I have to create a program in ruby on rails so that it will take less time to solve the particular condition. Now i am to getting the less response time for k=4 but response time is more in case of k>5
Problem:
Problem is response time.
When value of k is more than 5 (k>5) response time is too late for given below equation.
Input: K, N (where 0 < N < ∞, 0 < K < ∞, and K <= N)
Output: Number of possible equations of K numbers whose sum is N.
Example Input:
N=10 K=3
Example Output:
Total unique equations = 8
1 + 1 + 8 = 10
1 + 2 + 7 = 10
1 + 3 + 6 = 10
1 + 4 + 5 = 10
2 + 2 + 6 = 10
2 + 3 + 5 = 10
2 + 4 + 4 = 10
3 + 3 + 4 = 10
For reference, N=100, K=3 should have a result of 833 unique sets
Here is my ruby code
module Combination
module Pairs
class Equation
def initialize(params)
#arr=[]
#n = params[:n]
#k = params[:k]
end
#To create possible equations
def create_equations
return "Please Enter value of n and k" if #k.blank? && #n.blank?
begin
Integer(#k)
rescue
return "Error: Please enter any +ve integer value of k"
end
begin
Integer(#n)
rescue
return "Error: Please enter any +ve integer value of n"
end
return "Please enter k < n" if #n < #k
create_equations_sum
end
def create_equations_sum
aar = []
#arr = []
#list_elements=(1..#n).to_a
(1..#k-1).each do |i|
aar << [*0..#n-1]
end
traverse([], aar, 0)
return #arr.uniq #return result
end
#To check sum
def generate_sum(*args)
new_elements = []
total= 0
args.flatten.each do |arg|
total += #list_elements[arg]
new_elements << #list_elements[arg]
end
if total < #n
new_elements << #n - total
#arr << new_elements.sort
else
return
end
end
def innerloop(arrayOfCurrentValues)
generate_sum(arrayOfCurrentValues)
end
#Recursive method to create dynamic nested loops.
def traverse(accumulated,params, index)
if (index==params.size)
return innerloop(accumulated)
end
currentParam = params[index]
currentParam.each do |currentElementOfCurrentParam|
traverse(accumulated+[currentElementOfCurrentParam],params, index+1)
end
end
end
end
end
run the code using
params = {:n =>100, :k =>4}
c = Combination::Pairs::Equation.new(params)
c.create_equations
Here are two ways to compute your answer. The first is simple but not very efficient; the second, which relies on an optimization technique, is much faster, but requires considerably more code.
Compact but Inefficient
This is a compact way to do the calculation, making use of the method Array#repeated_combination:
Code
def combos(n,k)
[*(1..n-k+1)].repeated_combination(3).select { |a| a.reduce(:+) == n }
end
Examples
combos(10,3)
#=> [[1, 1, 8], [1, 2, 7], [1, 3, 6], [1, 4, 5],
# [2, 2, 6], [2, 3, 5], [2, 4, 4], [3, 3, 4]]
combos(100,4).size
#=> 832
combos(1000,3).size
#=> 83333
Comment
The first two calculations take well under one second, but the third took a couple of minutes.
More efficient, but increased complexity
Code
def combos(n,k)
return nil if k.zero?
return [n] if k==1
return [1]*k if k==n
h = (1..k-1).each_with_object({}) { |i,h| h[i]=[[1]*i] }
(2..n-k+1).each do |i|
g = (1..[n/i,k].min).each_with_object(Hash.new {|h,k| h[k]=[]}) do |m,f|
im = [i]*m
mxi = m*i
if m==k
f[mxi].concat(im) if mxi==n
else
f[mxi] << im if mxi + (k-m)*(i+1) <= n
(1..[(i-1)*(k-m), n-mxi].min).each do |j|
h[j].each do |a|
f[mxi+j].concat([a+im]) if
((a.size==k-m && mxi+j==n) ||
(a.size<k-m && (mxi+j+(k-m-a.size)*(i+1))<=n))
end
end
end
end
g.update({ n=>[[i]*k] }) if i*k == n
h.update(g) { |k,ov,nv| ov+nv }
end
h[n]
end
Examples
p combos(10,3)
#=> [[3, 3, 4], [2, 4, 4], [2, 3, 5], [1, 4, 5],
# [2, 2, 6], [1, 3, 6], [1, 2, 7], [1, 1, 8]]
p combos(10,4)
#=> [[2, 2, 3, 3], [1, 3, 3, 3], [2, 2, 2, 4], [1, 2, 3, 4], [1, 1, 4, 4],
# [1, 2, 2, 5], [1, 1, 3, 5], [1, 1, 2, 6], [1, 1, 1, 7]]
puts "size=#{combos(100 ,3).size}" #=> 833
puts "size=#{combos(100 ,5).size}" #=> 38224
puts "size=#{combos(1000,3).size}" #=> 83333
Comment
The calculation combos(1000,3).size took about five seconds, the others were all well under one second.
