Non Overlapping Intervals Minimum Removal in DSA Go - Time & Space Complexity

Time Complexity: Non Overlapping Intervals Minimum Removal

O(n log n)

Understanding Time Complexity

We want to find out how long it takes to remove the fewest intervals so none overlap.

How does the time needed grow as the number of intervals grows?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

func eraseOverlapIntervals(intervals [][]int) int {
    if len(intervals) == 0 {
        return 0
    }
    sort.Slice(intervals, func(i, j int) bool {
        return intervals[i][1] < intervals[j][1]
    })
    count := 0
    end := intervals[0][1]
    for i := 1; i < len(intervals); i++ {
        if intervals[i][0] < end {
            count++
        } else {
            end = intervals[i][1]
        }
    }
    return count
}
    

This code sorts intervals by their end time, then counts how many overlap to remove the minimum.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Sorting the intervals and then looping through them once.
• How many times: Sorting takes multiple comparisons (depends on n log n), then one loop runs n-1 times.

How Execution Grows With Input

Sorting grows faster than looping as input grows.

Input Size (n)	Approx. Operations
10	About 10 * log(10) + 10 = 43
100	About 100 * log(100) + 100 = 700
1000	About 1000 * log(1000) + 1000 = 10000

Pattern observation: Operations grow a bit faster than the input size because of sorting.

Final Time Complexity

Time Complexity: O(n log n)

This means as the number of intervals grows, the time needed grows a bit faster than just counting them because of sorting.

Common Mistake

[X] Wrong: "The loop alone decides the time, so it is O(n)."

[OK] Correct: Sorting the intervals takes more time than the loop, so it dominates the total time.

Interview Connect

Understanding how sorting affects time helps you explain your solution clearly and shows you know what matters most in performance.

Self-Check

"What if the intervals were already sorted by end time? How would the time complexity change?"

Practice

1. What is the main goal of the Non Overlapping Intervals Minimum Removal problem?

easy

A. To remove the minimum number of intervals so that the remaining intervals do not overlap

B. To find the maximum number of overlapping intervals

C. To merge all intervals into one big interval

D. To sort intervals by their start time

Solution

1. Step 1: Understand the problem goal

   The problem asks to remove as few intervals as possible so that no two intervals overlap.

2. Step 2: Clarify what non-overlapping means

   Non-overlapping means intervals do not share any common points except possibly at the boundary.

3. Final Answer:

   To remove the minimum number of intervals so that the remaining intervals do not overlap -> Option A

4. Quick Check:

   Minimum removal = no overlaps [OK]

Hint: Focus on removing intervals, not merging or counting overlaps [OK]

Common Mistakes:

• Thinking the goal is to merge intervals
• Confusing maximum overlaps with minimum removals
• Sorting intervals without removing any

2. Which of the following Go code snippets correctly sorts intervals by their end time for solving the minimum removal problem?

easy

A. sort.Slice(intervals, func(i, j int) bool { return intervals[i][0] > intervals[j][0] })

B. sort.Slice(intervals, func(i, j int) bool { return intervals[i][0] < intervals[j][0] })

C. sort.Slice(intervals, func(i, j int) bool { return intervals[i][1] < intervals[j][1] })

D. sort.Slice(intervals, func(i, j int) bool { return intervals[i][1] > intervals[j][1] })

Solution

1. Step 1: Identify sorting by end time

   We need to sort intervals by their end time ascending to apply greedy approach.

2. Step 2: Check the Go sort.Slice syntax

   sort.Slice(intervals, func(i, j int) bool { return intervals[i][1] < intervals[j][1] }) sorts by intervals[i][1] < intervals[j][1], which is correct for end time ascending.

