How to find two most distant points?

Question

This is a question that I was asked on a job interview some time ago. And I still can\'t figure out sensible answer.

Question is:

you are given set of points

Answer 1

This seems easy if the points are given in Cartesian coordinates. So easy that I'm pretty sure that I'm overlooking something. Feel free to point out what I'm missing!

1. Find the points with the max and min values of their x, y, and z coordinates (6 points total). These should be the most "remote" of all the boundary points.
2. Compute all the distances (30 unique distances)
3. Find the max distance
4. The two points that correspond to this max distance are the ones you're looking for.

Answer 2

This question is introduced at Introduction to Algorithm. It mentioned 1) Calculate Convex Hull O(NlgN). 2) If there is M vectex on Convex Hull. Then we need O(M) to find the farthest pair.

I find this helpful links. It includes analysis of algorithm details and program. http://www.seas.gwu.edu/~simhaweb/alg/lectures/module1/module1.html

Wish this will be helpful.

Answer 3

EDIT: One way is to find the convex hull http://en.wikipedia.org/wiki/Convex_hull of the set of points and then the two distant points are vertices of this.

Possibly answered here: Algorithm to find two points furthest away from each other

Also:

• http://mukeshiiitm.wordpress.com/2008/05/27/find-the-farthest-pair-of-points/

Answer 4

Given a set of points {(x1,y1), (x2,y2) ... (xn,yn)} find 2 most distant points.

My approach:

1). You need a reference point (xa,ya), and it will be:
xa = ( x1 + x2 +...+ xn )/n
ya = ( y1 + y2 +...+ yn )/n

2). Calculate all distance from point (xa,ya) to (x1,y1), (x2,y2),...(xn,yn)
The first "most distant point" (xb,yb) is the one with the maximum distance.

3). Calculate all distance from point (xb,yb) to (x1,y1), (x2,y2),...(xn,yn)
The other "most distant point" (xc,yc) is the one with the maximum distance.

So you got your most distant points (xb,yb) (xc,yc) in O(n)

For example, for points: (0,0), (1,1), (-8, 5)

1). Reference point (xa,ya) = (-2.333, 2)

2). Calculate distances:
from (-2.333, 2) to (0,0) : 3.073
from (-2.333, 2) to (1,1) : 3.480
from (-2.333, 2) to (-8, 5) : 6.411
So the first most distant point is (-8, 5)

3). Calculate distances:
from (-8, 5) to (0,0) : 9.434
from (-8, 5) to (1,1) : 9.849
from (-8, 5) to (-8, 5) : 0
So the other most distant point is (1, 1)

Answer 5

A stochastic algorithm to find the most distant pair would be

• Choose a random point
• Get the point most distant to it
• Repeat a few times
• Remove all visited points
• Choose another random point and repeat a few times.

You are in O(n) as long as you predetermine "a few times", but are not guaranteed to actually find the most distant pair. But depending on your set of points the result should be pretty good. =)

Answer 6

You are looking for an algorithm to compute the diameter of a set of points, Diam(S). It can be shown that this is the same as the diameter of the convex hull of S, Diam(S) = Diam(CH(S)). So first compute the convex hull of the set.

Now you have to find all the antipodal points on the convex hull and pick the pair with maximum distance. There are O(n) antipodal points on a convex polygon. So this gives a O(n lg n) algorithm for finding the farthest points.

This technique is known as Rotating Calipers. This is what Marcelo Cantos describes in his answer.

If you write the algorithm carefully, you can do without computing angles. For details, check this URL.