#2580

Medium

LeetCode Problem Workspace

Count Ways to Group Overlapping Ranges

Count Ways to Group Overlapping Ranges is a medium difficulty problem focused on array manipulation and sorting to count valid range groupings.

array sorting

Problem Statement

You are given a list of integer ranges, where each range is represented as an array [start, end] indicating the start and end of the range. You need to group these ranges into two (or more) groups based on whether the ranges overlap. Two ranges are considered to overlap if they share at least one common integer.

Your task is to calculate how many valid ways you can split the ranges into two groups such that each group contains non-overlapping ranges. The number of possible groupings depends on how you identify the overlaps between ranges, and requires sorting and array manipulation.

Examples

Example 1

Input: ranges = [[6,10],[5,15]]

Output: 2

The two ranges are overlapping, so they must be in the same group. Thus, there are two possible ways:

Put both the ranges together in group 1.
Put both the ranges together in group 2.

Example 2

Input: ranges = [[1,3],[10,20],[2,5],[4,8]]

Output: 4

Ranges [1,3], and [2,5] are overlapping. So, they must be in the same group. Again, ranges [2,5] and [4,8] are also overlapping. So, they must also be in the same group. Thus, there are four possible ways to group them:

All the ranges in group 1.
All the ranges in group 2.
Ranges [1,3], [2,5], and [4,8] in group 1 and [10,20] in group 2.
Ranges [1,3], [2,5], and [4,8] in group 2 and [10,20] in group 1.

Constraints

1 <= ranges.length <= 105
ranges[i].length == 2
0 <= starti <= endi <= 109

Solution Approach

Sort Ranges Based on Start Points

First, sort the ranges based on their start values. This allows for a more efficient way of checking if any two ranges overlap. Sorting makes it easier to traverse through the ranges and compare consecutive ones to determine if they overlap.

Check for Overlaps and Grouping

Once the ranges are sorted, iterate through them and check for overlaps. If the current range overlaps with the previous one, they must be in the same group. This helps in grouping all overlapping ranges together. You then count the number of valid groupings based on this overlap detection.

Counting Valid Groupings

After identifying which ranges are overlapping, count the valid ways to split them into groups. The number of ways depends on the distinct groups formed by non-overlapping ranges. Use dynamic programming or a combinatorial approach to calculate the total valid groupings.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity is O(n log n) due to the sorting step, where n is the number of ranges. After sorting, the algorithm checks each range for overlaps in linear time, making the overall time complexity dominated by the sorting step. The space complexity is O(n) to store the ranges and possibly additional data structures for groupings.

What Interviewers Usually Probe

Can the candidate use sorting efficiently to simplify the range checking process?
Does the candidate recognize the need for efficient overlap detection?
Is the candidate able to count valid groupings after determining overlap regions?

Common Pitfalls or Variants

Common pitfalls

Failing to sort the ranges before checking overlaps, which leads to inefficient or incorrect results.
Not correctly identifying all overlapping ranges, which can result in fewer valid groupings than expected.
Overcomplicating the solution with unnecessary computations or missing the core combinatorial counting aspect.

Follow-up variants

Can the candidate solve the problem with more than two groups of non-overlapping ranges?
What if ranges could be merged into more than one valid group?
How would the approach change if you needed to track the exact groupings rather than just counting them?

FAQ

What is the core pattern for solving Count Ways to Group Overlapping Ranges?

The core pattern is sorting the ranges and then checking for overlaps, followed by calculating the valid ways to group the ranges based on these overlaps.

How do I count the number of valid groupings?

After sorting and detecting overlaps, you calculate the valid groupings by considering each non-overlapping range as a separate group and counting the possible splits.

What is the time complexity of the optimal solution?

The optimal solution has a time complexity of O(n log n) due to sorting, followed by a linear scan to check overlaps.

Can this problem be solved without sorting the ranges?

Sorting the ranges is essential to efficiently detect overlaps and reduce the complexity of the solution. Without sorting, overlap detection becomes much more difficult and inefficient.

What are the common mistakes when solving Count Ways to Group Overlapping Ranges?

Common mistakes include not sorting the ranges first, failing to properly detect all overlaps, and overcomplicating the counting of valid groupings.

terminal

Solution

Solution 1: Sorting + Counting + Fast Power

We can first sort the intervals in the range, merge the overlapping intervals, and count the number of non-overlapping intervals, denoted as $cnt$.

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class Solution:
    def countWays(self, ranges: List[List[int]]) -> int:
        ranges.sort()
        cnt, mx = 0, -1
        for start, end in ranges:
            if start > mx:
                cnt += 1
            mx = max(mx, end)
        mod = 10**9 + 7
        return pow(2, cnt, mod)

Solution 2

#### Python3

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class Solution:
    def countWays(self, ranges: List[List[int]]) -> int:
        ranges.sort()
        cnt, mx = 0, -1
        for start, end in ranges:
            if start > mx:
                cnt += 1
            mx = max(mx, end)
        mod = 10**9 + 7
        return pow(2, cnt, mod)

Continue Practicing

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