Create Maximum Number in Python with Time Complexity Analysis

You are given two integer arrays nums1 and nums2 of lengths m and n respectively. nums1 and nums2 represent the digits of two numbers. You are also given an integer k.

Create the maximum number of length k <= m + n from digits of the two numbers. The relative order of the digits from the same array must be preserved.

Return an array of the k digits representing the answer.

Example 1:

Input: nums1 = [3,4,6,5], nums2 = [9,1,2,5,8,3], k = 5
Output: [9,8,6,5,3]

Example 2:

Input: nums1 = [6,7], nums2 = [6,0,4], k = 5
Output: [6,7,6,0,4]

Example 3:

Input: nums1 = [3,9], nums2 = [8,9], k = 3
Output: [9,8,9]

Constraints:

m == nums1.length
n == nums2.length
1 <= m, n <= 500
0 <= nums1[i], nums2[i] <= 9
1 <= k <= m + n

Understanding the Problem

The core challenge of this problem is to create the largest possible number of length k by selecting digits from two given arrays while maintaining the relative order of digits from each array. This problem is significant in scenarios where we need to merge and maximize values from different sources while preserving their order.

Potential Pitfalls and Misconceptions

Assuming that the largest digits should always come from one array.
Not preserving the relative order of digits from the same array.

Approach

To solve this problem, we need to consider the following steps:

Generate all possible subarrays of different lengths from both arrays.
Merge these subarrays to form the largest possible number of length k.
Compare and select the maximum number formed.

Naive Solution

A naive solution would involve generating all possible combinations of subarrays and merging them, which is computationally expensive and not feasible for larger inputs.

Optimized Solution

We can optimize the solution by breaking it down into two main parts:

Finding the maximum subarray of a given length from a single array.
Merging two subarrays to form the largest possible number while maintaining order.

Finding the Maximum Subarray

To find the maximum subarray of length t from an array, we can use a greedy approach with a stack to ensure the largest possible digits are selected while maintaining the order.

Merging Two Subarrays

To merge two subarrays, we can use a two-pointer technique to compare and select the larger digit from the two subarrays at each step.

Algorithm

Step-by-Step Breakdown

Define a function to find the maximum subarray of a given length using a stack.
Define a function to merge two subarrays into the largest possible number.
Iterate over all possible lengths for subarrays from both arrays that sum to k.
For each combination, find the maximum subarrays, merge them, and keep track of the largest number formed.

Code Implementation

def maxNumber(nums1, nums2, k):
    # Function to get the maximum subarray of length t from nums
    def maxSubarray(nums, t):
        stack = []
        drop = len(nums) - t
        for num in nums:
            while drop and stack and stack[-1] < num:
                stack.pop()
                drop -= 1
            stack.append(num)
        return stack[:t]

    # Function to merge two subarrays into the largest possible number
    def merge(sub1, sub2):
        return [max(sub1, sub2).pop(0) for _ in range(len(sub1) + len(sub2))]

    max_num = []
    for i in range(max(0, k - len(nums2)), min(k, len(nums1)) + 1):
        sub1 = maxSubarray(nums1, i)
        sub2 = maxSubarray(nums2, k - i)
        max_num = max(max_num, merge(sub1, sub2))
    return max_num

# Example usage
nums1 = [3, 4, 6, 5]
nums2 = [9, 1, 2, 5, 8, 3]
k = 5
print(maxNumber(nums1, nums2, k))  # Output: [9, 8, 6, 5, 3]

Complexity Analysis

The time complexity of the solution is O((m+n)^3) due to the nested loops and merging operations. The space complexity is O(m+n) for storing subarrays and the result.

Edge Cases

Arrays with all identical digits.
Arrays with digits in descending order.
Arrays with digits in ascending order.

Testing

To test the solution comprehensively, consider the following test cases:

Simple cases with small arrays.
Cases with maximum possible lengths for arrays and k.
Edge cases with identical digits and varying orders.

Thinking and Problem-Solving Tips

Break down the problem into smaller, manageable parts.
Use greedy algorithms for selecting maximum values while maintaining order.
Practice merging techniques and two-pointer approaches.

Conclusion

Understanding and solving this problem involves breaking it down into subproblems and using efficient algorithms to handle each part. Practice and familiarity with greedy algorithms and merging techniques are crucial for solving such problems effectively.