Closest Pair Of Points

Brute force solution: n²

Better solution:

• Given points P
• Sort points by x coordinate into X
• Sort points by y coordinate into Y
• Execute Closest-Pair(P,X,Y)

Closest-Pair(P, X, Y)
$\;\;\;\;\;$if $\vert P\vert \leq 3$
$\;\;\;\;\;$then compute closest pair and return $\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$; O(1)
$\;\;\;\;\;$divide points evenly along x axis at x = l into P_L(X_L,Y_L) and P_R(X_R,X_R)
$\;\;\;\;\;$d_L = distance(Closest-Pair(P_L,X_L,Y_L))
$\;\;\;\;\;$d_R = distance(Closest-Pair(P_R,X_R,Y_R))
$\;\;\;\;\;$d = min(d_L,d_R)
$\;\;\;\;\;$foreach point p in (l - d) $\leq$ x $\leq$ (l + d)
$\;\;\;\;\;$ $\;\;\;\;\;$check 7 points p' closest to p by y-coordinate
$\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$d' = distance(p,p')
$\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$if d' < d
$\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$then retain new closest pair
$\;\;\;\;\;$return closest pair