Problem Description

In this problem, you have two integer arrays nums1 and nums2, both containing n elements, indexed from 0. Your goal is to create a third array nums3, also with n elements, where each element at index i is either the element from nums1[i] or nums2[i]. You want to fill nums3 such that it contains the longest non-decreasing subarray possible; basically, you want nums3 to have the longest sequence where elements from left to right are not in descending order. The desired output is the length of this longest non-decreasing subarray in nums3.

A non-decreasing subarray is a contiguous sequence of elements that go from left to right without decreasing in value. In other words, every element is greater than or equal to the element that comes before it.

Intuition

The intuition behind the solution involves dynamic programming, a method that solves problems by combining solutions to subproblems and using these solutions to construct an answer for the larger problem.

Since for each index i, you can choose either nums1[i] or nums2[i] for the nums3 array, there can be many potential combinations to consider. However, we only need to keep track of the longest subarray lengths up to the current position for both potential choices. Here we define two states: f to represent the length of the longest non-decreasing subarray that ends with an element from nums1, and g to represent the length of the longest non-decreasing subarray that ends with an element from nums2.

To arrive at the solution, we start by initializing f and g to 1, as the longest subarray at the start can only be of length 1. As we iterate through each index starting from 1, we determine the possible lengths of f and g by checking if the current elements in nums1 and nums2 can extend the non-decreasing subarrays ending at the previous index.

For each element at index i, if nums1[i] is greater than or equal to the previous elements (nums1[i-1] or nums2[i-1]), we can extend the non-decreasing subarray ending with nums1[i-1] or nums2[i-1]. The same applies for nums2[i]. At each step, we update f and g to the longer of the current value of f or g and the length of the new non-decreasing subarray, which can be an extension of f or g from the previous step. Finally, the answer is the maximum value of f and g after traversing all elements.

This dynamic approach ensures that we do not need to evaluate all possible combinations of elements from nums1 and nums2; instead, we can efficiently calculate the length of the longest non-decreasing subarray by considering each position's optimal choice based on the previous computations.

Learn more about Dynamic Programming patterns.

Solution Approach

The implementation of the solution makes use of a dynamic programming strategy. It involves tracking the potential longest non-decreasing subarray lengths dynamically as we iterate through the arrays. Here's a breakdown of how the algorithm works and the concepts it uses:

1. Initialization of States: We start with two variables, f and g, both set to 1 because, at the very beginning, the longest non-decreasing subarray that can be formed will just include the first element from either nums1 or nums2. The variable ans is also set to 1, to track the length of the longest subarray found so far.

2. Iteration through Arrays: We loop through the arrays starting from index 1, as the 0th index is already considered in our initial state.

3. State Transition: For each index i, we compute two temporary variables ff and gg which represent the potential new values of f and g based on whether the current elements at index i can extend the longest non-decreasing subarray up to index i - 1. This involves four scenarios for updating ff and gg:

   • If nums1[i] is not smaller than nums1[i - 1], the subarray ending with nums1[i - 1] can be extended, so we update ff to max(ff, f + 1).
   • If nums1[i] is not smaller than nums2[i - 1], the subarray ending with nums2[i - 1] can be extended, so we update ff to max(ff, g + 1).
   • If nums2[i] is not smaller than nums1[i - 1], the subarray ending with nums1[i - 1] can be extended, so we update gg to max(gg, f + 1).
   • If nums2[i] is not smaller than nums2[i - 1], the subarray ending with nums2[i - 1] can be extended, so we update gg to max(gg, g + 1).

4. State Update: After calculating ff and gg, we set f and g to these new values, as they represent the updated longest non-decreasing subarray lengths ending at index i for nums1 and nums2 respectively.

5. Answer Calculation: With each iteration, the ans variable is updated to reflect the maximum value among ans, f, and g. This ensures that after traversing the entire array, ans holds the length of the longest non-decreasing subarray that can be formed.

The code uses a simple for loop for iteration and max function calls for comparison to execute this algorithm effectively. By only maintaining the necessary states, we optimize space complexity while ensuring that the program runs in linear time relative to the size of the given arrays.

Example Walkthrough

Let's walk through a small example to illustrate the solution approach.

Assume our input arrays are nums1 = [3, 4, 2, 6] and nums2 = [1, 2, 3, 4], and we wish to construct nums3 with the longest non-decreasing subarray.

Initially, we set f = 1, g = 1, and ans = 1, since the longest non-decreasing subarray at the beginning can only include one of the first elements.

Now we proceed to iterate through each index, starting from 1.

For i = 1:

• We compute temporary variables ff and gg; at the start of every iteration, they are set to zero.
• Since nums1[1] (4) is not smaller than nums1[i - 1] (3), we can extend the subarray ending with nums1[i - 1]. Thus, ff = max(ff, f + 1) => ff = max(0, 1 + 1) => ff = 2.
• Since nums1[1] (4) is also not smaller than nums2[i - 1] (1), we can extend the subarray ending with nums2[i - 1]. Thus, ff = max(ff, g + 1) => ff = max(2, 1 + 1) => ff = 2.
• nums2[1] (2) is not smaller than nums1[i - 1] (3), so we cannot extend from nums1[i - 1] this time; gg remains the same.
• Since nums2[1] (2) is not smaller than nums2[i - 1] (1), we can extend the subarray ending with nums2[i - 1]. Thus, gg = max(gg, g + 1) => gg = max(0, 1 + 1) => gg = 2.
• We now perform the state update, f = ff and g = gg, setting both f and g to 2.

Now, ans is updated to be the max of ans, f, and g, which is max(1, 2, 2), so ans becomes 2.

Repeating this process for the rest of the indices:

For i = 2:

• nums1[2] (2) is smaller than nums1[i - 1] (4), so ff does not update based on f.
• But nums1[2] (2) is not smaller than nums2[i - 1] (2), so ff = max(ff, g + 1) => ff = max(0, 2 + 1) => ff = 3.
• nums2[2] (3) is not smaller than nums1[i - 1] (4), so gg does not update based on f.
• nums2[2] (3) is larger than nums2[i - 1] (2), so gg = max(gg, g + 1) => gg = max(0, 2 + 1) => gg = 3.
• Reset f and g to their new values ff and gg, respectively, both are now 3.
• ans = max(ans, f, g) => ans = max(2, 3, 3) => ans = 3.

For i = 3:

• nums1[3] (6) is larger than both nums1[i - 1] (2) and nums2[i - 1] (3), so ff becomes max(0, f + 1) and max(0, g + 1), resulting in ff = 4.
• nums2[3] (4) is larger than nums1[i - 1] (2) but not nums2[i - 1] (3), so gg only updates from f, giving gg = max(0, f + 1) => gg = 4.
• State update puts f = 4 and g = 4.
• ans updates to max(ans, f, g) => ans = max(3, 4, 4) => ans = 4.

In the end, the longest non-decreasing subarray that can be created (nums3) is of length 4, which is our answer. Thus, nums3 can be [3, 4, 3, 4] or [3, 4, 2, 4], both of which have the longest non-decreasing subarray of length 4.

Solution Implementation

Time and Space Complexity

The time complexity of this code is O(n), where n is the length of the input arrays nums1 and nums2. The code consists of a single loop that iterates from 1 to n-1, performing a constant amount of work for each element with no nested loops, resulting in a linear time complexity.

The space complexity of the code is O(1), as it uses a fixed number of integer variables (f, g, ff, gg, ans, and n). No additional space that grows with the input size is utilized, resulting in constant space complexity.

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