Problem:
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Distance II

Your task is to calculate the distance between two $n$ dimensional vectors $x = \{x_1, x_2, ..., x_n\}$ and $y = \{y_1, y_2, ..., y_n\}$.

The Minkowski's distance defined below is a metric which is a generalization of both the Manhattan distance and the Euclidean distance.

\[
D_{xy} = (\sum_{i=1}^n |x_i - y_i|^p)^{\frac{1}{p}}
\]

It can be the Manhattan distance

\[
D_{xy} = |x_1 - y_1| + |x_2 - y_2| + ... + |x_n - y_n|
\]

where $p = 1 $.

It can be the Euclidean distance

\[
D_{xy} = \sqrt{(|x_1 - y_1|)^{2} + (|x_2 - y_2|)^{2} + ... + (|x_n - y_n|)^{2}}
\]

where $p = 2 $.

Also, it can be the Chebyshev distance

\[
D_{xy} = max_{i=1}^n (|x_i - y_i|)
\]

where $p = \infty$

Write a program which reads two $n$ dimensional vectors $x$ and $y$, and calculates Minkowski's distance where $p = 1, 2, 3, \infty$ respectively.

Input

In the first line, an integer $n$ is given. In the second and third line, $x = \{x_1, x_2, ... x_n\}$ and $y = \{y_1, y_2, ... y_n\}$ are given respectively. The elements in $x$ and $y$ are given in integers.

Output

Print the distance where $p = 1, 2, 3$ and $\infty$ in a line respectively.
The output should not contain an absolute error greater than 10-5.

Constraints

$1 \leq n \leq 100$

$0 \leq x_i, y_i \leq 1000$

Sample Input

3
1 2 3
2 0 4

Sample Output

4.000000
2.449490
2.154435
2.000000

Solution in Ruby (judge verdict: Accepted)

#!/usr/bin/ruby
# -*- coding: utf-8 -*-
=begin
ITP1_10-D

Math Functions - Distance II
Time Limit : 1 sec, Memory Limit : 65536 KB
??????????????????????????¢

???????????????????????????????????????????????????????????????????????¢??§??¬???????????????????????????????????°???????????????????§?????????¨????????§????????????????????????????????§??????????????? n
?¬???????????????? x={x1,x2,...,xn} ??¨ y={y1,y2,...,yn}

????????¢????¨????????????????????????????

????????????????????????????????¢?????¬???????¨?????????¨?????¨??????????¬??????????????????????????????¢?????\?????????????????????
Dxy=(???i=1n|xi???yi|p)1p

p=1 ?????¨???
Dxy=|x1???y1|+|x2???y2|+...+|xn???yn|

??¨?????????????????????????????????????????¢??¨?????°????????????

p=2 ?????¨???
Dxy=(|x1???y1|)2+(|x2???y2|)2+...+(|xn???yn|)2?????????????????????????????????????????????????????????????????????????????????????????????????????????

??¨???????????????????????¬?????????????????????????????????????????¢??????????????????

p=??? ?????¨???

Dxy=maxni=1(|xi???yi|)



??¨???????????????????????§????????§????????¢??¨?????°????????????

????????? n
?¬???????????????????????????????????????§???p ??????????????? 1???2???3??????

?????????????????????????????¢????±???????????????°????????????????????????????????????
Input

??????????????´??° n
???????????????????????????????????????????????? x ???????´? {x1,x2,...xn}??????????????????????????? y ???????´? {y1,y2,...yn}

????????????????????§???????????????????????\?????????????????´??°?????§??????
Output

p
??????????????? 1???2???3??????

??????????????????????????????????????¢??????????????????????????????????????????0.00001 ??\????????????????????£????????????????????¨????????????
Constraints

    1???n???100

0???xi,yi???1000

Sample Input

3
1 2 3
2 0 4

Sample Output

4.000000
2.449490
2.154435
2.000000

=end
class Dist
  def initialize(f0,f1)
    @len = f0.length
    @v0 = f0
    @v1 = f1
  end

  def dis1
    sum = 0.0
    (0..(@len-1)).each {|i| sum += (@v0[i] - @v1[i]).abs}
    sum
  end

  def dis2
    sum = 0.0
    (0..(@len-1)).each {|i| sum += (@v0[i] - @v1[i])**2}
    Math.sqrt(sum)
  end

  def dis3
    sum = 0.0
    (0..(@len-1)).each {|i| sum += (@v0[i] - @v1[i]).abs**3}
    sum**(1.0/3)
  end

  def disLarge
    l=99
    sum = 0.0
    (0..(@len-1)).each {|i| sum += (@v0[i] - @v1[i]).abs**l}
    sum**(1.0/l)
  end

  def disInf
    v2 = []
    (0..(@len-1)).each {|i| v2[i] = (@v0[i] - @v1[i]).abs}
    v2.max
  end

end

class Main
  def initialize
    fh = STDIN
    fh.gets
    n = $_.chomp.to_i
    fl0 = fh.gets.chomp.split.map{|x| x.to_f}
    fl1 = fh.gets.chomp.split.map{|x| x.to_f}
    d = Dist.new(fl0,fl1)
    puts d.dis1
    puts d.dis2
    puts d.dis3
    #      puts d.disLarge
    puts d.disInf
  end

  def stddev(f)
    n = f.length
    sum = f.inject(0.0){|acc,x| acc + x}
    ave = sum / n
    sq_sum = f.inject(0.0){|acc,x| acc + (x - ave)**2}
    Math.sqrt(sq_sum / n) 
  end
end

if $0==__FILE__ then
  Main.new()
end