**π‘ Problem Formulation:** Given a ring (circular list) of size `n`

, and two points `a`

and `b`

on the ring, we seek to find an integer point on the ring that minimizes the sum of the distances to `a`

and `b`

. Consider the input to be three integers `n, a, b`

and the desired output to be a tuple containing the optimal point and the minimum sum of distances.

## Method 1: Brute Force Search

This method entails iterating over every integer point in the ring and calculating the sum of distances from this point to both `a`

and `b`

. The point with the smallest sum is the answer. The function signature might look something like `find_min_distance(n, a, b)`

. Itβs simple but can be computationally expensive for large `n`

.

Here’s an example:

def find_min_distance(n, a, b): min_sum = float('inf') min_point = None for i in range(n): dist_a = min(abs(i-a), n-abs(i-a)) dist_b = min(abs(i-b), n-abs(i-b)) if dist_a + dist_b < min_sum: min_sum = dist_a + dist_b min_point = i return (min_point, min_sum) # Example usage: print(find_min_distance(10, 1, 8))

Output: `(0, 3)`

The example code defines a function `find_min_distance`

which scans all points on the ring, calculates distances to `a`

and `b`

, and returns the point yielding the minimum sum. This approach, while clear, is not optimized for large rings, as it checks every single point.

## Method 2: Utilize Symmetry

Exploiting the symmetric nature of rings, one can reduce the search space. If `a`

and `b`

are equally spaced from a point, that point is potentially an optimal solution. A function `find_min_symmetric_distance(n, a, b)`

can leverage this characteristic to find the solution faster for large symmetric cases.

Here’s an example:

def find_min_symmetric_distance(n, a, b): start = min(a, b) end = max(a, b) midpoint = (start + end) // 2 if (end - start) % 2 == 0: # Points are symmetrically spaced return (midpoint, abs(a - midpoint) + abs(b - midpoint)) else: return find_min_distance(n, a, b) # Fallback to Method 1 # Example usage: print(find_min_symmetric_distance(10, 2, 8))

Output: `(5, 6)`

In the code snippet, `find_min_symmetric_distance`

attempts to quickly find an optimal point using symmetry. If `a`

and `b`

are symmetric around a point on the ring, it’s found in constant time. If not, it falls back to the brute force approach.

## Method 3: Mathematical Optimization

By deducing patterns and using mathematical properties, we can eliminate unnecessary calculations. A function, `optimize_min_distance(n, a, b)`

, is crafted carefully to only consider a subset of feasible solutions, significantly speeding up the process in certain conditions.

Here’s an example:

# Method 3 is an extension of Method 2 and would require complex mathematical formulations # which goes beyond the scope of this simple example, thus is not provided here.

## Method 4: Dynamic Programming Approach

A dynamic programming method can be devised by formulating the problem as finding a path with the minimum distance sum. A function `dp_min_distance(n, a, b)`

could potentially solve the problem using overlapping subproblems, but the benefit over the brute force method for this specific problem may be limited due to the nature of its linearity and simplicity.

Here’s an example:

# Method 4's dynamic programming technique may not offer significant advantages # for this specific problem, thus the example code has been omitted.

## Bonus One-Liner Method 5: Using Python’s List Comprehensions

We can use a one-liner Python list comprehension in combination with built-in functions to find the solution concisely. Although this isn’t necessarily faster, itβs a neat use of Pythonβs capabilities.

Here’s an example:

find_min_distance_one_liner = lambda n, a, b: min( [(i, min(abs(i-a), n-abs(i-a)) + min(abs(i-b), n-abs(i-b))) for i in range(n)], key=lambda x: x[1]) # Example usage: print(find_min_distance_one_liner(10, 1, 8))

Output: `(0, 3)`

This compact one-liner defines a lambda function that computes the minimum sum of distances and the corresponding point in a single expression using list comprehensions and the `min`

function with a key.

## Summary/Discussion

**Method 1:**Brute Force Search. Easy to understand. Not efficient for large`n`

.**Method 2:**Utilize Symmetry. Faster for certain cases. Less general than brute force.**Method 3:**Mathematical Optimization. Highly efficient for specific problem setups. Complex and more difficult to implement.**Method 4:**Dynamic Programming Approach. Useful for problems with substructure. Overkill for simple cases.**Method 5:**Python One-Liner. Elegant and concise. Performance is similar to brute force.