Arithmetic Slices II - Subsequence

Arithmetic Slices II - Subsequence

What is this problem about?

The Arithmetic Slices II - Subsequence interview question is a significantly harder version of the "Arithmetic Slices" problem. Instead of contiguous subarrays, you must find the number of arithmetic subsequences (not necessarily contiguous) of at least three elements. This Arithmetic Slices II - Subsequence coding problem requires much more complex tracking because elements can be skipped.

Why is this asked in interviews?

This is a HARD-level question frequently used by Google, Meta, and Amazon. It tests a candidate's mastery of Dynamic Programming with Hash Maps. Since the differences between elements can be large and sequences can overlap in complex ways, you can't just use a simple array for DP; you need a more flexible state representation.

Algorithmic pattern used

The Array, Dynamic Programming interview pattern used here involves creating a DP table where each entry dp[i] is a Hash Map. The key in the map is the common difference d, and the value is the number of arithmetic subsequences ending at index i with difference d.

Example explanation

Given [2, 4, 6, 8]:

1. At index 1 (value 4), difference 4-2=2. Subsequences ending at index 1 with d=2 is 1 ([2,4]).
2. At index 2 (value 6), difference 6-4=2. We look at index 1 for d=2 (found 1) and add that to our total. Also 6-2=4 gives another subsequence [2,6].
3. As you move through, you build on previous counts. The "at least 3 elements" condition is handled by only adding to the global total when you're extending a sequence that already had at least 2 elements.

Common mistakes candidates make

• Integer Overflow: In many languages, the difference between two elements can exceed the range of a 32-bit integer.
• Memory Limits: Creating too many Hash Maps or maps that are too large can lead to Memory Limit Exceeded errors.
• Incorrect Counting: Accidentally including subsequences of length 2 in the final total.

Interview preparation tip

Master the use of "DP on Hash Maps." It's a common technique for problems where the DP state depends on a value (like a difference or a sum) that has a very large potential range but only a few actually occur in the input.