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: Accepted)

import Data.List

type Point = (Integer, Integer)

windows :: Integer -> [a] -> [[a]]
windows k xs
    | length xs < fromIntegral k = []
    | otherwise = (take (fromIntegral k) xs) : windows k (tail xs)

minArea :: [Point] -> Integer -> (Integer, Integer) -> Integer
minArea points k (x1, x2) =
    let insidePoints = filter (\(x, _) -> x1 <= x && x <= x2) points
        sortedPoints = sortBy (\(_, y1) (_, y2) -> compare y1 y2) insidePoints
        groups = windows k sortedPoints
    in if null groups
       then 40000000000000000000
       else minimum $ map area groups
    where area ps = xDist * (yDist ps)
          xDist = x2 - x1
          yDist ps = (snd $ last ps) - (snd $ head ps)

solve :: [Point] -> Integer -> Integer
solve points k =
    let xs = sort $ map fst points
        pairs = [(x1, x2) | x1 <- xs, x2 <- xs, x1 < x2]
    in minimum $ map (minArea points k) pairs

makePoint :: [Integer] -> Point
makePoint (x:y:_) = (x, y)

readIntegers :: IO [Integer]
readIntegers = map read . words <$> getLine :: IO [Integer]

readPoint :: IO Point
readPoint = makePoint <$> readIntegers

readPoints :: Integer -> IO [Point]
readPoints n
    | n <= 0 = return []
    | otherwise = do
        currPoint <- makePoint <$> readIntegers
        otherPoints <- readPoints (n - 1)
        return (currPoint : otherPoints)

main = do
    [n, k] <- readIntegers
    points <- readPoints n
    putStrLn . show $ solve points k