Given list of 2d points, find the point closest to all other points

Question

Input: list of 2d points (x,y) where x and y are integers.

Distance: distance is defined as the Manhattan distance. 
    ie:
    def dist(p1,p2) 
         return ab         


        

Answer 1

Starting from your initial algortihm, there is an optimization possible:

minDist=inf
bestPoint = null
for p1 in points:
    dist = 0
    for p2 in points:
        dist+=distance(p1,p2)
        //This will weed out bad points rather fast
        if dist>=minDist then continue(p1)
    /*
    //Unnecessary because of new line above
    minDist = min(dist,minDist)
    bestPoint = argmin(p1, bestPoint)
    */
    bestPoint = p1

The idea is, to throw away outliers as fast as possible. This can be improved more:

• start p1 loop with a heuristic "inner" point (This creates a good minDist first, so worse points get thrown away faster)
• start p2 loop with heuristic "outer" points (This makes dist rise fast, possibly triggering the exit condition faster

If you trade size for speed, you can go another route:

//assumes points are numbered 0..n
dist[]=int[n+1]; //initialized to 0
for (i=0;i

which needs more space for the dist array, but calculates each distance only once. This has the advantage to be better suited to a "get the N best points" sort of problem and for calculation-intensive metrices. I suspect it would bring nothing for the manhattan metric, on an x86 or x64 architecture though: The memory access would heavily dominate the calculation.