CSES Solution - Line Segment Intersection

There are two line segments: the first goes through the points (x_1,y₁) and (x₂,y₂), and the second goes through the points (x₃,y₃) and (x₄,y₄).

Your task is to determine if the line segments intersect, i.e., they have at least one common point.

Example:

Input: points = {1, 1}, {5, 3}, {1, 2}, {4, 3}
Output: No

Input: points = {1, 1}, {5, 3}, {1, 1}, {4, 3}
Output: No

Approach:

• The main idea is to use the concept of cross products to determine if the line segments intersect.
• Calculate the cross products of the vectors AB and AC, and CD and AB.
• Check if the cross products have different signs, If the signs of the cross products are different, it means that the line segments intersect.
• Check if any of the cross products is zero, then check if the point C or D lies on the line segment AB or point A or B lies on the line segment CD. If so, the line segments intersect.
• Return "YES" if the line segments intersect, and "NO" otherwise.

Steps-by-step approach:

• Calculate the cross products of AB and CD, AB and DA, CD and AB, and CD and BA.
• Check if any of the cross products are zero and if the corresponding points are midpoints.
  • Return true
• Check if the signs of the cross products are different for AB and CD, and for CD and AB.
  • Return true
• Otherwise, Return false

Below are the implementation of the above approach:

NO

Time complexity: O(n log n).
Auxiliary Space: O(1)