Count Prime-Gap Balanced Subarrays

This problem involves counting prime-gap balanced subarrays within an array. A subarray is prime-gap balanced if the differences between consecutive prime numbers in the subarray are constant and equal to a given value k. You’ll need to use a sliding window approach while efficiently calculating prime numbers within the range using the sieve method.

Problem Statement

You are given an integer array nums and an integer k. A subarray is defined as prime-gap balanced if the differences between consecutive prime numbers in the subarray are equal to k. Your task is to return the count of prime-gap balanced subarrays in nums.

To solve the problem, you will need to iterate through the array using a sliding window approach. For each subarray, check the gap between consecutive prime numbers and ensure it matches the value k. Efficient prime number calculation can be achieved using the sieve method, which helps improve the performance of this approach.

Examples

Example 1

Input: nums = [1,2,3], k = 1

Output: 2

Prime-gap balanced subarrays are: Thus, the answer is 2.

Example 2

Input: nums = [2,3,5,7], k = 3

Output: 4

Prime-gap balanced subarrays are: Thus, the answer is 4.

Constraints

• 1 <= nums.length <= 5 * 104
• 1 <= nums[i] <= 5 * 104
• 0 <= k <= 5 * 104

Solution Approach

Prime number calculation with sieve

Use the Sieve of Eratosthenes to precompute all prime numbers up to the maximum value in the array. This allows quick lookup to identify prime numbers during the sliding window process.

Sliding window approach

Implement a sliding window technique to iterate through the array, maintaining the state of the current subarray. As you expand and contract the window, check if the subarray meets the prime-gap balanced condition.

Efficient gap comparison

As you slide the window, calculate the differences between consecutive primes. If the differences match the given value k, count the subarray. This requires careful tracking of gaps between primes as the window moves.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity depends on the approach used for prime number generation (Sieve of Eratosthenes, which is O(n log log n)) and the sliding window traversal (O(n)). Overall, the time complexity will be dominated by these factors. The space complexity is determined by the need to store primes and the sliding window state, making it O(n).

What Interviewers Usually Probe

• Ability to efficiently calculate primes using the Sieve of Eratosthenes.
• Proficiency with sliding window algorithms for dynamic subarray problems.
• Skill in managing running state and gap comparisons as part of a window traversal.

Common Pitfalls or Variants

Common pitfalls

• Incorrectly calculating prime gaps within subarrays.
• Failure to optimize prime number lookup, leading to inefficient solutions.
• Not properly handling edge cases, such as when the array has no valid prime-gap balanced subarrays.

Follow-up variants

• Optimize prime number generation for larger values.
• Consider variations in gap comparisons other than a fixed k.
• Handle arrays with no valid subarrays efficiently.

FAQ

What is a prime-gap balanced subarray?

A prime-gap balanced subarray is a subarray where the differences between consecutive prime numbers are equal to a given value k.

How can the Sieve of Eratosthenes help in this problem?

The Sieve of Eratosthenes helps precompute prime numbers up to the maximum value in the array, allowing efficient prime checking during the sliding window process.

What is the time complexity of the sliding window solution?

The time complexity is primarily O(n log log n) for the prime number generation using the sieve, and O(n) for the sliding window traversal.

How do I efficiently calculate prime gaps in subarrays?

By maintaining a running window and calculating the gaps between consecutive primes within that window, you can efficiently check if the subarray meets the prime-gap balanced condition.

What are some edge cases to consider in this problem?

Consider edge cases where no valid prime-gap balanced subarrays exist or when k is very large relative to the values in the array.