3. Final Answer:

   sort.Slice(intervals, func(i, j int) bool { return intervals[i][1] < intervals[j][1] }) -> Option C

4. Quick Check:

   Sort by end time ascending = sort.Slice(intervals, func(i, j int) bool { return intervals[i][1] < intervals[j][1] }) [OK]

Hint: Sort intervals by end time ascending for greedy removal [OK]

Common Mistakes:

• Sorting by start time instead of end time
• Sorting in descending order by mistake
• Using wrong slice indices

3. What is the output of the following Go code snippet that finds the minimum number of intervals to remove?

package main
import (
  "fmt"
  "sort"
)
func eraseOverlapIntervals(intervals [][]int) int {
  if len(intervals) == 0 {
    return 0
  }
  sort.Slice(intervals, func(i, j int) bool { return intervals[i][1] < intervals[j][1] })
  count := 0
  end := intervals[0][1]
  for i := 1; i < len(intervals); i++ {
    if intervals[i][0] < end {
      count++
    } else {
      end = intervals[i][1]
    }
  }
  return count
}
func main() {
  intervals := [][]int{{1,3},{2,4},{3,5}}
  fmt.Println(eraseOverlapIntervals(intervals))
}

medium

A. 2

B. 1

C. 0

D. 3

Solution

1. Step 1: Sort intervals by end time

   Sorted intervals: [1,3], [2,4], [3,5]

2. Step 2: Count overlaps

   Start with end=3. Interval [2,4] starts at 2 < 3, so count=1. Interval [3,5] starts at 3 ≥ 3, update end=5.

3. Final Answer:

   1 -> Option B

4. Quick Check:

   One overlap removed = 1 [OK]

Hint: Sort by end time, count intervals starting before current end [OK]

Common Mistakes:

• Not sorting intervals before counting
• Counting intervals that do not overlap
• Incorrectly updating end pointer

4. The following Go code is intended to solve the minimum removal problem but has a bug. What is the bug?

func eraseOverlapIntervals(intervals [][]int) int {
  if len(intervals) == 0 {
    return 0
  }
  sort.Slice(intervals, func(i, j int) bool { return intervals[i][1] < intervals[j][1] })
  count := 0
  end := intervals[0][1]
  for i := 1; i < len(intervals); i++ {
    if intervals[i][0] < end {
      count++
    } else {
      end = intervals[i][0]
    }
  }
  return count
}

medium

A. Updating end with intervals[i][0] instead of intervals[i][1] causes wrong comparisons

B. Missing check for empty intervals slice causes runtime error

C. Sorting by start time instead of end time causes incorrect removal count

D. Using count++ inside the loop is incorrect; should decrement count

Solution

1. Step 1: Check sorting and end update

   Sorting by end time is correct, but updating end incorrectly with intervals[i][0] is wrong.

2. Step 2: Explain why end update is wrong

   End should track the end time of last non-overlapping interval, so intervals[i][1] is correct.

3. Final Answer:

   Updating end with intervals[i][0] instead of intervals[i][1] causes wrong comparisons -> Option A

4. Quick Check:

   End must be interval end, not start [OK]

Hint: Update end with interval's end time, not start time [OK]

Common Mistakes:

• Confusing interval start and end when updating end
• Ignoring empty slice check
• Misunderstanding sorting impact

5. Given intervals [[1,2],[2,3],[3,4],[1,3]], what is the minimum number of intervals to remove to make the rest non-overlapping?

hard

A. 2

B. 3

C. 0

D. 1

Solution

1. Step 1: Sort intervals by end time

   Sorted intervals: [1,2], [1,3], [2,3], [3,4]

2. Step 2: Count overlaps and removals

   Start end=2. Interval [1,3] starts at 1 < 2, count=1. Interval [2,3] starts at 2 ≥ 2, update end=3. Interval [3,4] starts at 3 ≥ 3, update end=4.

3. Final Answer:

   1 -> Option D

4. Quick Check:

   One removal needed = 1 [OK]

Hint: Sort by end, remove intervals starting before current end [OK]

Common Mistakes:

• Not sorting intervals before counting
• Removing more intervals than needed
• Miscounting overlaps with boundary intervals