Explain the concept of optimal substructure in the context of the Maximum Product Subarray problem.

In the context of the Maximum Product Subarray problem, the concept of optimal substructure refers to the property that an optimal solution to the problem can be constructed from optimal solutions to its subproblems.

The Maximum Product Subarray problem involves finding the contiguous subarray within an array of integers that has the largest product. To solve this problem using dynamic programming, we can define a subproblem as finding the maximum product subarray ending at a particular index.

To understand the optimal substructure, let's consider an example. Suppose we have an array [2, -3, 4, -2, 1]. We can calculate the maximum product subarray ending at each index as follows:

- For index 0, the maximum product subarray is [2], which has a product of 2.
- For index 1, the maximum product subarray is [-3], which has a product of -3.
- For index 2, the maximum product subarray is [4], which has a product of 4.
- For index 3, the maximum product subarray is [4, -2], which has a product of -8.
- For index 4, the maximum product subarray is [4, -2, 1], which has a product of -8.

From these subproblems, we can observe that the maximum product subarray ending at index i depends on the maximum product subarray ending at index i-1. In other words, the optimal solution for the subproblem at index i can be obtained by considering the maximum product subarray ending at index i-1 and extending it with the element at index i.

Using this observation, we can define a recurrence relation for the maximum product subarray problem:

max_product(i) = max(nums[i], max_product(i-1) * nums[i])

Here, max_product(i) represents the maximum product subarray ending at index i, and nums[i] represents the element at index i.

By solving these subproblems in a bottom-up manner, starting from the base case of max_product(0) = nums[0], we can find the maximum product subarray for the entire array.

In summary, the concept of optimal substructure in the context of the Maximum Product Subarray problem means that the optimal solution for the problem can be constructed by considering the optimal solutions for its subproblems. This allows us to solve the problem efficiently using dynamic programming.