Understanding the Problem

Finding adjacent identical substrings in a string is a common problem in computer science, often encountered in text processing or pattern matching tasks. The goal is to determine if a string contains any two identical substrings that are contiguous. In other words, we're looking for a pattern "XX" where "X" is any non-empty substring.

Technical Description

Given a string `S` of length `n`, the task is to check if there exist two non-empty substrings `S[i:j]` and `S[j:k]` such that `S[i:j] == S[j:k]` and `j = k - (j-i)`, which implies the substrings are adjacent. The complexity of directly checking all possible pairs of substrings can be overwhelming, especially for longer strings, since a naive approach would take $O(n^3)$ time.

Efficient Algorithm

To solve this problem efficiently, a useful approach involves using rolling hashing—a technique that allows comparing strings in a constant time after pre-processing. Here's how we can implement it:

1. Hash Function: Calculate a hash value for strings based on an arbitrary but large base (often a prime number, such as 31 or 101) and perform modulus operation with another large prime to avoid overflow. This hash can be represented as: $$\text{hash}(S) = (S[0] \times \text{base}^{n-1} + S[1] \times \text{base}^{n-2} + \ldots + S[n-1] \times \text{base}^0) \bmod \text{modulus}$$
2. Rolling Hash: As you move along the string, compute the hash dynamically using the previous hash value—a concept similar to a sliding window: $$\text{new\_hash} = (\text{old\_hash} - S_{\text{left}} \times \text{base}^{L-1}) \times \text{base} + S_{\text{right}}$$ Here, $S_{\text{left}}$ is the character going out of the window, and $S_{\text{right}}$ is the character coming in.
3. Implementation Steps: • Calculate the hash for the first substring of length `len`. • Slide the window of length `len` through the string, updating hash values using the rolling hash formula. • Check for hash collisions; if two adjacent substrings have the same hash, compare them directly to confirm they're identical (to avoid false positives due to hash collisions).
4. Edge Cases: Be sure to consider strings with special characters or non-Alphabetical inputs, and adjust the base and modulus to prevent hash collision probabilities.

Example

Consider the string `S = "abcabcx"`, and you want to check substrings of length 3:

1. Calculate the initial hash of `abc` as described.
2. As the window slides, you'll compare hash for `abc` and `cab`.
3. When the window reaches `abcx`, hash of `abc` after `c` matches the first window hash, confirming they are identical.

Additional Considerations

Complexity Analysis

Using rolling hashes, the time complexity turns into $O(n \cdot \log n)$ to find a solution by checking all possible window sizes (using binary search over lengths) up to the string's half-length.

Applications

• Text Editors: Detect duplicate consecutive blocks of text. • Compression Algorithms: Identify redundancy very quickly for optimization. • DNA Sequencing: Find repeat patterns in sequences for biological research.

Table: Key Points Summary

This algorithm efficiently finds adjacent identical substrings, offering optimized performance through rolling hash methods while being versatile enough to handle a wide range of applications across computational disciplines.