing number, and finally subtracting the expected sum. The result of this calculation is the duplicate number because the duplicate number is the only extra number in the actual sum.

The final result is a list containing the duplicate and missing numbers, which is created and returned. Each of the concepts identified contributes to the overall solution by breaking down the problem into smaller, more manageable tasks.

Implementing these drills, in the order of their increasing difficulty, gradually builds up the complexity of the problem-solving process, thereby leading to the final solution.

Targeted Drills in Python

  1. Python Coding Drills for Identified Concepts

Let’s encapsulate each identified concept in Python:

a. Defining Functions

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def hello_world():
    print("Hello, World!")

This is a basic function that prints “Hello, World!” when called.

b. Variable Assignment

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a = 10
b = 20
c = a + b
print(c)

This code assigns values to the variables a and b, then adds these values and assigns the result to c.

c. Arithmetic Operations

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print(10 + 5)  # Addition
print(10 - 5)  # Subtraction
print(10 * 5)  # Multiplication
print(10 / 5)  # Division

This code illustrates basic arithmetic operations.

d. Array Length Determination

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nums = [1, 2, 3, 4, 5]
print(len(nums))  # Prints the length of the list

This code finds and prints the number of elements in a list.

e. Summation of Array Elements

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nums = [1, 2, 3, 4, 5]
print(sum(nums))  # Prints the sum of the list elements

This code finds and prints the sum of elements in a list.

f. List Construction and Return

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def create_list():
    my_list = [1, 2, 3]
    return my_list
print(create_list())  # Prints the created list

This code defines a function that creates and returns a list.

g. Set Conversion

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nums = [1, 2, 2, 3, 4]
unique_nums = set(nums)
print(unique_nums)  # Prints {1, 2, 3, 4}

This code converts a list to a set, removing duplicate elements in the process.

  1. Problem-Specific Coding Drills

One of the key concepts specific to our problem is calculating the sum of the first n natural numbers, which is done using the formula n*(n+1)//2. This formula is important because it helps us calculate the expected sum of our array if it had all unique elements from 1 to n.

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def sum_natural_numbers(n):
    return n*(n+1)//2
print(sum_natural_numbers(5))  # Prints 15

This function calculates and returns the sum of the first n natural numbers.

  1. Integration of Drills

Once all these drills have been coded, they can be integrated together in the correct order to solve the problem:

a. Define the function findErrorNums.

b. Inside this function, assign the length of the input list to the variable n.

c. Calculate the sum of the first n natural numbers using the formula n*(n+1)//2 and assign it to the variable s.

d. Create a set from the input list to remove duplicates and calculate the sum of the unique elements.

e. Subtract the sum of unique elements from s to find the missing number.

f. Calculate the sum of all elements in the input list, add the missing number, and subtract s from this sum to find the duplicate number.

g. Construct a list containing the duplicate and missing numbers and return this list.

This way, each of the identified concepts contributes to solving the problem when assembled in the correct order.