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 Ruby (2.3.3) (judge verdict: Time Limit Exceeded)

require "set"
n, K = gets.chomp.split.map(&:to_i)
@ary = n.times.map do
  gets.chomp.split.map(&:to_i)
end

@ary.sort_by!(&:first)

x_set = @ary.map(&:first).uniq
y_set = @ary.map(&:last).uniq

def count_dots x_range, y_range
  x_range.sort!
  s_idx = @ary.index{|dot| dot[0] == x_range[0]}
  e_idx = @ary.rindex{|dot| dot[0] == x_range[1]}
  y_range.sort!
  @ary[s_idx..e_idx].count do |dot|
    y_range[0] <= dot[1]  && dot[1] <= y_range[1]
  end
end

if K == 50
  puts((x_set.max - x_set.min) * (y_set.max - y_set.min))
else
  ans_list = Set.new({})
  x_set.combination(2).each do |x_combi|
    y_s = if K < 25
      y_set.combination(2).sort_by{|y0, y1| (y0 - y1).abs}.find do |y_combi|
        count_dots(x_combi, y_combi) >= K
      end
    else
      y_set.combination(2).sort_by{|y0, y1| -(y0 - y1).abs}.find do |y_combi|
        count_dots(x_combi, y_combi) >= K
      end
    end
    next unless y_s
    ans_list.add(((x_combi[1] - x_combi[0]) * (y_s[1] - y_s[0])).abs)
  end
  puts ans_list.min
end