Check if it is possible to move from X to Z using point Y

Given three points X, Y, and Z of a 2D plane, the task is to check if it is possible to move from X to Z using point Y with at most one turn of 90 degrees. If possible return "YES" else return "NO".

Note: Diagonal moves are not allowed, only vertical or horizontal moves are allowed. 

Examples:

Input: X = {1, 3}, Y{4, 3}, Z = {4, 5}
Output: YES
Explanation:

👁 Image

Input: X = {1, 1}, Y = {2, 2}, Z = {3, 3}
Output: NO
Explanation: It can be verified that we can't reach point Z from X using point Y. 

Brute Force Approach :

We can check all possible paths from X to Z that involve at most one turn of 90 degrees at Y. We can start by moving from X to Y in a straight line, and then from Y to Z in a straight line, and check if this path is valid. If not, we can check all paths that involve one turn of 90 degrees at Y by considering all possible directions of the turn (left, right, up, or down), and checking if each path is valid. If we find a valid path, we can return "YES", otherwise we return "NO".

Below is the Implementation of the above approach :

Output :

YES

Complexity Analysis :

The time complexity of this solution is O(1) for the initial check if X and Z are on the same line, and O(9) for the nested loops that consider all possible paths with one turn at Y. Therefore, the overall time complexity is O(1) + O(9) = O(1).

The auxiliary space of this solution is O(1) as it only uses a constant amount of additional space to store the three points and the intermediate point after the turn.

Approach: Implement the idea below to solve the problem:

The problem is observation based and can be solved via implementing those observations. For more clarification see the Concept of approach section below.  

Concept of approach:

It should be noted that Reaching from X to Z is only possible when Y is an intermediate point between X and Z. We can move in the horizontal or vertical direction, So that either the Y should be in between x coordinate of  X and Z or y coordinate of  X and Z also. So the conditions at which reaching is possible are:

• x2 ? min(x1, x3) && x2 ? max(x1, x3) && (y2 == y1 || y2 == y3)
• y2 ? min(y1, y3) && y2 ? max(y1, y3) && (x2 == x1 || x2 == x3)

Where X = {x1, y1}, Y = {x2, y2}, Z = {x3, y3}. All other cases will have no path with at most one turn.    

Follow the below steps to solve the problem:

• Check for the conditions:
  • if (x2 ? min(x1, x3) && x2 ? max(x1, x3) && (y2 == y1 ||y2 == y3))
  • else if (y2 ? min(y1, y3) && y2 ? max(y1, y3) && (x2 == x1 || x2 == x3)), Only at these conditions a path will exist.
• Otherwise, no path is possible.

Below is the Implementation of the above approach:

YES

Time Complexity: O(1)
Auxiliary Space: O(1)