Problem Description

The given problem presents us with an array of integers, nums. Our task is to find the length of the longest subarray where the product of all the elements is positive. A subarray is defined as a contiguous part of the original array and can contain zero or more elements. The output should be the length of the longest such subarray.

To solve this problem, we must consider that the product of elements in a subarray is positive if there are an even number of negative numbers (including zero) because the product of two negatives is positive. We must also remember that the presence of a zero resets the product because any number multiplied by zero is zero.

Intuition

The intuition behind the solution is based on dynamic programming. We can keep track of the maximum subarray length with a positive product in two situations:

• The current number is positive
• The current number is negative

We do this through two variables, f1 and f2, where f1 represents the length of the subarray ending at the current position with a positive product, and f2 (if non-zero) represents the length of a subarray ending at the current position with a negative product.

Here's the approach to arrive at the solution:

1. Initialize f1 and f2. If the first number is positive, f1 starts at 1, and if it's negative, f2 starts at 1. This sets up our initial condition.
2. Iterate through the array, starting from the second element, and update f1 and f2 based on the current number:

• If the current number is positive:
  • Increment f1 because a positive number won't change the sign of the product.
  • If f2 is non-zero, increment it as well, as a positive number won't affect the sign of an already negative product.
• If the current number is negative:
  • Assign f1 the value of f2 + 1, because a negative number changes the sign of the product.
  • Assign f2 the value of f1 (before updating f1) + 1, again because a negative number changes the sign.
• If the current number is zero, reset both f1 and f2 to zero because the product would be zero regardless of the preceding values.

3. At each step, compare f1 with our current maximum length (res) and update res if f1 is larger.
4. Once we've iterated through the array, return res as it contains the maximum length of a subarray with a positive product.

Thus, using this approach, we are able to avoid calculating the product of subarray elements and can find the maximum length of a subarray with a positive product efficiently.

Learn more about Greedy and Dynamic Programming patterns.

Solution Approach

The solution uses a dynamic programming approach that revolves around two main concepts: tracking the longest subarray length where the product is positive (f1), and tracking the longest subarray length where the product is negative (f2). To implement this, we use a simple linear scan of the input array with constant space complexity, updating both f1 and f2 based on the sign and value of the current element.

Here is a step-by-step explanation of the algorithm:

1. Initialize f1 and f2:

   • f1 is set to 1 if the first element of nums is positive, indicating the subarray starts with a positive product.
   • f2 is set to 1 if the first element of nums is negative, indicating the subarray starts with a negative product.
   • res is initialized to f1 to keep track of the maximum length.

2. Iterate over the array starting from the second element:

   • For each element, there are three cases to consider: a. The current element is positive:
     • Increment f1 as the positive number will not change the sign of the product.
     • Increment f2 only if it is positive, meaning there was a previous negative number and this positive number extends the subarray with a now positive product. b. The current element is negative:
     • Swap f1 and f2 and increment f2, as the negative number makes a previously positive product negative.
     • Only increment f1 if f2 was originally positive, as this indicates we had a negative product and now have made it positive. c. The current element is zero:
     • Reset f1 and f2 to 0 as any subarray containing a zero has a product of zero, which is neither positive nor negative.

3. After considering the current number, update the maximum length res with the current value of f1 if f1 is greater than the current res.

4. Continue the loop until the end of the array.

5. Return the final value of res, which contains the maximum length of the subarray with a positive product.

The code uses constant extra space (only needing f1, f2, and res), and it runs in O(n) time, where n is the length of the input array nums. This is an efficient solution because it avoids the need to calculate products directly and uses the properties of positive and negative numbers to infer the sign of the subarray products.

Example Walkthrough

Let's walk through an example to illustrate the solution approach using the following array of integers:

nums = [1, -2, -3, 4]

Our goal is to find the length of the longest subarray where the product of all elements is positive.

1. Initialize f1, f2, and res:

   • The first element is positive, so f1 = 1.
   • There are no negative elements yet, so f2 = 0.
   • Initialize res with the value of f1, hence res = 1.

2. Iterate over the array:

   a. Second element, -2 (current element is negative):

   • Since f1 = 1 and f2 = 0, we update f2 to f1 + 1, resulting in f2 = 2.
   • There is no previously recorded negative subarray (f2 was 0), so f1 remains the same.
   • No update to res as f1 has not increased.

   b. Third element, -3 (current element is negative):

   • Swap f1 and f2. Before the swap, f1 = 1 and f2 = 2. After the swap, f1 is now 2.
   • Increment f2 to become f1(pre-swap) + 1, hence f2 = 2.
   • Update res with the current value of f1 which is now 2, as it is greater than the previous res.

   c. Fourth element, 4 (current element is positive):

   • Increment f1 to f1 + 1, so f1 becomes 3.
   • Increment f2 as well since it's non-zero, so f2 becomes 3.
   • Update res with the current value of f1 which is 3 as it is greater than the previous res.

3. After iterating through the array, the maximum length res is 3. We have found the longest subarray [1, -2, -3, 4], which indeed has a positive product and its length is 3.

Therefore, the length of the longest subarray with a positive product is 3.

Solution Implementation

Time and Space Complexity

The provided code maintains two variables f1 and f2 to keep track of the length of the longest subarray with a positive product and the length of the longest subarray with a negative product, respectively.

Time Complexity:

The function iterates through the list nums once. For each element in nums, it performs a constant number of computations: updating f1, f2, and res based on the sign of the current element. Therefore, the time complexity is O(n), where n is the number of elements in nums.

Space Complexity:

The space complexity is O(1) because the code uses a fixed amount of space: variables f1, f2, res, pf1, and pf2. The space used does not grow with the size of the input array.

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