Unique Substrings in Wraparound String

Unique Substrings in Wraparound String

What is this problem about?

The Unique Substrings in Wraparound String interview question involves a special string $S$ that is an infinite wraparound of the English alphabet (...abcd...xyzabc...). You are given a string p, and you need to find how many unique non-empty substrings of p are present in $S$. A substring of p is in $S$ if its characters are consecutive in the alphabet, including the z to a transition.

Why is this asked in interviews?

This Unique Substrings in Wraparound String coding problem is a favorite at Google because it requires a clever observation to avoid $O(n^2)$ complexity. It tests whether a candidate can use Dynamic Programming to aggregate results effectively. The challenge lies in counting only unique substrings, even if they appear multiple times in p or are parts of longer consecutive sequences.

Algorithmic pattern used

The core String, Dynamic Programming interview pattern involves tracking the maximum length of a consecutive sequence ending at each character ('a' through 'z'). We iterate through p. If the current character is the next in sequence from the previous one (or is 'a' after 'z'), we increment the current consecutive length. We then update an array count[26] with the maximum length found so far for that ending character. The total number of unique substrings is the sum of all values in the count array.

Example explanation

Suppose p = "abcabcd".

1. "abc": length 3 ending in 'c'. count['c'] = max(0, 3) = 3. (This accounts for "c", "bc", "abc")
2. "abcd": length 4 ending in 'd'. count['d'] = max(0, 4) = 4. (This accounts for "d", "cd", "bcd", "abcd") The count array ensures we don't double-count "abc" when we process "abcd".

Common mistakes candidates make

Many candidates try to use a Set to store every unique substring they find. This leads to $O(n^2)$ time and space complexity, which will fail for large inputs. Another mistake is failing to handle the z to a wraparound correctly. The key is to realize that if you know the longest consecutive string ending at 'c', you automatically know all shorter consecutive strings ending at 'c'.

Interview preparation tip

When asked to count unique substrings with a specific property, try to find a way to "group" them by their ending character or starting character. This often allows you to use a small fixed-size array (like 26 for the alphabet) to store your DP state, drastically improving efficiency.