Optimizing the Minimum Value of a - b, b - c, c - a Across Three Arrays: An Efficient Algorithm

To find the minimum value of the expression a - b, b - c, c - a for a ∈ A, b ∈ B, c ∈ C, we can simplify the expression first. This optimization problem can be approached using a combination of sorting and a three-pointer technique, significantly improving the efficiency of the algorithm.

Simplification

The expression a - b, b - c, c - a can be rewritten using mathematical simplifications:

a - b, b - c, c - a 2 ? max(a, b, c) - 2 ? min(a, b, c)

This simplification reveals that our goal is to minimize the difference between the maximum and minimum of the selected values from arrays A, B, and C.

Approach

To efficiently solve this problem, follow these steps:

Sort the Arrays: Begin by sorting the arrays A, B, and C. Sorting helps in utilizing a three-pointer technique to identify potential minimum values efficiently. Use a Three-Pointer Technique: Utilize three pointers, one for each sorted array. Start all pointers at the beginning of their respective arrays. Iterate and Update Pointers: At each step, calculate the current values of a, b, and c pointed to by the three pointers. Compute the current minimum value of the expression:

current_value  abs(a - b), abs(b - c), abs(c - a)

Move the Pointer: To minimize the expression, move the pointer that points to the smallest value among a, b, and c. This is because increasing the smallest value may help reduce the overall range. Repeat: Continue this process until one of the pointers reaches the end of its respective array.

Algorithm

Here’s the step-by-step outline of the algorithm:

def min_expression(A, B, C):
    # Sort the arrays
    ()
    ()
    ()
    # Initialize pointers and the minimum value
    i, j, k  0, 0, 0
    min_value  float('inf')
    while i 

Time Complexity

The time complexity of this algorithm can be broken down as follows:

Sorting the Arrays: Sorting each array takes O(n log n) where n is the length of the longest array. Three-Pointer Traversal: The three-pointer traversal can take up to O(n) in the worst case where n is the total number of elements across the arrays.

The overall time complexity is O(n log n), making it a highly efficient solution.

Conclusion

This algorithm efficiently finds the minimum value of the given expression by leveraging sorting and a three-pointer technique, yielding a complexity of O(n log n). By using this approach, you can optimize the solution significantly, ensuring a high performance and lower computational costs.