The problem of finding the number of repeated subarrays in an array is an intriguing computational challenge that arises in the realm of algorithms and data structures. In this article, we'll explore the technical details of this problem, walk through examples, and use suitable data structures to solve it efficiently.

Problem Definition

Given an array arr, the goal is to find how many subarrays appear more than once in the array. A subarray is any contiguous segment of the array.

Detailed Explanation

To solve the problem efficiently, especially for large arrays, a naive approach of comparing all possible subarrays is computationally expensive. Instead, we can leverage hashing techniques combined with algorithms like the Rabin-Karp algorithm to find repeated subarrays efficiently.

Key Concepts

• Subarray: A subarray of an array is a contiguous portion of the array. For example, in the array [1, 2, 3, 4], [2, 3] is a subarray.
• Hashing: To efficiently compare subarrays, we utilize hashing techniques to map a subarray into a hash value, which enables quick comparison.
• Rabin-Karp Algorithm: This algorithm is an efficient way to find patterns in a text using a rolling hash. It's applicable here to compute hash values for subarrays.

Steps to Find Repeated Subarrays

1. Initialize hash structures: Use a hash map to store the frequency of hash values, where each hash represents a subarray.
2. Compute hash values: Calculate the hash of each subarray using the Rabin-Karp rolling hash technique. This allows adjustment of the hash when the subarray moves forward.
3. Count frequencies: Update the hash map with the frequency of each hash value.
4. Extract repeats: Iterate over the hash map to count how many subarrays have a frequency greater than one.

Example

Consider an array arr = [1, 2, 3, 1, 2, 4, 1, 2, 3].

• Compute hash values for all possible subarrays.
• Maintain a hash table to record how many times each hash value appears.
• After processing, check the hash table for hash values with frequencies greater than one to count the repeated subarrays.

Implementation in Python

Here's a Python code snippet that follows the outlined approach: