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Im trying to decipher why the div method is faster than the div2 method, and I cant find the reason.
def div2(num)
[*1..num].select do |n|
n if num % n == 0
end
end
p div2(58463982)
def div(num)
result = []
(1..num).each do |n|
break if result.include?(num / n)
result.concat([n, num / n]).uniq! if num % n == 0
end
result.sort!
end
p div(58463982)
I will let others explain why div is faster than div2. I want to show how to compute the factors of the given natural number in a way that is considerably faster.
Every integer can be expressed as the product of a collection of prime numbers, each taken to a power of one or more. We can use the method Prime::prime_division to obtain those prime numbers and powers. For example,
require 'prime'
arr = Prime.prime_division(58463982)
#=> [[2, 1], [3, 2], [53, 1], [61283, 1]]
This means that:
(2**1) * (3**2) * (53**1) * (61283**1)
#=> 58463982
One divisor of 58463982 equals, for example:
(2**1) * (3**2) * (53**0) * (61283**1)
#=> 2 * 9 * 1 * 61283
#=> 1103094
To confirm:
58463982 % 1103094
#=> 0
Another would be
(2**0) * (3**1) * (53**1) * (61283**0)
#=> 1 * 3 * 53 * 1
#=> 159
We find that all factors of a given number can be computed (combinatorially) as follows, using the methods Array#product and Enumerable#reduce (a.k.a. inject).
def all_factors(n)
primes, exponents = Prime.prime_division(n).transpose
first_exp_range, *rest_exp_range = exponents.map { |e| [*0..e] }
first_exp_range.product(*rest_exp_range).map do |exps|
primes.zip(exps).reduce(1) { |t,(p,e)| t*(p**e) }
end.sort
end
Depending on requirements, .sort at the end may not be required.
We may test:
require 'time'
t = Time.now
p all_factors(58463982)
p Time.now - t
#=> [1, 2, 3, 6, 9, 18, 53, 106, 159, 318, 477, 954, 61283, 122566,
# 183849, 367698, 551547, 1103094, 3247999, 6495998, 9743997,
# 19487994, 29231991, 58463982]
#
#=> 0.001405 (seconds)
By constrast, computing the factors of 58463982 with div2 and div required 4.467112 and 0.021103 seconds, respectively.
This is clearly much faster than those methods.
We may step through the example to view the calculations being performed.
n = 58463982
then
primes, exponents = Prime.prime_division(n).transpose
#=> [[2, 3, 53, 61283], [1, 2, 1, 1]]
so
primes
#=> [2, 3, 53, 61283]
exponents
#=> [1, 2, 1, 1]
Then,
first_exp_range, *rest_exp_range = exponents.map { |e| [*0..e] }
#=> [[0, 1], [0, 1, 2], [0, 1], [0, 1]]
so
first_exp_range
#=> [0, 1]
rest_exp_range
#=> [0, 1, 2], [0, 1], [0, 1]
Then
a = first_exp_range.product(*res_exp_range)
#=> [[0, 0, 0, 0], [0, 0, 0, 1], [0, 0, 1, 0], [0, 0, 1, 1],
# [0, 1, 0, 0], [0, 1, 0, 1], [0, 1, 1, 0], [0, 1, 1, 1],
# [0, 2, 0, 0], [0, 2, 0, 1], [0, 2, 1, 0], [0, 2, 1, 1],
# [1, 0, 0, 0], [1, 0, 0, 1], [1, 0, 1, 0], [1, 0, 1, 1],
# [1, 1, 0, 0], [1, 1, 0, 1], [1, 1, 1, 0], [1, 1, 1, 1],
# [1, 2, 0, 0], [1, 2, 0, 1], [1, 2, 1, 0], [1, 2, 1, 1]]
Then,
b = a.map { |exps| primes.zip(exps).reduce(1) { |t,(p,e)| t*(p**e) } }
#=> [1, 61283, 53, 3247999, 3, 183849, 159, 9743997, 9, 551547,
# 477, 29231991, 2, 122566, 106, 6495998, 6, 367698, 318,
# 19487994, 18, 1103094, 954, 58463982]
To view the result sorted,
b.sort
#=> [1, 2, 3, 6, 9, 18, 53, 106, 159, 318, 477, 954, 61283, 122566,
# 183849, 367698, 551547, 1103094, 3247999, 6495998, 9743997,
# 19487994, 29231991, 58463982]
The div2 method create a list from 1 to num then iterates over all of the elements in it.
