Set Intersection Size At Least Two

Identifying Problem Isomorphism

“Set Intersection Size At Least Two” can be mapped to “Interval List Intersections”.

The rationale behind this is that both problems deal with the notion of finding intersections. In the first problem, you are given a collection of sets and you need to find a set that has at least two common elements with all the other sets. In the second problem, you are given two lists of intervals and you need to find their intersections.

The methods to solve these problems are similar. For “Interval List Intersections”, you iterate through the two lists of intervals, comparing each pair of intervals to see if they intersect. For “Set Intersection Size At Least Two”, you would iterate through the collection of sets, comparing each pair to see if they have at least two elements in common.

“Interval List Intersections” is simpler because it only requires you to find whether intervals overlap, whereas “Set Intersection Size At Least Two” involves checking that the overlap is of a certain size (two or more). Additionally, the structure of intervals (having a distinct start and end) could simplify the intersection logic.

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class Solution:
   def intersectionSizeTwo(self, intervals: List[List[int]]) -> int:
      sorted_intervals = sorted(intervals, key=lambda x: (x[1], -x[0]))
      last_two_endpoints = [-1, -1]
      containing_set_size = 0

      for start, end in sorted_intervals:
         if start > last_two_endpoints[1]:
            last_two_endpoints = [end - 1, end]
            containing_set_size += 2
         elif start > last_two_endpoints[0]:
            last_two_endpoints = [last_two_endpoints[1], end]
            containing_set_size += 1
      return containing_set_size

Problem Classification

This problem is a kind of computational mathematics problem under the broader category of optimization problems. It’s about finding the smallest set that meets a certain condition: each interval from a given list of intervals must have at least two integers in the set.

Here are the ‘What’ components of the problem:

You are given a 2D integer array intervals where each sub-array represents a range of integers.
A ‘containing set’ is defined as an array where each interval from intervals has at least two integers in the set.
The task is to find and return the minimum possible size of a containing set.

This problem can be classified as a greedy algorithm problem as well as an interval scheduling problem. The greedy algorithm is used to ensure that the solution we find is the smallest possible set, while interval scheduling is a common technique used when dealing with problems involving ranges or periods of time (in this case, ranges of integers).

Computational mathematics: The problem involves working with numeric data and mathematical concepts (e.g., sets, intervals).
Optimization: The goal is to find the smallest possible set that meets a certain condition.
Greedy algorithm: This approach involves making the locally optimal choice at each stage with the hope of finding a global optimum.
Interval scheduling: This technique involves organizing and dealing with data that is structured as intervals or ranges.

Thought Process

Language Agnostic Coding Drills

Defining a class and function: The first coding concept present in the provided code is defining a class Solution and its function intersectionSizeTwo(). This is a fundamental concept common to many programming languages that use classes and methods.
Sorting: In the function, the list of intervals is sorted based on the ending point and then the starting point in descending order. Sorting is an important concept, and understanding how to use sorting functions, as well as the key parameter, is essential.
Initializing variables: Three variables, sorted_intervals, last_two_endpoints, and containing_set_size are initialized. Understanding variable initialization and assignment is crucial for coding.
Looping and condition checking: A for loop is used to iterate over sorted_intervals. Within this loop, if-else statements are used to check conditions and modify variables accordingly. Looping and condition checking are fundamental coding concepts.
List Manipulation: Within the for loop, the list last_two_endpoints is updated based on conditions. This concept involves understanding how to manipulate lists, such as assigning new values to list elements.

Order of concepts in increasing difficulty:

Defining a class and function (Beginner)
Initializing variables (Beginner)
Looping and condition checking (Intermediate)
Sorting (Intermediate)
List Manipulation (Advanced)

Problem-Solving Approach: This problem requires a grasp of the greedy algorithm. Greedy algorithms make the best choice at each decision point with the hope that these local decisions lead to a global optimum.

The problem-solving approach for this task would be as follows:

Sort the intervals based on the ending value and starting value in reverse. The purpose of this sorting is to ensure that we always select the smallest endpoint possible.
Initiate an empty list to keep track of the last two endpoints and a counter to keep track of the size of the containing set.
Iterate over the sorted intervals. For each interval, check whether the start of the interval is greater than the last two endpoints. If it is, update the last two endpoints to be the last two numbers in the current interval and increase the containing set size by 2. If the start is only greater than the first of the last two endpoints, update the last two endpoints to be the last endpoint and the end of the current interval, and increase the containing set size by 1.
Return the size of the containing set after iterating over all intervals.

Each coding concept contributes to the overall solution, and understanding them individually can lead to a better grasp of the solution as a whole.

Targeted Drills in Python

These exercises will be designed for beginners, focusing on the fundamentals:

Defining a class and function:

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class MyClass:
    def my_function(self):
        print("Hello, World!")

Initializing variables:

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my_var = 5
my_str = "Hello, World!"
my_list = [1, 2, 3, 4, 5]

Looping and condition checking:

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for i in range(10):
    if i % 2 == 0:
        print(f"{i} is even.")
    else:
        print(f"{i} is odd.")

Sorting:

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my_list = [5, 3, 4, 1, 2]
sorted_list = sorted(my_list)
print(sorted_list)  # Output: [1, 2, 3, 4, 5]

List Manipulation:

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my_list = [1, 2, 3, 4, 5]
my_list[0] = 10  # Changing an element
print(my_list)  # Output: [10, 2, 3, 4, 5]

Problem-specific Concepts:

Understanding intervals and set containment: To solve this problem, it’s crucial to understand what an interval is and what it means for one set to contain another.

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# Defining two sets:
set_A = {1, 2, 3, 4, 5}
set_B = {2, 3}

# Checking if set_A contains set_B:
print(set_B.issubset(set_A))  # Output: True

Now, let’s discuss how to integrate these concepts to form a complete solution.

Define a class and a function with the appropriate parameters.
Initialize your variables, including your intervals and the tracking variables for the last two endpoints and the size of the containing set.
Sort the intervals in the desired order.
Begin a loop to iterate over these sorted intervals.
Use conditionals to check if the start of the current interval is greater than the second number in your tracking variable. If so, update your tracking variable to the last two numbers of the current interval and increase your size by 2.
If not, check if the start of the current interval is greater than the first number in your tracking variable. If so, update your tracking variable to its second number and the end of the current interval, and increase your size by 1.
After iterating over all intervals, return the size of the containing set. This is your final result.

Each of these coding drills, when understood and mastered individually, will allow one to understand and implement the final solution more efficiently and effectively.