CSES Solutions - Polygon Lattice Points

Given a polygon, your task is to calculate the number of lattice points inside the polygon and on its boundary. A lattice point is a point whose coordinates are integers.

The polygon consists of n vertices (x₁,y₁),(x₂,y₂),....,(x_n,y_n). The vertices (x_i,y_i) and (x_i+1,y_i+1) are adjacent for i=1,2,....,n-1, and the vertices (x₁,y₁) and (x_n,y_n) are also adjacent.

Example:

Input: n = 4, vertices = {{1, 1}, {5, 3}, {3, 5}, {1, 4}}
Output: 6 8

Input: n = 4, vertices = {{2, 4}, {3, 2}, {1, 3}, {2, 4}}
Output: 1 3

Approach:

The idea is to use calculate the area of the polygon using the shoelace formula and Pick's theorem, which states that the number of lattice points inside a polygon is given by the following formula:

interior = (area + 2 - boundary) / 2

where:

• interior is the number of lattice points inside the polygon
• area is the area of the polygon
• boundary is the number of lattice points on the boundary of the polygon

Steps-by-step approach:

• Calculates the area of the polygon using the shoelace formula.
• Calculates the number of boundary points:
  • Iterates over each point in the polygon. The variable i is the index of the current point, and n is the total number of points.
  • Inside the loop, the code checks the relationship between the current point (x[i], y[i]) and the next point (x[i+1], y[i+1])
  • If x[i + 1] == x[i], it means that the two points are vertically aligned (i.e., they form a vertical edge). The number of boundary points on this edge is the absolute difference in their y coordinates, abs(y[i + 1] - y[i])
  • If y[i + 1] == y[i], it means that the two points are horizontally aligned (i.e., they form a horizontal edge). The number of boundary points on this edge is the absolute difference in their x coordinates, abs(x[i + 1] - x[i])
  • If neither of the above conditions is true, it means that the two points form a diagonal edge. The number of boundary points on this edge is the greatest common divisor (gcd) of the absolute differences in their x and y coordinates, __gcd(abs(x[i + 1] - x[i]), abs(y[i + 1] - y[i])).
  • The number of boundary points calculated for each edge is added to the boundary variable.
• Uses Pick's theorem to calculate the number of interior points.
• Prints the number of interior and boundary points.

Note: why __gcd(abs(x[i + 1] - x[i]), abs(y[i + 1] - y[i])) function is used to calculate the number of boundary points on a diagonal edge of the polygon !!

• When two points form a diagonal edge, the number of boundary points on this edge is the number of grid points that the line segment between these two points passes through.
• To calculate this, we use the greatest common divisor (gcd) of the absolute differences in their x and y coordinates. The gcd of two numbers is the largest number that divides both of them without leaving a remainder. In this context, it represents the number of grid points that the diagonal line segment intersects.
• For example, consider two points (3, 3) and (6, 6). The line segment between these points is a diagonal line that passes through (4, 4) and (5, 5). So, there are 3 boundary points on this edge: (3, 3), (4, 4), and (5, 5). The absolute differences in the x and y coordinates are 3 and 3, respectively, and their gcd is 3, which is the correct number of boundary points.
• So, the __gcd function is used to correctly calculate the number of boundary points on a diagonal edge. This ensures that the total number of boundary points on the polygon is accurate.

Below are the implementation of the above approach:

6 8

Time complexity: O(nlog(n)), where n is the number of vertices in the polygon, log(n) due to gcd function.
Auxiliary Space: O(n)