Explanation
This method employs dynamic programming to compute a solution. The state variable is the largest positive integer used to compute arrays with sizes no more than k whose elements sum to no more than n. Begin with the largest integer equal to one. The next step is compute all combinations of k or fewer elements that include the numbers 1 and 2, then 1, 2 and 3, and so on, until we have all combinations of k or fewer elements that include the numbers 1 through n. We then select all combinations of k elements that sum to n from the last calculation.
Suppose
k => 3
n => 7
then
h = (1..k-1).each_with_object({}) { |i,h| h[i]=[[1]*i] }
#=> (1..2).each_with_object({}) { |i,h| h[i]=[[1]*i] }
#=> { 1=>[[1]], 2=>[[1,1]] }
This reads, using the only the number 1, [[1]] is the array of all arrays that sum to 1 and [[1,1]] is the array of all arrays that sum to 2.
Notice that this does not include the element 3=>[[1,1,1]]. That's because, already having k=3 elments, if cannot be combined with any other elements, and sums to 3 < 7.
We next execute:
enum = (2..n-k+1).each #=> #<Enumerator: 2..5:each>
We can convert this enumerator to an array to see what values it will pass into its block:
enum.to_a #=> [2, 3, 4, 5]
As n => 7 you may be wondering why this array ends at 5. That's because there are no arrays containing three positive integers, of which at least one is a 6 or a 7, whose elements sum to 7.
The first value enum passes into the block, which is represented by the block variable i, is 2. We will now compute a hash g that includes all arrays that sum to n => 7 or less, have at most k => 3 elements, include one or more 2's and zero or more 1's. (That's a bit of a mouthful, but it's still not precise, as I will explain.)
enum2 = (1..[n/i,k].min).each_with_object(Hash.new {|h,k| h[k]=[]})
#=> (1..[7/2,3].min).each_with_object(Hash.new {|h,k| h[k]=[]})
#=> (1..3).each_with_object(Hash.new {|h,k| h[k]=[]})
Enumerable#each_with_object creates an initially-empty hash that is represented by the block variable f. The default value of this hash is such that:
f[k] << o
is equivalent to
(f[k] |= []) << o
meaning that if f does not have a key k,
f[k] = []
is executed before
f[k] << o
is performed.
enum2 will pass the following elements into its block:
enum2.to_a #=> => [[1, {}], [2, {}], [3, {}]]
(though the hash may not be empty when elements after the first are passed into the block). The first element passed to the block is [1, {}], represented by the block variables:
m => 1
f => Hash.new {|h,k| h[k]=[]}
m => 1 means we will intially construct arrays that contain one (i=) 2.
im = [i]*m #=> [2]*1 => [2]
mxi = m*i #=> 2*1 => 2
As (m == k) #=> (1 == 3) => false, we next execute
f[mxi] << im if mxi + (k-m)*(i+1) <= n
#=> f[2] << [2] if 2 + (3-1)*(1+1) <= 7
#=> f[2] << [2] if 8 <= 7
This considers whether [2] should be added to f[2] without adding any integers j < i = 2. (We have yet to consider the combining of one 2 with integers less than 2 [i.e., 1].) As 8 <= 7, we do not add [2] to f[2]. The reason is that, for this to be part of an array of length k=3, it would be of the form [2,x,y], where x > 2 and y > 2, so 2+x+y >= 2+3+3 = 8 > n = 7. Clear as mud?
