Intervals Between Identical Elements

Intervals Between Identical Elements

1. What is this problem about?

The Intervals Between Identical Elements interview question asks you to find the sum of absolute differences between the indices of identical elements in an array. For each index $i$, you need to calculate $\sum |i - j|$ for all $j$ such that arr[j] == arr[i]. The output is an array of these sums.

2. Why is this asked in interviews?

Companies like Google ask this to evaluate your ability to optimize an $O(N^2)$ distance calculation into $O(N)$. It tests your knowledge of Hash Table interview patterns and the Prefix Sum interview pattern. It’s a great example of how mathematical derivation can simplify a problem.

3. Algorithmic pattern used

This problem follows the Prefix Sum and Frequency Counting pattern.

1. Group Indices: First, use a Hash Map to store all indices for each unique value (e.g., value -> [idx1, idx2, idx3]).
2. Formula Optimization: For a list of indices $P$, the sum of distances for $P[i]$ is: (i * P[i] - prefixSum[i]) + (suffixSum[i] - (count - i - 1) * P[i]).
   • The first part is the sum of distances to all identical elements to the left.
   • The second part is the sum of distances to all identical elements to the right.
3. Linear Pass: Iterate through each list of indices in the map and apply this formula in $O(1)$ per index using pre-calculated prefix/suffix sums of that specific list.

4. Example explanation

Array: [2, 1, 3, 1, 2, 1]. Let's look at '1' (indices: [1, 3, 5]).

• For index 3:
  • Left index: 1. Distance $|3-1| = 2$.
  • Right index: 5. Distance $|3-5| = 2$.
  • Total for index 3 is 4. The algorithm calculates this for every '1' and every '2' efficiently.

5. Common mistakes candidates make

• $O(N^2)$ approach: Trying to iterate through the entire array for every index. This will time out for $N=10^5$.
• Ignoring Math: Failing to realize that $\sum |i - j|$ can be split into two groups (left and right) to remove the absolute value bars and enable prefix sum logic.
• Integer Overflow: The sums can be quite large, so ensure you use 64-bit integers.

6. Interview preparation tip

Whenever you see a sum of absolute differences $\sum |i - j|$, your first goal should be to remove the absolute value. This is almost always done by Sorting the indices and using Prefix Sums.