Merge Intervals

This requires merging overlapping intervals. Here’s a step-by-step guide to solve this problem.

  1. Sort the Intervals: First, sort the intervals based on their starting time. This ensures that intervals that could possibly overlap are next to each other.

  2. Initialize the Result: Create a list to store the merged intervals. Initialize it with the first interval in the sorted list.

  3. Iterate through Intervals: Loop through the sorted intervals starting from the second interval. For each interval, compare it with the last interval in the result list.

    • If Overlapping: If the current interval overlaps with the last interval in the result list (i.e., its start time is less than or equal to the end time of the last interval), merge them by updating the end time of the last interval in the result list to be the maximum of the end times of the two overlapping intervals.
    • If Not Overlapping: If the current interval does not overlap with the last interval in the result list, simply add the current interval to the result list.

  4. Return the Result: The result list now contains the merged non-overlapping intervals.

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26

class Solution:
    def merge(self, intervals: List[List[int]]) -> List[List[int]]:
        if not intervals:
            return []

        # Step 1: Sort the intervals by their start times
        intervals.sort(key=lambda x: x[0])

        # Step 2: Initialize the result with the first interval
        result = [intervals[0]]

        # Step 3: Iterate through the sorted intervals
        for i in range(1, len(intervals)):
            current_start, current_end = intervals[i]
            last_start, last_end = result[-1]

            # Check if the current interval overlaps with the last interval in the result
            if current_start <= last_end:
                # Merge the overlapping intervals
                result[-1] = [last_start, max(last_end, current_end)]
            else:
                # Add the non-overlapping interval to the result
                result.append(intervals[i])

        # Step 4: Return the result
        return result

This code ensures that all overlapping intervals are merged, and it returns the non-overlapping intervals that cover all the intervals in the input. The time complexity is (O(n \log n)) due to the sorting step, and the space complexity is (O(n)), where (n) is the number of intervals.

Identifying Problem Isomorphism

“Insert Interval” is an isomorphic problem to “Merge Intervals”.

In both problems, the task is to handle overlapping intervals. “Merge Intervals” asks you to merge all overlapping intervals. In “Insert Interval”, you have a similar task but with an additional step. Given a set of non-overlapping intervals and a new interval to insert, you have to merge the new interval with the existing ones if it overlaps with them and insert it at the correct position if it doesn’t.

They both require you to understand and handle overlapping intervals.

In terms of mapping the given problem to “Insert Interval”, consider each meeting room as a non-overlapping interval and the meetings as new intervals to be inserted. The process of allocating a meeting room then becomes a process of inserting a new interval.

This is an approximate mapping, and the constraints and edge cases in the original problems may not be exactly the same. However, the task of managing overlapping intervals is common in both problems, which makes this an isomorphism.

10 Prerequisite LeetCode Problems

Here are ten problems for understanding problem 56, Merge Intervals:

  1. 252. Meeting Rooms: This problem introduces the basic concept of interval scheduling and manipulation.

  2. 253. Meeting Rooms II: A slight step-up from the previous problem, you need to find out the minimum number of conference rooms required, which is done by sorting the intervals.

  3. 435. Non-overlapping Intervals: This problem is closely related to interval merging and scheduling. You need to find out the minimum number of intervals you need to remove to make the rest non-overlapping.

  4. 646. Maximum Length of Pair Chain: This problem is another example of interval scheduling where you are required to find out the maximum number of pairs you can form.

  5. 986. Interval List Intersections: A more direct preparation for merge intervals, this problem requires you to find out the intersection of two lists of intervals.

  6. 729. My Calendar I: This problem is a practical implementation of interval scheduling where you need to book appointments without conflict.

  7. 57. Insert Interval: This problem is almost similar to Merge Intervals but with an extra step of inserting a new interval before merging.

  8. 763. Partition Labels: A variant of the interval problem where you need to partition labels (characters in a string) such that each letter appears in at most one partition.

  9. 1288. Remove Covered Intervals: This problem requires understanding of intervals and manipulation where you need to remove covered intervals.

  10. 228. Summary Ranges: This problem helps in understanding how to handle consecutive ranges which is quite similar to the merge intervals problem.

These cover interval manipulation, sorting based on custom conditions, and the greedy approach.