Next,
enum3 = (1..[(i-1)*(k-m), n-mxi].min).each
#=> = (1..[2,5].min).each
#=> = (1..2).each
#=> #<Enumerator: 1..2:each>
which passes the values
enum3.to_a #=> [1, 2]
into its block, represented by the block variable j, which is the key of the hash h. What we will be doing here is combine one 2 (m=1) with arrays of elements containing integers up to 1 (i.e., just 1) that sum to j, so the elements of the resulting array will sum to m * i + j => 1 * 2 + j => 2 + j.
The reason enum3 does not pass values of j greater than 2 into its block is that h[l] is empty for l > 2 (but its a little more complicated when i > 2).
For j => 1,
h[j] #=> [[1]]
enum4 = h[j].each #=> #<Enumerator: [[1]]:each>
enum4.to_a #=> [[1]]
a #=> [1]
so
f[mxi+j].concat([a+im]) if
((a.size==k-m && mxi+j==n) || (a.size<k-m && (mxi+j+(k-m-a.size)*(i+1))<=n))
#=> f[2+1].concat([[1]+[2]) if ((1==2 && 2+1==7) || (1<=3-1 && (2+1+(1)*(3)<=7))
#=> f[3].concat([1,2]) if ((false && false) || (1<=2 && (6<=7))
#=> f[3] = [] << [[1,2]] if (false || (true && true)
#=> f[3] = [[1,2]] if true
So the expression on the left is evaluated. Again, the conditional expressions are a little complex. Consider first:
a.size==k-m && mxi+j==n
which is equivalent to:
([2] + f[j]).size == k && ([2] + f[j]).reduce(:+) == n
That is, include the array [2] + f[j] if it has k elements that sum to n.
The second condition considers whether the array the arrays [2] + f[j] with fewer than k elements can be "completed" with integers l > i = 2 and have a sum of n or less.
Now, f #=> {3=>[[1, 2]]}.
We now increment j to 2 and consider arrays [2] + h[2], whose elements will total 4.
For j => 2,
h[j] #=> [[1, 1]]
enum4 = h[j].each #=> #<Enumerator: [[1, 1]]:each>
enum4.to_a #=> [[1, 1]]
a #=> [1, 1]
f[mxi+j].concat([a+im]) if
((a.size==k-m && mxi+j==n) || (a.size<k-m && (mxi+j+(k-m-a.size)*(i+1)<=n))
#=> f[4].concat([1, 1, 2]) if ((2==(3-1) && 2+2 == 7) || (2+2+(3-1-2)*(3)<=7))
#=> f[4].concat([1, 1, 2]) if (true && false) || (false && true))
#=> f[4].concat([1, 1, 2]) if false
so this operation is not performed (since [1,1,2].size => 3 = k and [1,1,2].reduce(:+) => 4 < 7 = n.
We now increment m to 2, meaning that we will construct arrays having two (i=) 2's. After doing so, we see that:
f={3=>[[1, 2]], 4=>[[2, 2]]}
and no other arrays are added when m => 3, so we have:
g #=> {3=>[[1, 2]], 4=>[[2, 2]]}
The statement
g.update({ n=>[i]*k }) if i*k == n
#=> g.update({ 7=>[2,2,2] }) if 6 == 7
adds the element 7=>[2,2,2] to the hash g if the sum of its elements equals n, which it does not.
We now fold g into h, using Hash#update (aka Hash#merge!):
h.update(g) { |k,ov,nv| ov+nv }
#=> {}.update({3=>[[1, 2]], 4=>[[2, 2]]} { |k,ov,nv| ov+nv }
#=> {1=>[[1]], 2=>[[1, 1]], 3=>[[1, 2]], 4=>[[2, 2]]}
Now h contains all the arrays (values) whose keys are the array totals, comprised of the integers 1 and 2, which have at most 3 elements and sum to at most 7, excluding those arrays with fewer than 3 elements which cannot sum to 7 when integers greater than two are added.