The div method can break early, and so does not have to iterate as many times.
I'm trying to write a generalized cartesian product, where input data of [n1, n2, ...ni] produces output data that is an array of [m1, m2, ...mi] for all mj such that 0 <= mj < nj. I understand the routine below would produce a somewhat folded version of that, but I'm trying to keep the example code as simple as possible. My immediate problem is that the second block variable (accumulator), which I understand is supposed to update for each iteration of the block, is not doing so:
#!/usr/bin/ruby
def gcp(dims)
first = dims.shift
dims.each_with_object((0...first).to_a) do |dim, v|
puts "\nv: #{v}, dim: #{dim}"
p v.product((0...dim).to_a)
end
end
gcp([3,2,4])
This produces the following output:
v: [0, 1, 2], dim: 2
[[0, 0], [0, 1], [1, 0], [1, 1], [2, 0], [2, 1]]
v: [0, 1, 2], dim: 4
[[0, 0], [0, 1], [0, 2], [0, 3], [1, 0], [1, 1], [1, 2], [1, 3], [2, 0], [2, 1], [2, 2], [2, 3]]
The p method is a passthrough, so the return value of the block should be [[0, 0], [0, 1], [1, 0], [1, 1], [2, 0], [2, 1]] on the first iteration, and that should be the value of v on the second iteration, unless I gravely misunderstand each_with_object.
Each iteration gets the same object, so you either need to mutate the object inside the block, or use reduce.
def gcp(dims)
first = dims.shift
dims.reduce((0...first).to_a) do |v, dim|
puts "\nv: #{v}, dim: #{dim}"
p v.product((0...dim).to_a)
end
end
gcp([3,2,4])
Results in:
v: [0, 1, 2], dim: 2
[[0, 0], [0, 1], [1, 0], [1, 1], [2, 0], [2, 1]]
v: [[0, 0], [0, 1], [1, 0], [1, 1], [2, 0], [2, 1]], dim: 4
[[[0, 0], 0], [[0, 0], 1], [[0, 0], 2], [[0, 0], 3], [[0, 1], 0], [[0, 1], 1], [[0, 1], 2], [[0, 1], 3], [[1, 0], 0], [[1, 0], 1], [[1, 0], 2], [[1, 0], 3], [[1, 1], 0], [[1, 1], 1], [[1, 1], 2], [[1, 1], 3], [[2, 0], 0], [[2, 0], 1], [[2, 0], 2], [[2, 0], 3], [[2, 1], 0], [[2, 1], 1], [[2, 1], 2], [[2, 1], 3]]
I confess I don't fully understand the question, but I've addressed a similar problem that may explain why v is not being updated by your code.
Let's step through your code, returning the desired result rather than displaying it along the way.
dims = [3,2,4]
first = dims.shift
#=> 3
dims
#=> [2, 4] dims
The expression
dims.each_with_object((0...first).to_a) do |dim, v|
v.product((0...dim).to_a)
end
is effectively the same as
v = []
dims.each do |dim|
v.product((0...dim).to_a)
end
v #=> []
That v is still an empty array at the end should not be a surprise, as the value of v is not altered within the loop. The return value of v.product((0...dim).to_a) is shot out into space, never to be seen again. You need an assignment statement within the loop.
Now consider the following.
dims = [3,2,4]
v = []
dims.each do |n|
v << (0...n).to_a
end
v #=> [[0, 1, 2], [0, 1], [0, 1, 2, 3]]
(or v.push((0..n).to_a)). To use Enumerable#each_with_object we would modify the above code by removing the first (v = []) and last (v) statements, changing each to each_with_object([]) (the argument being the initial value of the object the method will return) and add a block variable v, which holds the object:
dims.each_with_object([]) do |n,v|
v << (0...n).to_a
end
#=> [[0, 1, 2], [0, 1], [0, 1, 2, 3]]
We can simplify this using Emumerable#map:
dims.map do |n|
(0...n).to_a
end
#=> [[0, 1, 2], [0, 1], [0, 1, 2, 3]]
Depending on your needs, you may prefer using Emumerable#flat_map:
dims.flat_map do |n|
(0...n).to_a
end
#=> [0, 1, 2, 0, 1, 0, 1, 2, 3]
Say I have the following input:
inp = [2, 9, 3]
I need output as all tuples in mixed counting, like this:
outp = [[0, 0, 0], [0, 0, 1], [0, 0, 2], [0, 1, 0], [0, 1, 1], ..., [1, 8, 2]]
I know algorithm from Knuth vol 4a as direct loop solution, but I've heard ruby has some magic inside.
