Problem Description

In this problem, you're given an integer array nums of length n. The task is to find the smallest index i (where 0 <= i <= n - 2) such that splitting the array into two parts at index i results in the product of the first i + 1 elements being coprime with the product of the remaining elements. Two numbers are coprime if their greatest common divisor (GCD) is equal to 1.

A split is considered valid if and only if the aforementioned condition of coprimality is satisfied. If no such index exists, the function should return -1.

For example:

• With nums = [4,7,8,15,3,5], a valid split occurs at index i = 2 where the products are 4*7*8 = 224 and 15*3*5 = 225, and gcd(224, 225) = 1.
• With nums = [4,7,15,8,3,5], no valid split can be found, and the function should return -1.

Intuition

The general approach to solving this problem is to iterate over the nums array and at each index, we need to track the prime factors of the cumulative product of elements up to that index, as well as maintain a mapping of the last index at which each prime factor appears.

Here's a step-by-step breakdown on how we arrive at the solution:

1. Prime Factorization for each element: We iterate through the array and perform prime factorization for each element. This will help us in understanding the prime factors that make up each number in the array.

2. Tracking First and Last Occurrences: While doing the prime factorization, we maintain two data structures. One dictionary (first) to store the first occurrence of a prime factor and another list (last) to note down the last occurrence of a prime factor that has been seen so far. If a prime factor appears again at a later index, we update the last list at the index of the first occurrence of this prime factor to the current index.

3. Finding the Split Index: After completing the prime factorization and tracking, we iterate over the last list to find the smallest index i where last[i] < i. This indicates that splitting the array at index i will have no common prime factors for the product of the first i + 1 numbers and the product of the remaining numbers.

The index mx is used to track the maximum index in last that we have seen so far. If at any point mx < i, we immediately return mx as the answer - this is our split point. If we finish iterating over the list and don't find such an index, we return -1, which means no valid split exists.

This solution is efficient since it avoids explicitly calculating the product of the subarrays, which could result in very large numbers, and it allows us to determine if the array can be split validly based on the presence of common prime factors between the two subarrays.

Learn more about Math patterns.

Solution Approach

The solution to this problem uses prime factorization and dictionary mapping to efficiently find the smallest valid split index.

Here's how the implementation works:

• Prime Factorization: For each number x in the array nums, the algorithm finds its prime factors. Starting with the smallest prime number 2, the code iterates through potential factors increasing j until j exceeds the square root of x. If x is divisible by j, it is repeatedly divided by j until it is no longer divisible. In this way, each element x is broken down into its prime factors.

• Dictionary Mapping (first occurrence): The first dictionary keeps track of the first occurrence of each prime factor throughout the array. When a new prime factor is encountered, it's added to the dictionary with the current index as its value.

• List Tracking (last occurrence): Simultaneously, the list last is updated to track the last occurrence of a previously seen prime factor. If a prime factor that's already in the first dictionary is found again, the last array at the index of the first occurrence of this factor (obtained from the first dictionary) is updated to the current index.

• Finding the Valid Split: After populating the first and last data structures, the algorithm searches for the smallest index i at which the split can occur. To keep track of the split index, the algorithm uses the variable mx to store the maximum value found so far in last. It iterates through the last array while updating mx to the current maximum index. If at any index i, mx is found to be less than i, it means that the common prime factors (if any) of nums[0] to nums[i] and nums[i+1] to nums[n-1] are only found before index i. Hence, mx is the valid smallest split index, and it is returned.

• Returning the Result: The iteration continues until the end of the list. If no valid split is found, the function returns -1.

The algorithm is efficient because it avoids the need to calculate the actual products of the subarrays, which can be very large and could lead to integer overflow. Instead, it relies on the presence of common prime factors to determine whether a split is valid. The use of a dictionary and list to track the first and last occurrence of prime factors allows for fast lookups and updates, resulting in an efficient solution.

The time complexity for this algorithm can be approximated as O(n * sqrt(m)), where n is the length of the array, and m is the maximum value in the array, since for each element we might need to iterate up to its square root to find all prime factors.

Example Walkthrough

Let's consider a small example where nums = [6, 5, 10]:

1. Initialize first dictionary and last array:

first = {}
last = [-1, -1, -1]  // -1 indicates that we haven't seen the prime factor yet

2. Perform prime factorization on each number in nums and update first and last:

• For nums[0] = 6, its prime factors are 2 and 3.

  first = {2: 0, 3: 0}
  last = [0, 0, -1]  // We've seen 2 and 3 at index 0

• For nums[1] = 5, its only prime factor is 5.

  first = {2: 0, 3: 0, 5: 1}
  last = [0, 0, -1]  // No need to update as 5 is a new prime factor

• For nums[2] = 10, its prime factors are 2 and 5. Factor 2 has been encountered before, so we update its last occurrence. Factor 5 has also been encountered before, but since its first and last occurrence is at the same index, there is no need to update.

  first = {2: 0, 3: 0, 5: 1}
  last = [2, 0, -1]  // Update last occurrence of 2 to index 2 where it's seen again

3. Search for the smallest index i as a valid split:

• Iterate through last. We use mx to store the maximum index found so far.

  • At i = 0, last[0] = 2, so mx = 2.
  • At i = 1, last[1] = 0, but mx is still 2 (mx remains the max of last[0] and last[1]).

• We encounter i < mx at i = 1. This means that all prime factors seen in the first i+1 elements (up to index 1) and all following elements (from index 2) are unique.

Thus, the valid split index found is i = 1.

The algorithm concludes that the products of the first half 6 * 5 and the second half 10 do not have any common prime factors, so they are coprime. The smallest valid split index returned is 1.

Solution Implementation

Time and Space Complexity

The provided code block has two main parts to consider when analyzing the time complexity:

1. The first loop where we iterate over nums and factorize each number, updating first and last based on the factors.
2. The second loop where we iterate over last to find the valid split.

For the first part, in the worst case, the factorization of each number can take O(sqrt(x)) time where x is the value of the number being factorized. Since we are doing this for every number in nums, and with n being the size of nums, the total time complexity for this part is O(n * sqrt(max(nums))).

The second part is a linear scan over the array last, having a time complexity of O(n).

Therefore, the overall time complexity is O(n * sqrt(max(nums))) + O(n) which simplifies to O(n * sqrt(max(nums))) as sqrt(max(nums)) is the dominating factor.

The space complexity of the code is affected by the first and last data structures, which hold at most n elements, giving us a space complexity of O(n).

Learn more about how to find time and space complexity quickly using problem constraints.