The operations performed are as follows:
i m j f
h #=> { 1=>[[1]], 2=>[[1,1]] }
2 1 1 {3=>[[1, 2]]}
2 1 2 {3=>[[1, 2]]}
2 2 1 {3=>[[1, 2]], 4=>[[2, 2]]}
{3=>[[1, 2]], 4=>[[2, 2]]}
3 1 1 {}
3 1 2 {}
3 1 3 {}
3 1 4 {7=>[[2, 2, 3]]}
3 2 1 {7=>[[2, 2, 3], [1, 3, 3]]}
g before g.update: {7=>[[2, 2, 3], [1, 3, 3]]}
g after g.update: {7=>[[2, 2, 3], [1, 3, 3]]}
h after h.update(g): {1=>[[1]],
2=>[[1, 1]],
3=>[[1, 2]],
4=>[[2, 2]],
7=>[[2, 2, 3], [1, 3, 3]]}
4 1 1 {}
4 1 2 {}
4 1 3 {7=>[[1, 2, 4]]}
g before g.update: {7=>[[1, 2, 4]]}
g after g.update: {7=>[[1, 2, 4]]}
h after h.update(g): {1=>[[1]],
2=>[[1, 1]],
3=>[[1, 2]],
4=>[[2, 2]],
7=>[[2, 2, 3], [1, 3, 3], [1, 2, 4]]}
5 1 1 {}
5 1 2 {7=>[[1, 1, 5]]}
g before g.update: {7=>[[1, 1, 5]]}
g after g.update: {7=>[[1, 1, 5]]}
h after h.update(g): {1=>[[1]],
2=>[[1, 1]],
3=>[[1, 2]],
4=>[[2, 2]],
7=>[[2, 2, 3], [1, 3, 3], [1, 2, 4], [1, 1, 5]]}
And lastly,
h[n].select { |a| a.size == k }
#=> h[7].select { |a| a.size == 3 }
#=> [[2, 2, 3], [1, 3, 3], [1, 2, 4], [1, 1, 5]]
#Cary's answer is very in-depth and impressive, but it appears to me that there is a much more naive solution, which proved to be much more efficient as well - good old recursion:
def combos(n,k)
if k == 1
return [n]
end
(1..n-1).flat_map do |i|
combos(n-i,k-1).map { |r| [i, *r].sort }
end.uniq
end
This solution simply reduces the problem each level by taking decreasing the target sum by each number between 1 and the previous target sum, while reducing k by one. Now make sure you don't have duplicates (by sort and uniq) - and you have your answer...
This is great for k < 5, and is much faster than Cary's solution, but as k gets larger, I found that it makes much too many iterations, sort and uniq took a very big toll on the calculation.
So I made sure that won't be needed, by making sure I get only sorted answers - each recursion should check only numbers larger than those already used:
def combos(n,k,min = 1)
if n < k || n < min
return []
end
if k == 1
return [n]
end
(min..n-1).flat_map do |i|
combos(n-i,k-1, i).map { |r| [i, *r] }
end
end
This solution is on par with Cary's on combos(100, 7):
user system total real
My Solution 2.570000 0.010000 2.580000 ( 2.695615)
Cary's 2.590000 0.000000 2.590000 ( 2.609374)
But we can do better: caching! This recursion does many calculations again and again, so caching stuff we already did will save us a lot of work when dealing with long sums:
def combos(n,k,min = 1, cache = {})
if n < k || n < min
return []
end
cache[[n,k,min]] ||= begin
if k == 1
return [n]
end
(min..n-1).flat_map do |i|
combos(n-i,k-1, i, cache).map { |r| [i, *r] }
end
end
end
This solution is mighty fast and passes Cary's solution for large n by light-years:
Benchmark.bm do |bm|
bm.report('Uri') { combos(1000, 3) }
bm.report('Cary') { combos_cary(1000, 3) }
end
user system total real
Uri 0.200000 0.000000 0.200000 ( 0.214080)
Cary 7.210000 0.000000 7.210000 ( 7.220085)
And is on par with k as high as 9, and I believe it is still less complicated than his solution.