I am mostly C++ developer. My direct solution now looks like:
inparr = [2, 9, 3]
bmix = Array.new(inparr.size) { |i| 0 }
outp = Array.new
loop do
# some debug output
puts bmix.to_s
#visit next tuple
outp << bmix.clone
digit = inparr.size
while digit > 0 do
digit -= 1
if bmix[digit] + 1 < inparr[digit]
bmix[digit] += 1
break
end
bmix[digit] = 0
end
break if (bmix.select{|x| x != 0}.empty?)
end
How to rewrite it in several simple lines?
inp.
map { |i| (0...i).to_a }.
reduce(&:product).
map(&:flatten)
Used operations: Range, Enumerable#map, Enumerable#reduce, Array#product, Array#flatten.
You could use recursion.
def recurse(inp)
first, *rest = inp
rest.empty? ? [*0..first-1] : (0..first-1).flat_map do |e|
recurse(rest).map { |arr| [e, *arr] }
end
end
recurse [2, 4, 3]
#=> [[0, 0, 0], [0, 0, 1], [0, 0, 2],
# [0, 1, 0], [0, 1, 1], [0, 1, 2],
# [0, 2, 0], [0, 2, 1], [0, 2, 2],
# [0, 3, 0], [0, 3, 1], [0, 3, 2],
# [1, 0, 0], [1, 0, 1], [1, 0, 2],
# [1, 1, 0], [1, 1, 1], [1, 1, 2],
# [1, 2, 0], [1, 2, 1], [1, 2, 2],
# [1, 3, 0], [1, 3, 1], [1, 3, 2]]
If first, *rest = [2,4,3], then first #=> 2 and rest #=> [4,3].
See Enumerable#flat_map and Array#map. a ? b : c is called a ternery expression.
If e #=> 1 and arr #=> [2,1] then [e, *arr] #=> [1,2,1].
I will go to great lengths to avoid the use of Array#flatten. It's irrational, but to me it's an ugly method. That's usually possible using flat_map and/or the splat operator *.
Here's a mix of the 2 existing answers. It might be a bit more concise and readable:
head, *rest = inp.map{ |n| n.times.to_a }
head.product(*rest)
As an example:
inp = [2, 4, 3]
# => [2, 4, 3]
head, *rest = inp.map{ |n| n.times.to_a }
# => [[0, 1], [0, 1, 2, 3], [0, 1, 2]]
head.product(*rest)
# => [[0, 0, 0], [0, 0, 1], [0, 0, 2], [0, 1, 0], [0, 1, 1], [0, 1, 2], [0, 2, 0], [0, 2, 1], [0, 2, 2], [0, 3, 0], [0, 3, 1], [0, 3, 2], [1, 0, 0], [1, 0, 1], [1, 0, 2], [1, 1, 0], [1, 1, 1], [1, 1, 2], [1, 2, 0], [1, 2, 1], [1, 2, 2], [1, 3, 0], [1, 3, 1], [1, 3, 2]]
I want to program a counter which is represented by an array of numbers, starting with:
[0, 0, 0]
The constraint here is, that each position has a different cap, so it's not necessarily 9 or something else, but it is given. For instance:
[4, 2, 1]
Which would lead to the following incrementation sequence:
[0, 0, 0]
[0, 0, 1]
[0, 1, 0]
[0, 1, 1]
[0, 2, 0]
[0, 2, 1]
[1, 0, 0]
.
.
.
Of course I can think of a solution using modulo and adding each carryover onto the next position. But has someone an idea how to implement this efficiently, respectively with nice Ruby syntax without cluttering it too much?