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13

class Solution:
    def merge(self, intervals: List[List[int]]) -> List[List[int]]:
        intervals = sorted(intervals, key=lambda x: x [0])

        ans = []

        for interval in intervals:
            if not ans or ans[-1][1] < interval[0]:
                ans.append(interval)
            else:
                ans[-1][1] = max(ans[-1][1], interval[1])
        
        return ans

Problem Classification

The given problem falls under interval manipulation and computational geometry. It involves analyzing intervals and merging overlapping ones.

What Components:

  1. Input: An array of intervals, with each interval represented by a pair of integers denoting the start and end time.
  2. Output: An array of non-overlapping intervals obtained by merging any overlapping intervals from the input.
  3. Constraints: The length of intervals, the start and end times are bounded.
  • Problem Type: This is an optimization problem, where the goal is to reduce the number of intervals by merging overlapping ones.
  • Data Structure: The problem is using arrays to represent intervals.
  • Methods Involved: It involves sorting, iteration, and comparison of elements within the data structure.
  • Complexity Level: The problem is moderately complex, given the need to identify and merge overlapping intervals while maintaining order.

The problem’s core involves the manipulation of intervals, making it part of computational geometry. By examining intervals and merging those that overlap, the problem requires an understanding of sorting and iteration, plus comparison to optimize the interval representation. The constraints provide bounds that simplify the problem’s scope, but the need to understand and correctly manage overlapping intervals adds a layer of complexity to the problem. Therefore, the problem can be categorized as an optimization problem within the computational geometry domain.

Language Agnostic Coding Drills

  1. Understanding Data Types: Before diving into any code, it’s important to understand what kind of data you’re working with. In this case, we’re dealing with a list of lists (a 2D list) where each inner list contains two integer elements.

  2. List Manipulation: Working with lists, including creation, indexing, and slicing, is an essential concept in this code. For instance, ans is a list that you append elements to, and ans[-1][1] is an example of indexing to access a specific element.

  3. Using For Loops: This code includes a for loop which iterates over the list intervals. Understanding how to setup and use for loops is an important concept to grasp.

  4. Conditionals (If statements): Within the for loop, there is an if statement to decide whether to append a new list to ans or modify the existing last list in ans.

  5. Sorting: The first line inside the function sorts the intervals list based on the first element of each inner list. Understanding how to use sorting functions is crucial.

  6. Lambda Functions: The sorting operation uses a lambda function to specify the sorting key. Grasping lambda (anonymous) functions, which are small, one-line functions, is necessary.

  7. Class and Function Definition: The code is organized as a method inside a class. Understanding how to define classes and methods, and the role of the self keyword, is important.

  8. Working with 2D Lists: Given that intervals is a list of lists, understanding how to create, access, and manipulate 2D lists is key.

Here’s the ordered list of concepts:

  1. Understanding Data Types
  2. List Manipulation
  3. Using For Loops
  4. Conditionals (If statements)
  5. Sorting
  6. Lambda Functions
  7. Class and Function Definition
  8. Working with 2D Lists

Targeted Drills in Python

Here are the coding drills in Python corresponding to the concepts outlined:

  1. Understanding Data Types

    1
2
3
4
5
6

    x = 5
y = "Hello, World!"
z = [1, 2, 3]
print(type(x))
print(type(y))
print(type(z))

  2. List Manipulation

    1
2
3
4

    list = [1, 2, 3, 4, 5]
list.append(6)  # Append an element
list.insert(0, 0)  # Insert an element at specific position
print(list[-1])  # Print last element

  3. Using For Loops

    1
2

    for i in range(5):
    print(i)

  4. Conditionals (If statements)

    1
2
3
4
5

    x = 5
if x > 10:
    print("x is greater than 10")
else:
    print("x is less than or equal to 10")

  5. Sorting

    1
2
3

    list = [5, 1, 9, 3, 7]
sorted_list = sorted(list)
print(sorted_list)

  6. Lambda Functions

    1
2

    square = lambda x: x * x
print(square(5))

  7. Class and Function Definition

    1
2
3
4
5
6

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

obj = MyClass()
obj.my_function()

  8. Working with 2D Lists

    1
2
3
4
5

    matrix = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
print(matrix[0][1])  # Accessing an element
matrix[1] = [10, 11, 12]  # Changing a row
matrix[2][1] = 13  # Changing a specific element
print(matrix)

Remember, each of these drills focuses on a single concept, but the final code combines all these concepts. Thus, once you’re comfortable with all these drills, understanding and constructing the final code should be much easier.