Problem Description

In this problem, you need to work with a special array generation rule to find the maximum element in the generated array. An integer n is given which determines the size of an integer array nums of length n + 1. The array nums is constructed following specific rules:

• nums[0] is set to 0.
• nums[1] is set to 1.
• For each even index 2i where 2 <= 2i <= n, the value is nums[2i] = nums[i].
• For each odd index 2i + 1 where 2 <= 2i + 1 <= n, the value is nums[2i + 1] = nums[i] + nums[i + 1].

Your goal is to return the maximum integer value that exists in the array nums.

Intuition

Intuitively, the problem can be solved by directly applying the given generation rules to construct the array and then find the maximum value in it. Since the maximum number generated is a result of the addition in the sequence, one would expect it to appear at an odd index because all odd indices are created by summing two elements of the array.

We can observe that each element is derived from previously calculated values. This gives us a hint that we should approach the problem iteratively, constructing the array one number at a time and making use of previously computed elements. This iterative approach ensures that we use O(n) time complexity because each element calculation requires constant time.

Additionally, we can see that the array follows a repetitive pattern based on even and odd indices, which means we can use bitwise operations for efficiency. Notably, when working with an index i, i >> 1 is equivalent to i//2, which we need for calculating even indices. For odd indices, we use i >> 1 (the same as i//2) and (i >> 1) + 1 (the same as i//2 + 1).

Using this approach, we create and fill the array nums up to index n and simply return the maximum value in the array as our solution. Here space complexity is O(n) because we need to store n + 1 elements in the array nums.

Learn more about Dynamic Programming patterns.

Solution Approach

The implementation of the solution follows a straightforward approach based on the rules for generating the array. This solution uses the iterative method to populate the array based on the two conditions for even and odd indices. Let's break down the solution step by step:

1. Initialize the nums array with n + 1 elements, as the length of the array is determined by the input n. Initialize all elements to 0.

2. Since the first two elements nums[0] and nums[1] are already given by the problem (0 and 1, respectively), we manually set them. This serves as a base case for building up the rest of the array.

3. Iterate over the range from 2 to n, inclusive. We divide this process into two cases:

   • When i is even (i % 2 == 0), we set nums[i] to nums[i >> 1]. The bitwise right shift operation "i >> 1" effectively divides i by 2.
   • When i is odd, we set nums[i] to the sum of nums[i >> 1] and nums[(i >> 1) + 1], essentially summing the elements at indices i//2 and i//2 + 1.

4. After we've populated the nums array, we return the maximum integer from it using Python's built-in max() function.

In terms of data structures, we only use a list to store the sequence, and no additional data structures are needed. As the process involves simple arithmetic and assignment operations, the algorithm doesn't make use of complex patterns. It's an imperative, step-by-step generation of the sequence values using conditions for even and odd integers, followed by a quest for the maximum value.

The time complexity of this solution is O(n), as we iterate over the range once and the max function also traverses the list with complexity O(n). The space complexity is O(n) as well, due to the storage requirements for the nums array.

Here is the implementation of our solution:

class Solution:
    def getMaximumGenerated(self, n: int) -> int:
        if n < 2:
            return n
        nums = [0] * (n + 1)
        nums[1] = 1
        for i in range(2, n + 1):
            # Populate nums[i] based on even/odd index
            nums[i] = nums[i >> 1] if i % 2 == 0 else nums[i >> 1] + nums[(i >> 1) + 1]
        # Return the max value from the generated array
        return max(nums)

Example Walkthrough

Let's illustrate the solution approach with a small example. Suppose we have n = 7. We want to generate the nums array according to the rules and find the maximum element.

1. We initialize nums with n + 1 (8) elements: nums = [0, 0, 0, 0, 0, 0, 0, 0].

2. We set nums[0] to 0 and nums[1] to 1 as given: nums = [0, 1, 0, 0, 0, 0, 0, 0].

3. We start iterating from index 2 to 7:

   1. For i = 2 (even), we use the formula nums[2] = nums[2 >> 1], which is nums[2] = nums[1], so nums[2] = 1.
   2. For i = 3 (odd), nums[3] = nums[3 >> 1] + nums[(3 >> 1) + 1], which is nums[3] = nums[1] + nums[2], so nums[3] = 2.
   3. For i = 4 (even), nums[4] = nums[4 >> 1], which is nums[4] = nums[2], so nums[4] = 1.
   4. For i = 5 (odd), nums[5] = nums[5 >> 1] + nums[(5 >> 1) + 1], which is nums[5] = nums[2] + nums[3], so nums[5] = 3.
   5. For i = 6 (even), nums[6] = nums[6 >> 1], which is nums[6] = nums[3], so nums[6] = 2.
   6. For i = 7 (odd), nums[7] = nums[7 >> 1] + nums[(7 >> 1) + 1], which is nums[7] = nums[3] + nums[4], so nums[7] = 3.

4. Now our nums array looks like this: nums = [0, 1, 1, 2, 1, 3, 2, 3].

5. The final step is to return the maximum value in nums, which is 3.

This example follows the solution approach step by step to generate the nums array and then uses the built-in max() function to find the maximum element, which is the objective of our problem. The unit of work done for each iteration illustrates the O(n) time complexity of generating the array. The space complexity is also illustrated with the array holding n + 1 = 8 elements in this example.

Solution Implementation

Time and Space Complexity

The given Python code generates an array of integers according to specific generation rules and returns the maximum value in the array for a given n.

• Time Complexity: The time complexity of the code is O(n). This is because the for loop runs from 2 to n, and each operation inside the loop (calculating nums[i] and referencing previously computed values) takes constant time.

• Space Complexity: The space complexity of the code is also O(n). This is due to the allocation of a list nums of size n + 1, where each element is initialized and potentially modified as the for loop executes. No auxiliary space that depends on n is used besides this list.

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