That is my naive implementation:
max = [10, 1, 1, 1, 10]
counter = [0, 0, 0, 0, 0]
i = counter.length-1
while counter != max do
counter[i] = counter[i] + 1
while counter[i] > max[i]
counter[i] = 0
i = i - 1
counter[i] = counter[i] + 1
end
i = counter.length-1
end
I'm not sure about efficiency but here's my shot at it:
start = [0, 0, 0]
cap = [4, 2, 1]
start.zip(cap).map{ |i, c| (i..c).to_a }.reduce(&:product).map &:flatten
Produces something like:
[[0, 0, 0],
[0, 0, 1],
[0, 1, 0],
[0, 1, 1],
[0, 2, 0],
[0, 2, 1],
[1, 0, 0],
[1, 0, 1],
[1, 1, 0],
[1, 1, 1],
[1, 2, 0],
[1, 2, 1],
[2, 0, 0],
[2, 0, 1]...]
Edit: I was writing this before you made your edit. It seemed like you wanted a counter object, not just to output a list.
1) I would recommend specifying not the limits but (limit+1) of each of the digits. For example, for a [second, minute, hour, day, year] counter it makes more sense (to me) to write [60, 60, 24, 365] instead of [59,59,23,364].
2) You'll have to figure out what to do if your counter overflows the last limit of your array. I added an extra position that counts to infinity.
3) I would also recommend reversing the order of the array, at least in the internal representation to avoid inverting subscripts. If you don't want it like that, you can .reverse the bases in initialize and #digits in to_s
class MyCounter
def initialize bases
#bases = bases
#bases << 1.0/0 # Infinity
#digits = Array.new(bases.size, 0)
prod = 1
#digit_values = [1] + #bases[0..-2].map { |b| prod *= b }
end
attr_reader :digit_values
def to_s
#digits
end
def increment(digit=0)
v = #digits[digit] + 1
if v < #bases[digit]
#digits[digit] = v
else
#digits[digit] = 0
increment(digit+1)
end
self
end
def +(integer)
(#digits.size - 1).step(0,-1).each do |i|
#digits[i] += integer / #digit_values[i]
integer = integer % #digit_values[i]
end
self
end
end
c1 = MyCounter.new [2,3,5]
20.times { c1.increment; p c1 }
c2 = MyCounter.new [2,3,5]
c2 += 20
p c2
Create an array for each cap, with values from 0 upto cap. Take the first array and calculate the Cartesian product with the rest of the arrays.
caps = [4, 2, 1]
arrs = caps.map{|cap| (0..cap).to_a} #=>[[0, 1, 2, 3, 4], [0, 1, 2], [0, 1]]
p arrs.shift.product(*arrs)
# =>[[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [0, 2, 0], [0, 2, 1], ...
If you don't want a memory-consuming array with the results, then provide a block. product will yield each element to it, one by one.
arrs = caps.map{|cap| (0..cap).to_a}
arrs.shift.product(*arrs){|el| puts el.join} #no resulting array
#000
#001
#010
#011
#...
Code works, but feels very brute force, suggestions?
Goal of the code is to supply an array length, and then as fast as possible generate all possible unique binary combinations with that array length.
CODE:
class Array
def sequence(i = 0, *a)
return [a] if i == size
self[i].map {|x|
sequence(i+1, *(a + [x]))
}.inject([]) {|m, x| m + x}
end
end
[(0..1),(0..1),(0..1)].sequence
OUTPUTS:
[[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, 0, 0], [1, 0, 1], [1, 1, 0], [1, 1, 1]]
permutation and repeated_permutation are built in, so you can do:
def sequence(n)
[0, 1].repeated_permutation(n).to_a
end
p sequence(3) #=>[[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, 0, 0], [1, 0, 1], [1, 1, 0], [1, 1, 1]]
"All unique binary combinations" with n bits is nothing but (0...2**n), so the only task is to efficiently convert from an integer to its binary representation, and the following is a solution that does not rely on string generation/manipulation:
def sequence(n)
ret = []
(2**n).times do |number|
ret << []
(n - 1).downto(0) do |bit|
ret.last << number[bit]
end
end
ret
end
sequence(3)
# => [[0, 0, 0], [0, 0, 1], [0, 1, 0], [0, 1, 1], [1, 0, 0], [1, 0, 1], [1, 1, 0], [1, 1, 1]]
Or, if you prefer a version more oriented on list operations, this is pretty much the same:
def sequence(n)
(0...2**n).map {|number|
(1..n).map {|bit|
number[n-bit]
}
}
end