Insert Interval into Sorted List in DSA C# - Time & Space Complexity

Time Complexity: Insert Interval into Sorted List

O(n)

Understanding Time Complexity

We want to understand how the time needed to insert a new interval into a sorted list changes as the list grows.

Specifically, how does the number of steps grow when the list has more intervals?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

public IList<Interval> Insert(IList<Interval> intervals, Interval newInterval) {
    var result = new List<Interval>();
    int i = 0;
    while (i < intervals.Count && intervals[i].End < newInterval.Start) {
        result.Add(intervals[i]);
        i++;
    }
    while (i < intervals.Count && intervals[i].Start <= newInterval.End) {
        newInterval.Start = Math.Min(newInterval.Start, intervals[i].Start);
        newInterval.End = Math.Max(newInterval.End, intervals[i].End);
        i++;
    }
    result.Add(newInterval);
    while (i < intervals.Count) {
        result.Add(intervals[i]);
        i++;
    }
    return result;
}
    

This code inserts a new interval into a sorted list of intervals, merging overlapping intervals as needed.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Three while loops that each traverse parts of the intervals list.
• How many times: Each loop moves forward through the list without going back, together covering all intervals once.

How Execution Grows With Input

As the number of intervals (n) grows, the code checks each interval at most once.

Input Size (n)	Approx. Operations
10	About 10 checks and merges
100	About 100 checks and merges
1000	About 1000 checks and merges

Pattern observation: The number of operations grows roughly in direct proportion to the number of intervals.

Final Time Complexity

Time Complexity: O(n)

This means the time to insert grows linearly with the number of intervals in the list.

Common Mistake

[X] Wrong: "Since there are three loops, the time complexity is O(3n) or O(n^2)."

[OK] Correct: The loops run one after another and together cover the list only once, so the total work is proportional to n, not n squared.

Interview Connect

Understanding how to analyze loops that run sequentially is a key skill. It helps you explain your code clearly and shows you can reason about efficiency in real problems.

Self-Check

"What if the intervals list was not sorted? How would the time complexity change?"

Practice

1. What is the main goal when inserting a new interval into a sorted list of non-overlapping intervals?

easy

A. To sort the intervals again after insertion without merging

B. To add the new interval at the end without checking overlaps

C. To remove all intervals that come before the new interval

D. To merge overlapping intervals and keep the list sorted without overlaps

Solution

1. Step 1: Understand the problem context

   The list contains sorted, non-overlapping intervals. Inserting a new interval may cause overlaps.

2. Step 2: Identify the required action

   We must merge any overlapping intervals with the new one and keep the list sorted without overlaps.

3. Final Answer:

   To merge overlapping intervals and keep the list sorted without overlaps -> Option D

4. Quick Check:

   Insert and merge intervals = To merge overlapping intervals and keep the list sorted without overlaps [OK]

Hint: Merge overlapping intervals to keep list sorted and non-overlapping [OK]

Common Mistakes:

• Inserting without merging overlaps
• Appending without sorting
• Removing unrelated intervals

2. Which of the following C# code snippets correctly declares a list of intervals as List intervals?

easy

A. var intervals = new List { [1, 3], [6, 9] };

B. var intervals = new List { new int[] {1, 3}, new int[] {6, 9} };

C. List intervals = new List { (1, 3), (6, 9) };

D. List<int> intervals = new List<int> { {1, 3}, {6, 9} };

Solution

1. Step 1: Check correct syntax for List of int arrays

   List requires elements as int arrays, declared with new int[] {start, end}.

2. Step 2: Validate each option

   var intervals = new List { new int[] {1, 3}, new int[] {6, 9} }; uses new int[] correctly; others use invalid syntax like braces or tuples.

3. Final Answer:

   var intervals = new List { new int[] {1, 3}, new int[] {6, 9} }; -> Option B

4. Quick Check:

   List of int arrays syntax = var intervals = new List { new int[] {1, 3}, new int[] {6, 9} }; [OK]

Hint: Use new int[] {start, end} inside List initialization [OK]

Common Mistakes:

• Using curly braces instead of new int[]
• Using tuples instead of arrays
• Incorrect variable type declaration

3. Given the intervals [[1, 3], [6, 9]] and a new interval [2, 5], what is the resulting list after inserting and merging?

medium

A. [[2, 5], [6, 9]]

B. [[1, 3], [2, 5], [6, 9]]

C. [[1, 5], [6, 9]]

D. [[1, 9]]

Solution

1. Step 1: Identify overlapping intervals

   The new interval [2, 5] overlaps with [1, 3] because 2 ≤ 3.

2. Step 2: Merge overlapping intervals

   Merge [1, 3] and [2, 5] into [1, 5]. The interval [6, 9] remains unchanged.

3. Final Answer:

   [[1, 5], [6, 9]] -> Option C

4. Quick Check:

   Merged intervals = [[1, 5], [6, 9]] [OK]

Hint: Merge intervals if new start ≤ existing end [OK]

Common Mistakes:

• Not merging overlapping intervals
• Inserting new interval without merging
• Merging all intervals incorrectly

4. Consider this C# code snippet to insert an interval into a sorted list. What is the bug?

var intervals = new List { new int[] {1, 2}, new int[] {3, 5} };
var newInterval = new int[] {4, 7};

foreach(var interval in intervals) {
    if(interval[1] >= newInterval[0]) {
        interval[1] = Math.Max(interval[1], newInterval[1]);
        break;
    }
}

medium

A. It modifies only the first overlapping interval but misses merging subsequent overlaps

B. It throws a runtime exception due to invalid index access

C. It inserts the new interval at the wrong position

D. It does not check for any overlaps before merging

Solution

1. Step 1: Analyze the loop behavior

   The loop breaks after merging with the first overlapping interval, ignoring others.

2. Step 2: Identify the bug impact

   Intervals after the first overlap are not merged, causing incorrect final intervals.

3. Final Answer:

   It modifies only the first overlapping interval but misses merging subsequent overlaps -> Option A

4. Quick Check:

   Loop breaks early, missing merges = It modifies only the first overlapping interval but misses merging subsequent overlaps [OK]

Hint: Do not break loop early; merge all overlapping intervals [OK]

Common Mistakes:

• Breaking loop too soon
• Not merging all overlaps
• Assuming single overlap only

5. You have intervals [[1, 2], [3, 5], [6, 7], [8, 10], [12, 16]] and a new interval [4, 8]. After inserting and merging, what is the final list?

hard

A. [[1, 2], [3, 10], [12, 16]]

B. [[1, 2], [4, 8], [12, 16]]

C. [[1, 2], [3, 5], [6, 7], [8, 10], [12, 16]]

D. [[1, 2], [3, 8], [12, 16]]

Solution

1. Step 1: Find intervals overlapping with [4, 8]

   Intervals [3, 5], [6, 7], and [8, 10] overlap with [4, 8].

2. Step 2: Merge all overlapping intervals

   Merge [3, 5], [6, 7], [8, 10], and [4, 8] into [3, 10].

3. Step 3: Keep non-overlapping intervals unchanged

   Intervals [1, 2] and [12, 16] remain as is.

4. Final Answer:

   [[1, 2], [3, 10], [12, 16]] -> Option A

5. Quick Check:

   Merged overlapping intervals correctly = [[1, 2], [3, 10], [12, 16]] [OK]

Hint: Merge all intervals overlapping new interval into one [OK]

Common Mistakes:

• Merging only some overlapping intervals
• Not merging intervals that touch at boundaries
• Replacing intervals instead of merging