You want the number of integer partitions of n into exactly k summands. There is a (computationally) somewhat ugly recurrence for that number.
The idea is this: let P(n,k) be the number of ways to partition n into k nonzero summands; then P(n,k) = P(n-1,k-1) + P(n-k,k). Proof: every partition either contains a 1 or it doesn't contain a 1 as one of the summands. The first case P(n-1,k-1) calculates the number of cases where there is a 1 in the sum; take that 1 away from the sum and partition the remaining n-1 into the now available k-1 summands. The second case P(n-k,k) considers the case where every summand is strictly greater than 1; to do that, reduce all of the k summands by 1 and recurse from there. Obviously, P(n,1) = 1 for all n > 0.
Here's a link that mentions that probably, no closed form is known for general k.
def peel array
output = []
while ! array.empty? do
output << array.shift
mutate! array
end
output.flatten
end
I have not included the mutate! method, because I am only interested in removing the output variable. The mutate! call is important because we cannot iterate over the array using each because array is changing.
EDIT: I am getting an array as output, which is what I want. The method works correctly, but I think there is a way to collect the array.shift values without using a temp variable.
EDIT #2: OK, here is the mutate! method and test case:
def mutate! array
array.reverse!
end
a = (1..5).to_a
peel( a ).should == [ 1, 5, 2, 4, 3 ]
It doesn't matter if peel modifies array. I guess it should be called peel!. Yes, mutate! must be called after each element is removed.
All this reversing makes me dizzy.
def peel(array)
indices = array.size.times.map do |i|
i = -i if i.odd?
i = i/2
end
array.values_at(*indices) # indices will be [0, -1, 1, -2, 2] in the example
end
a = (1..5).to_a
p peel(a) #=>[1, 5, 2, 4, 3]
Another approach:
def peel(array)
mid = array.size/2
array[0..mid]
.zip(array[mid..-1].reverse)
.flatten(1)
.take(array.size)
end
Usage:
peel [1,2,3,4,5,6]
#=> [1, 6, 2, 5, 3, 4]
peel [1,2,3,4,5]
#=> [1, 5, 2, 4, 3]
Here's a way using parallel assignment:
def peel array
n = array.size
n.times {|i| (n-2-2*i).times {|j| array[n-1-j], array[n-2-j] = array[n-2-j], array[n-1-j]}}
array
end
peel [1,2,3,4,5] # => [1,5,2,4,3]
peel [1,2,3,4,5,6] # => [1,6,2,5,3,4]
What I'm doing here is a series of pairwise exchanges. By way of example, for [1,2,3,4,5,6], the first 6-2=4 steps (6 being the size of the array) alter the array as follows:
[1,2,3,4,6,5]
[1,2,3,6,4,5]
[1,2,6,3,4,5]
[1,6,2,3,4,5]
The 1, 6 and the 2 are in now the right positions. We repeat these steps, but this time only 6-4=2 times, to move the 5 and 3 into the correct positions:
[1,6,2,3,5,4]
[1,6,2,5,3,4]
The 4 is pushed to the end, it's correct position, so we are finished.
I was trying to see how splat operator worked with range in Ruby. To do so ran the below code in my IRB:
*a = (1..8)
#=> 1..8
When the above is fine, what happened with below? means why a gives []?
*a,b = (1..8)
#=> 1..8
b
#=> 1..8
a
#=> []
means why b gives []?
a,*b = (1..8)
#=> 1..8
a
#=> 1..8
b
#=> []
What precedence took place in the below Rvalues ?
a,*b = *(2..8),*3,*5
# => [2, 3, 4, 5, 6, 7, 8, 3, 5]
b
# => [3, 4, 5, 6, 7, 8, 3, 5]
a
# => 2
Here is another try to the splat operator(*) :-
While I know that in parallel assignment we couldn't use multiple splatted variable, but why not the same when splat is used with Rvalues?
*a,*b = [1,2,3,4,5]
SyntaxError: (irb):1: syntax error, unexpected tSTAR
*a,*b = [1,2,3,4,5]
^
from /usr/bin/irb:12:in `<main>'
The above is as expected.
a = *2,*3,*5
#=> [2, 3, 5]
But couldn't understand the above.
