356. Line Reflection - Detailed Explanation

Problem Statement

Description:
You are given an array of points where each point is represented by its coordinates ([x, y]). Your task is to determine if there exists a vertical line (parallel to the y-axis) such that the set of points is symmetric about this line.

What does it mean to be symmetric?
A set of points is symmetric with respect to a vertical line if for every point ((x, y)) in the set, there exists another point ((x', y)) in the set such that both points are mirror images of each other with respect to that line. Mathematically, if the line of reflection has the x-coordinate (c), then for every point ((x, y)) there must exist a point ((2c - x, y)) in the set.

Examples:

1. Example 1:

   • Input: points = [[1, 1], [-1, 1]]
   • Output: true
   • Explanation:
     The line of reflection is (x = 0). The point ([1, 1]) reflects to ([-1, 1]) and vice versa.

2. Example 2:

   • Input: points = [[1, 1], [2, 2]]
   • Output: false
   • Explanation:
     There is no vertical line such that every point has a mirror image. For instance, if you try to set a reflection line, the point ([1, 1]) would require a corresponding ([c \times 2 - 1, 1]) point which is not present.

Hints

• Vertical Line of Reflection:
  Since the reflection line is vertical, only the x-coordinates matter for symmetry. The y-coordinates of corresponding points must be the same.

• Candidate Reflection Line:
  A key insight is that if a reflection line exists, its x-coordinate (c) can be derived from the minimum and maximum x-values among the points. Specifically, (c = \frac{\min(x) + \max(x)}{2}).

• Check for Reflected Points:
  For each point ((x, y)), check if there is a corresponding point ((2c - x, y)) in the set. To do this efficiently, consider using a hash set (or dictionary) for constant-time lookups.

• Avoid Floating-Point Pitfalls:
  Instead of directly computing (c) as a float, you can compute the sum (s = \min(x) + \max(x)) and then for each point ((x, y)), check if ((s - x, y)) exists. This trick avoids precision issues since you’re working with integers.

Approaches

Approach 1: Hash Set with Computed Reflection Line

1. Determine the Reflection Line:

   • Find the minimum and maximum x-coordinates among all points.
   • Calculate (s = \text{min}_x + \text{max}_x). The candidate vertical line is at (c = \frac{s}{2}).

2. Use a Hash Set for Fast Lookups:

   • Insert every point (e.g., as a tuple ((x, y))) into a hash set.

3. Check Each Point:

   • For every point ((x, y)) in the array, compute its reflection as ((s - x, y)).
   • If the reflected point is not found in the set, return false.

4. Return Result:

   • If every point’s reflection exists in the set, return true.

Approach 2: Sorting-Based Verification (Alternative)

1. Sort the Points:
   • Sort the points primarily by their y-coordinate and secondarily by their x-coordinate.
2. Group by y-coordinate:
   • For each group (points sharing the same y-value), check if the points are symmetric about the candidate line.
3. Verification:
   • For every group, use two pointers (one at the beginning and one at the end of the sorted group) to check for symmetry.

Note: The hash set approach is typically simpler and runs in linear time relative to the number of points.