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Task

Provide a function to find the closest two points among a set of given points in two dimensions,   i.e. to solve the   Closest pair of points problem   in the   planar   case.

The straightforward solution is a   O(n2)   algorithm   (which we can call brute-force algorithm);   the pseudo-code (using indexes) could be simply:

bruteForceClosestPair of P(1), P(2), ... P(N)
if N < 2 then
  return ∞
else
  minDistance ← |P(1) - P(2)|
  minPoints ← { P(1), P(2) }
  foreach i ∈ [1, N-1]
    foreach j ∈ [i+1, N]
      if |P(i) - P(j)| < minDistance then
        minDistance ← |P(i) - P(j)|
        minPoints ← { P(i), P(j) } 
      endif
    endfor
  endfor
  return minDistance, minPoints
 endif

A better algorithm is based on the recursive divide&conquer approach,   as explained also at   Wikipedia's Closest pair of points problem,   which is   O(n log n);   a pseudo-code could be:

closestPair of (xP, yP)
               where xP is P(1) .. P(N) sorted by x coordinate, and
                     yP is P(1) .. P(N) sorted by y coordinate (ascending order)
if N ≤ 3 then
  return closest points of xP using brute-force algorithm
else
  xL ← points of xP from 1 to ⌈N/2⌉
  xR ← points of xP from ⌈N/2⌉+1 to N
  xm ← xP(⌈N/2⌉)x
  yL ← { p ∈ yP : px ≤ xm }
  yR ← { p ∈ yP : px > xm }
  (dL, pairL) ← closestPair of (xL, yL)
  (dR, pairR) ← closestPair of (xR, yR)
  (dmin, pairMin) ← (dR, pairR)
  if dL < dR then
    (dmin, pairMin) ← (dL, pairL)
  endif
  yS ← { p ∈ yP : |xm - px| < dmin }
  nS ← number of points in yS
  (closest, closestPair) ← (dmin, pairMin)
  for i from 1 to nS - 1
    k ← i + 1
    while k ≤ nS and yS(k)y - yS(i)y < dmin
      if |yS(k) - yS(i)| < closest then
        (closest, closestPair) ← (|yS(k) - yS(i)|, {yS(k), yS(i)})
      endif
      k ← k + 1
    endwhile
  endfor
  return closest, closestPair
endif



References and further readings

   Closest pair of points problem
   Closest Pair (McGill)
   Closest Pair (UCSB)
   Closest pair (WUStL)
   Closest pair (IUPUI) 


100 PROGRAM "Closestp.bas"
110 NUMERIC X(1 TO 10),Y(1 TO 10)
120 FOR I=1 TO 10
130   READ X(I),Y(I)
140   PRINT X(I),Y(I)
150 NEXT 
160 LET MN=INF
170 FOR I=1 TO 9
180   FOR J=I+1 TO 10
190     LET DSQ=(X(I)-X(J))^2+(Y(I)-Y(J))^2
200     IF DSQ<MN THEN LET MN=DSQ:LET MINI=I:LET MINJ=J
210   NEXT 
220 NEXT 
230 PRINT "Closest pair is (";X(MINI);",";Y(MINI);") and (";X(MINJ);",";Y(MINJ);")":PRINT "at distance";SQR(MN)
240 DATA 0.654682,0.925557
250 DATA 0.409382,0.619391
260 DATA 0.891663,0.888594
270 DATA 0.716629,0.996200
280 DATA 0.477721,0.946355
290 DATA 0.925092,0.818220
300 DATA 0.624291,0.142924
310 DATA 0.211332,0.221507
320 DATA 0.293786,0.691701
330 DATA 0.839186,0.728260