Question

Given 3 points in a grid, how would you find a point such that sum of distances of this point from all 3 points in minimized. An obvious answer to this problem is Fermat’s triangle. I am interested in knowing if we can locate Fermat’s point using a breadth first search algorithm in a graph.

struct node{
  int Person1X,Person1Y,Person2X,Person2Y,Person3X,Person3Y; //X and Y coordinates of all 3 persons
  int steps;   //sum of distances covered by all 3 person to reach this state
}

While doing the BFS we could put a constraint,
if steps>min(sum of any two edges of the triangle with 3 persons as vertices) return;

if(Person1X=Person2X=Person2X)AND(Person1Y=Person2Y=Person3Y) return steps;

Answer 1

No search is necessary.

Given “triangle” ABC:
SumOfDistances( p ) = dist( A, p ) + dist( B, p ) + dist( C, p )

where dist( q, p ) = |q_x-p_x| + |q_y-p_y| (Manhattan distance)

you can see that SumOfDistances( p ) = SumOfDistances_x( p ) + SumOfDistances_y( p )

So, you can minimize by the distance on the x and y axis independently.

So, the Fermat point’s x-coordinate is the median of the 3 given x-coordinates.
The Fermat point’s y-coordinate is the median of the 3 given y-coordinates.