Minimum Jumps to Reach End via Prime Teleportation

To solve the problem, we use breadth-first search (BFS) to find the minimum jumps needed to reach the last index. The key is considering prime numbers as valid jump distances. This problem tests your ability to implement BFS efficiently, utilizing prime number calculations within the traversal process.

Problem Statement

You are given an integer array nums of length n. Starting at index 0, you need to reach the last index, n - 1, with the fewest number of jumps. From any index i, you can jump to another index j such that nums[i] + i = j and nums[i] + i is a prime number. If no prime jump is possible, you must stay at the current index. Find the minimum number of jumps to reach the end.

Your goal is to return the minimum number of jumps needed to reach index n - 1, or -1 if it is not possible to reach the end. You need to ensure your solution works efficiently given that n can be as large as 10^5.

Examples

Example 1

Input: nums = [1,2,4,6]

Output: 2

One optimal sequence of jumps is: Thus, the answer is 2.

Example 2

Input: nums = [2,3,4,7,9]

Output: 2

One optimal sequence of jumps is: Thus, the answer is 2.

Example 3

Input: nums = [4,6,5,8]

Output: 3

Constraints

• 1 <= n == nums.length <= 105
• 1 <= nums[i] <= 106

Solution Approach

Breadth-First Search (BFS)

The optimal solution involves using a BFS approach where each node represents an index in the array. From each index, explore all possible jumps that lead to prime numbers and track the distance from the starting index.

Prime Number Calculation

You need to efficiently check whether the sum of an index and its value is a prime number. Use a sieve method or precompute primes to speed up the process of determining valid jumps.

Edge Cases and Bounds

Ensure to handle edge cases such as when no jumps are possible or when the array size is minimal. Additionally, handle large numbers carefully within the given constraints.

Complexity Analysis

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

The time complexity depends on the BFS traversal and prime number checks. The space complexity is primarily determined by the array and BFS queue, resulting in O(n) space. Prime calculation can be optimized using a sieve method, reducing time complexity for prime checks.

What Interviewers Usually Probe

• Candidate uses BFS for traversing the array efficiently.
• Candidate optimizes prime checking, ensuring good performance for large inputs.
• Candidate handles edge cases like unreachable nodes or small arrays.

Common Pitfalls or Variants

Common pitfalls

• Overlooking the need for efficient prime number checking in large arrays.
• Inefficient BFS implementation, leading to slow solutions on large inputs.
• Failing to account for edge cases where the end index is unreachable.

Follow-up variants

• Use dynamic programming to solve the problem, where each state tracks the minimum jumps to that index.
• Extend the problem by considering non-prime teleportation jumps.
• Change the problem to find the maximum number of jumps instead of the minimum.

FAQ

What is the best approach for solving the 'Minimum Jumps to Reach End via Prime Teleportation' problem?

The best approach is to use breadth-first search (BFS) with a focus on prime number jumps to find the minimum jumps needed.

How do I efficiently check if a number is prime for this problem?

You can precompute primes using the Sieve of Eratosthenes, which allows constant-time prime checking during BFS traversal.

What is the time complexity of the BFS approach?

The time complexity is O(n) for BFS traversal, but you need to factor in the time for prime checking, which can be optimized using a sieve.

What are common mistakes when solving this problem?

Common mistakes include inefficient prime checking, missing edge cases where no valid jumps exist, and incorrectly implementing BFS.

How can GhostInterview assist with this problem?

GhostInterview provides tools for efficiently implementing BFS, checking prime numbers, and handling edge cases, making it easier to implement and debug your solution.