I think of parallel assignment as setting an array of variables equal to another array with pattern matching.
One point is that a range is a single value until you convert it to an array or splat it. For instance [1..5] which is a one element array of the range 1..5 and not [1,2,3,4,5]. To get the array of ints you need to do (1..5).to_a or [*(1..5)]
The first one i think is the trickiest. If the splatted var is assigned to one element, the var itself must be a one-element array:
*a = 5
a
# => [ 5 ]
For the next two, splat takes 0 or more not already assigned values into an array. So the following makes sense:
*a, b = (1..8)
is like
*a, b = "hey"
which is like
*a, b = [ "hey" ]
so *a is [] and b is "hey" and by the same logic that if *a is nothing, a must be an empty array. Same idea for
a, *b = (1..5)
For the next one, the range is splatted, so the assignment makes a lot of sense again:
[*(2..4), 9, 5]
# => [2, 3, 4, 9, 5]
And parallel assignment with a splat again. Next one is similar:
[*3, *4, *5]
# => [3, 4, 5]
So that's like
a = 3, 4, 5
which is like
a = [3, 4, 5]
splat has a very low precedence, almost anything will be executed earlier than the splat.
The code is splatting but the result is thrown away: b = *a = (1..8); p b #=> [1, 2, 3, 4, 5, 6, 7, 8]
Array#drop removes the first n elements of an array. What is a good way to remove the last m elements of an array? Alternately, what is a good way to keep the middle elements of an array (greater than n, less than m)?
This is exactly what Array#pop is for:
x = [1,2,3]
x.pop(2) # => [2,3]
x # => [1]
You can also use Array#slice method, e.g.:
[1,2,3,4,5,6].slice(1..4) # => [2, 3, 4, 5]
or
a = [1,2,3,4,5,6]
a.take 3 # => [1, 2, 3]
a.first 3 # => [1, 2, 3]
a.first a.size - 1 # to get rid of the last one
The most direct opposite of drop (drop the first n elements) would be take, which keeps the first n elements (there's also take_while which is analogous to drop_while).
Slice allows you to return a subset of the array either by specifying a range or an offset and a length. Array#[] behaves the same when passed a range as an argument or when passed 2 numbers
this will get rid of last n elements:
a = [1,2,3,4,5,6]
n = 4
p a[0, (a.size-n)]
#=> [1, 2]
n = 2
p a[0, (a.size-n)]
#=> [1, 2, 3, 4]
regard "middle" elements:
min, max = 2, 5
p a.select {|v| (min..max).include? v }
#=> [2, 3, 4, 5]
I wanted the return value to be the array without the dropped elements. I found a couple solutions here to be okay:
count = 2
[1, 2, 3, 4, 5].slice 0..-(count + 1) # => [1, 2, 3]
[1, 2, 3, 4, 5].tap { |a| a.pop count } # => [1, 2, 3]
But I found another solution to be more readable if the order of the array isn't important (in my case I was deleting files):
count = 2
[1, 2, 3, 4, 5].reverse.drop count # => [3, 2, 1]
You could tack another .reverse on there if you need to preserve order but I think I prefer the tap solution at that point.
You can achieve the same as Array#pop in a non destructive way, and without needing to know the lenght of the array:
a = [1, 2, 3, 4, 5, 6]
b = a[0..-2]
# => [1, 2, 3, 4, 5]
n = 3 # if we want drop the last n elements
c = a[0..-(n+1)]
# => [1, 2, 3]
Array#delete_at() is the simplest way to delete the last element of an array, as so
arr = [1,2,3,4,5,6]
arr.delete_at(-1)
p arr # => [1,2,3,4,5]
For deleting a segment, or segments, of an array use methods in the other answers.
You can also add some methods
class Array
# Using slice
def cut(n)
slice(0..-n-1)
end
# Using pop
def cut2(n)
dup.tap{|x| x.pop(n)}
end
# Using take
def cut3(n)
length - n >=0 ? take(length - n) : []
end
end
[1,2,3,4,5].cut(2)
=> [1, 2, 3]