A tram with a very low unusefulness. Maybe Lund’s tram line
will be as appreciated as Istanbul’s, one day.
Photographer: Maj Stenmark
It is 1815 and the politicians in Lund have just decided
to build a tram line in Lund. Oh, sorry. That was wrong, let’s
start over. It is 2015 and the politicians in Lund have just
decided to build a tram line in Lund.
The politicians have already decided that the tram line should
run from southeast to northwest. In order not to cause too
many complaints from the citizens, they want to make the line
as useful as possible. Therefore they want to minimize the
total unusefulness of the tram.
The
unusefulness for citizen
$i$ is equal to the square of the
closest distance from the citizen’s home to the tram line. The
total unusefulness of the tram is the sum of all citizens’
unusefulnesses.
Given the coordinates of each citizen’s home, determine the
value
$a$ minimizing the
total unusefulnes, where the equation of the tram line is given
by
$y=x+a$.
Input
The first line of input consists of an integer, $1\leq N\leq 10^5$, the number of
citizens in Lund. Then follow $N$ lines, with two spaceseparated
integers $x_ i,y_ i$
($x_ i,y_ i\leq
10^6$), the coordinates of citizen $i$’s home.
Output
The output consists of a single number, $a$, minimizing the total
unusefulness. An answer will be accepted if it is within an
absolute or relative error of $10^{3}$.
Sample Input 1 
Sample Output 1 
3
1 1
2 2
3 3

0.000000

Sample Input 2 
Sample Output 2 
3
0 1
1 0
1 1

0.000000

Sample Input 3 
Sample Output 3 
3
0 2
1 1
1 0

0.333333
