Count Ways to Group Overlapping Ranges

Count Ways to Group Overlapping Ranges

What is this problem about?

The Count Ways to Group Overlapping Ranges coding problem gives you a set of intervals. You need to split these intervals into two groups (Group 1 and Group 2) such that no two intervals in different groups overlap. You need to return the total number of ways to perform this split modulo $10^9 + 7$.

Why is this asked in interviews?

Oracle and IBM ask this Array and Sorting interview pattern to test your ability to simplify a problem. The core insight is that overlapping intervals must belong to the same "cluster." Once you find the number of disjoint clusters, the problem becomes a simple counting task. It evaluates your skills in interval merging and basic combinatorics.

Algorithmic pattern used

1. Interval Merging: Sort the intervals by their start time and merge all overlapping ones.
2. Cluster Counting: Count how many disjoint intervals exist after merging. Let this number be $m$.
3. Power of 2: Each cluster can independently be assigned to either Group 1 or Group 2. Therefore, the total number of ways is $2^m$.
4. Return $2^m \pmod{10^9 + 7}$.

Example explanation

Intervals: [[1, 3], [2, 5], [8, 10]]

1. Sort: [1, 3], [2, 5], [8, 10].
2. Merge: [1, 3] and [2, 5] overlap, they become [1, 5].
3. Disjoint clusters: {[1, 5], [8, 10]}. $m = 2$.
4. Ways: Each of the 2 clusters has 2 choices. $2^2 = 4$. The ways are: (Both G1), (Both G2), (C1 in G1, C2 in G2), (C1 in G2, C2 in G1).

Common mistakes candidates make

• O(n^2) Merging: Trying to check every pair of intervals for overlaps instead of sorting first.
• Calculating Power of 2 manually: Using a loop for $2^m$ when $m$ is large, instead of using modular exponentiation or bit shifting.
• Miscounting Clusters: Failing to handle the case where the end of one interval matches the start of another (check if the problem says "inclusive").

Interview preparation tip

Interval problems almost always start with sorting by the start or end time. If you're stuck, try sorting and see if a greedy or merging strategy emerges.