Count of Unique Direct Path Between N Points On a Plane
Given N points on a plane, where each point has a direct path connecting it to a different point, the task is to count the total number of unique direct paths between the points.
Note: The value of N will always be greater than 2.
Input: N = 4
Explanation: Think of 4 points as a 4 sided polygon. There will 4 direct paths (sides of the polygon) as well as 2 diagonals (diagonals of the polygon). Hence the answer will be 6 direct paths.
Input: N = 3
Explanation: Think of 3 points as a 3 sided polygon. There will 3 direct paths (sides of the polygon) as well as 0 diagonals (diagonals of the polygon). Hence the answer will be 3 direct paths.
Approach: The given problem can be solved using an observation that for any N-sided there are (number of sides + number of diagonals) direct paths. For any N-sided polygon, there are N sides and N*(N – 3)/2 diagonals. Therefore, the total number of direct paths is given by N + (N * (N – 3))/2.
Below is the implementation of the above approach:
Time Complexity: O(1)
Auxiliary Space: O(1)