Problem:
Score : 400 points

Problem Statement

We have N points in a two-dimensional plane.

The coordinates of the i-th point (1 \leq i \leq N) are (x_i,y_i).

Let us consider a rectangle whose sides are parallel to the coordinate axes that contains K or more of the N points in its interior.

Here, points on the sides of the rectangle are considered to be in the interior.

Find the minimum possible area of such a rectangle.

Constraints

2 \leq K \leq N \leq 50

-10^9 \leq x_i,y_i \leq 10^9 (1 \leq i \leq N)

x_i≠x_j (1 \leq i<j \leq N)

y_i≠y_j (1 \leq i<j \leq N)

All input values are integers. (Added at 21:50 JST)

Input

Input is given from Standard Input in the following format:

N K
x_1 y_1
:
x_{N} y_{N}

Output

Print the minimum possible area of a rectangle that satisfies the condition.

Sample Input 1

4 4
1 4
3 3
6 2
8 1

Sample Output 1

21

One rectangle that satisfies the condition with the minimum possible area has the following vertices: (1,1), (8,1), (1,4) and (8,4).

Its area is (8-1) × (4-1) = 21.

Sample Input 2

4 2
0 0
1 1
2 2
3 3

Sample Output 2

1

Sample Input 3

4 3
-1000000000 -1000000000
1000000000 1000000000
-999999999 999999999
999999999 -999999999

Sample Output 3

3999999996000000001

Watch out for integer overflows.

Solution in Haskell (GHC 7.10.3) (judge verdict: Time Limit Exceeded)

{-# OPTIONS_GHC -O2 -funbox-strict-fields #-}
import           Control.Monad (replicateM)
import           Data.Int      (Int64)
import           Data.List     (minimum, sort)

type Point = (Int64, Int64)
type Rect = (Int64, Int64, Int64, Int64)

main :: IO ()
main = do
  [n, k] <- readInts
  xys <- replicateM n readPoint
  print $ solve n k xys

solve :: Int -> Int -> [Point] -> Int64
solve n k xys = minimum $ map area rs where
  rs = [(x1, x2, y1, y2) | x1 <- init xs
                         , x2 <- tail xs
                         , y1 <- init ys
                         , y2 <- tail ys
                         , isValid (x1, x2, y1, y2)
                         ]
  (xs, ys) = (sort (map fst xys), sort (map snd xys))
  isValid r = length (filter (isIncluded r) xys) >= k

isIncluded :: Rect -> Point -> Bool
isIncluded (x1, x2, y1, y2) (a, b) = x1 <= a && a <= x2 && y1 <= b && b <= y2

area :: Rect -> Int64
area (x1, x2, y1, y2) = (x2 - x1) * (y2 - y1)

readInts :: Read a => IO [a]
readInts = map read . words <$> getLine

readPoint :: IO Point
readPoint = do
  [x, y] <- readInts
  return (x, y)