Problem Description

In this problem, we are provided m separate arrays, each being sorted in ascending order. Our task is to find the maximum distance between any two integers where each integer is taken from a different array. The distance is defined as the absolute difference between those two integers. That is, if we take one integer a from one array and another integer b from a different array, the distance is |a - b|. We need to compute and return the maximum such distance possible.

Intuition

To solve this problem efficiently, we leverage the fact that each of the m arrays is sorted in ascending order. Given this property, the smallest element of each array will be at the 0-th index and the largest element at the last index.

To find the maximum distance, we need to make the difference as large as possible. This means we should consider the potential largest distance by taking the smallest element from one array and the largest from another array. So to maximize the distance, we always consider the minimum element from one array and the maximum element from the other.

The approach is to iterate through the arrays while tracking the smallest and largest elements we have seen so far. For each new array, we consider the distance between the current array's smallest element and the maximum element seen so far, and vice versa - the distance between the current array's largest element and the smallest element seen so far. The answer is the largest of all these distances.

Here is the step-by-step reasoning:

1. Initialize ans to 0, which will keep track of the maximum distance.
2. Take the smallest (mi) and largest (mx) elements from the first array to initiate tracking.
3. Iterate through the remaining arrays starting from the second one. For each array:
   • Calculate the absolute difference between the current array's smallest element and mx (the largest element seen so far).
   • Calculate the absolute difference between the current array's largest element and mi (the smallest element seen so far).
   • Update ans with the largest value between ans, the first calculated difference, and the second calculated difference.
   • Update mi with the smallest value between mi (the smallest value found so far) and the smallest element of the current array.
   • Update mx with the largest value between mx (the largest value found so far) and the largest element of the current array.
4. After finishing the iteration, return ans as the maximum distance found.

By only comparing the extreme ends of each array, we ensure that we are always considering the likely pairs to give us the maximum distance while maintaining a linear time complexity.

Learn more about Greedy patterns.

Solution Approach

The solution follows a greedy strategy by keeping track of the smallest and the largest element found so far across all arrays. Here is how the implementation reflects the approach:

• An initial answer ans is set to 0, which will hold the maximum distance.
• The smallest element mi and the largest element mx from the first array are used to initialize the tracking.
• Then, for every subsequent array, the implementation performs the following steps:
  1. Calculates two distances: the distance between the smallest element of the current array and the largest element mx seen so far (a = abs(arr[0] - mx)), and the distance between the largest element of the current array and the smallest element mi seen so far (b = abs(arr[-1] - mi)).
  2. The answer ans is updated to be the maximum of ans, a, and b.
  3. The current array's smallest element is considered to update mi using mi = min(mi, arr[0]) if it is smaller than the current mi.
  4. Similarly, the current array's largest element is used to update mx using mx = max(mx, arr[-1]) if it is larger than the current mx.

By using min() and max() functions, the elements mi and mx are continuously updated to always represent the smallest and largest element found up to the current point in iteration.

This approach guarantees each array is only visited once, making the overall time complexity O(m), where m is the number of arrays. There is no use of additional data structures, keeping space complexity to O(1) as it only uses a few variables for tracking the minimum, maximum, and the current answer.

The code directly follows this strategy and does not use any complex data structures or algorithms beyond basic conditional statements and math operations.

Here's a snippet of the main logic using the algorithm described:

class Solution:
    def maxDistance(self, arrays: List[List[int]]) -> int:
        ans = 0
        mi, mx = arrays[0][0], arrays[0][-1]
        for arr in arrays[1:]:
            a, b = abs(arr[0] - mx), abs(arr[-1] - mi)
            ans = max(ans, a, b)
            mi = min(mi, arr[0])
            mx = max(mx, arr[-1])
        return ans

Example Walkthrough

Let's say we have the following 3 sorted arrays:

• Array 1: [1, 2, 3]
• Array 2: [4, 5]
• Array 3: [1, 5, 9]

Our goal is to find the maximum distance between any two integers from two different arrays.

1. Initialize ans to 0, which will be used to keep track of the maximum distance found so far.
2. We start with the first array and take its smallest and largest elements as the initial values for mi and mx respectively. In this case, mi is 1 and mx is 3.

Now, we iterate through the remaining arrays to calculate potential distances and update mi and mx.

3. Moving to the second array [4, 5], we calculate the distance between the smallest element in this array 4 and the current mx 3 which is abs(4 - 3) = 1. We also calculate the distance between the largest element in this array 5 and the current mi 1 which is abs(5 - 1) = 4. We update ans to the maximum of ans, which is 0, and these new distances. ans is now 4. We also update mi and mx if necessary. Since 4 is greater than mi and 5 is greater than mx, we only update mx to 5.

4. Next, we go to the third array [1, 5, 9]. We calculate the distance between 1 (the smallest element of this array) and the current largest mx, which is 5. The distance is abs(1 - 5) = 4. Then, we calculate the distance between 9 (the largest element of this array) and the current smallest mi, which is 1. The distance here is abs(9 - 1) = 8. We update ans to the maximum of ans and these new distances. ans is now updated to 8, which is the maximum distance found. We check if we need to update mi or mx. Since 1 is equal to current mi and 9 is greater than current mx, we update mx to 9.

Finally, after iterating through all arrays, we have the largest distance ans as 8, which is the correct maximum distance between two integers from two different arrays. Thus, we return 8.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the given code is primarily determined by the loop that iterates over all elements of the arrays list, except the first element. In the worst case, where there are n arrays inside the arrays list, we iterate n - 1 times. Inside the loop, we execute a constant number of operations for each array: calculating the absolute difference between the first and last elements with mi and mx, updating ans, and updating mi and mx with the current array's first and last elements, respectively. Since all these operations inside the loop have a constant time complexity, the overall time complexity of the loop is O(n).

Hence, the total time complexity of the code is O(n).

Space Complexity

The space complexity is determined by the additional memory used by the program which is not part of the input. In the given code, we use a fixed number of variables (ans, mi, mx, a, b) that do not depend on the size of the input. No other dynamic data structures or recursive calls which could use additional space are involved. Therefore, the space complexity is O(1), which represents constant space usage.

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