I'm trying to write a method that returns the nth prime number.
I've worked out a solution but the problem is in my method. I create a large array of numbers that seems to process super slow. (1..104729).to_a to be exact. I chose 104729 because the max n can be is 10000 and the 10000th integer is 104729. I'm looking for a way to optimize my method.
Is 104729 is too large a value? Is there a way to write this so that I'm not creating a large array?
Here's the method:
def PrimeMover(num)
def is_prime(x)
i = 0
nums = (2..x).to_a
while nums[i] < nums.max
if x % nums[i] != 0
i += 1
else
return false
end
end
return true
end
primes_arr = (3..104729).to_a.select {|y| is_prime(y)}
primes_arr[num]
end
require "prime"
def find_prime(nth)
Prime.take(nth).last
end
Combine Ruby's built-in prime library, and a lazy enumerator for performance:
require 'prime'
(1...100_000).lazy.select(&:prime?).take(100).to_a
Or simply, as highlighted by Arturo:
Prime.take(100)
You can use Ruby's built in #prime? method, which seems pretty efficient.
The code:
require 'prime'
primes_arr = (3..104729).to_a.select &:prime?
runs in 2-3 seconds on my machine, which I find somewhat acceptable.
If you need even better performance or if you really need to write your own method, try implementing the Sieve of Erathostenes. Here are some Ruby samples of that: http://rosettacode.org/wiki/Sieve_of_Eratosthenes#Ruby
Here's an optimal a trial division implementation of is_prime without relying on the Prime class:
A prime number is a whole number divisible only by 1 and itself, and 1 is not prime. So we want to know if x divides into anything less than x and greater than 1. So we start the count at 2, and we end at x - 1.
def prime?(x)
return false if x < 2
2.upto(x - 1) do |n|
return false if (x % n).zero?
end
true
end
As soon as x % n has a remainder, we can break the loop and say this number is not prime. This saves you from looping over the entire range. If all the possible numbers were exhausted, we know the number is prime.
This is still not optimal. For that you would need a sieve, or a different detection algorithm to trial division. But it's a big improvement on your code. Taking the nth up to you.
Related
I am doing a ruby problem that wants a method to find all divisors of a number except itself with the output being a sorted array. If the number is prime, list that it is prime.
I am currently trying to teach myself recursion. Simple recursive problems like finding the factorial of a number is pretty basic to understand but I wanted to know if this particular problem could be done recursively. It seems it fits the criteria of one that could but I could not figure it out.
Example n = 15, divisors besides itself are [3,5].
My code that solved the problem.
require 'prime'
def divisors(n)
return "#{n} is prime" if Prime.prime?(n)
x = n/2
arr = []
until x == 1
arr << x if n % x == 0
x -= 1
end
arr.sort
end
Any help doing this recursively would be great or just letting me know it's not a problem that can be done this way would be helpful too.
def divisors(n, x=nil)
return "#{n} is prime" if Prime.prime?(n)
x ||= n/2
arr = []
return arr if x == 1
if n % x == 0
arr << x
end
(arr.concat divisors(n, x - 1)).sort
end
The function is refactored to handle three things:
the initial call (x ||= /2)
base cases (early returns)
iteration logic done through recursion.
An important thing is that the variable which changes during the iteration (x) is placed as a parameter for the method (with a default value, so it can essentially be used as a private parameter)
By the way, I personally found learning Elixir very helpful in understanding recursion. With pattern matching and multiple functional clauses, the initial call, base case, and iteration can be split into their own methods.
I am trying to write a code for the following problem:
Input
The input begins with the number t of test cases in a single line (t<=10). In each of the next t lines there are two numbers m and n (1 <= m <= n <= 1000000000, n-m<=100000) separated by a space.
Output
For every test case print all prime numbers p such that m <= p <= n, one number per line, test cases separated by an empty line.
Sample Input:
2
1 10
3 5
Sample Output:
2
3
5
7
3
5
My code:
def prime?(number)
return false if number == 1
(2..number-1).each do |n|
return false if number % n == 0
end
true
end
t = gets.strip.to_i
for i in 1..t
mi, ni = gets.strip.split(' ')
mi = mi.to_i
ni = ni.to_i
i = mi
while i <= ni
puts i if prime?(i)
i += 1
end
puts "\n"
end
The code is running fine, only problem I am having is that it is taking a lot of time when run against big input ranges as compared to other programming languages.
Am I doing something wrong here? Can this code be further optimized for faster runtime?
I have tried using a for loop, normal loop, creating an array and then printing it.
Any suggestions.
Ruby is slower than some other languages, depending on what language you compare it to; certainly slower than C/C++. But your problem is not the language (although it influences the run-time behavior), but your way of finding primes. There are many better algorithms for finding primes, such as the Sieve of Eratosthenes or the Sieve of Atkin. You might also read the “Generating Primes” page on Wikipedia and follow the links there.
By the way, for the Sieve of Eratosthenes, there is even a ready-to-use piece of code on Stackoverflow. I'm sure a little bit of googling will turn up implementations for other algorithms, too.
Since your problem is finding primes within a certain range, this is the Sieve of Eratosthenes code found at the above link modified to suit your particular problem:
def better_sieve_upto(first, last)
sieve = [nil, nil] + (2..last).to_a
sieve.each do |i|
next unless i
break if i*i > last
(i*i).step(last, i) {|j| sieve[j] = nil }
end
sieve.reject {|i| !i || i < first}
end
Note the change from "sieve.compact" to a complexer "sieve.reject" with a corresponding condition.
Return true if the number is 2, false if the number is evenly divisible by 2.
Start iterating at 3, instead of 2. Use a step of two.
Iterate up to the square root of the number, instead of the number minus one.
def prime?(number)
return true if number == 2
return false if number <= 1 or number % 2 == 0
(3..Math.sqrt(number)).step(2) do |n|
return false if number % n == 0
end
true
end
This will be much faster, but still not very fast, as #Technation explains.
Here's how to do it using the Sieve of Eratosthenes built into Ruby. You'll need to precompute all the primes up to the maximum maximum, which will be very quick, and then select the primes that fall within each range.
require 'prime'
ranges = Array.new(gets.strip.to_i) do
min, max = gets.strip.split.map(&:to_i)
Range.new(min, max)
end
primes = Prime.each(ranges.map(&:max).max, Prime::EratosthenesGenerator.new)
ranges.each do |range|
primes.each do |prime|
next if prime < range.min
break if prime > range.max
puts prime
end
primes.rewind
puts "\n"
end
Here's how the various solutions perform with the range 50000 200000:
Your original prime? function: 1m49.639s
My modified prime? function: 0m0.687s
Prime::EratosthenesGenerator: 0m0.221s
The more ranges being processed, the faster the Prime::EratosthenesGenerator method should be.
I am trying to solve a problem from http://www.beatmycode.com/challenge/5/take, and I wrote a script:
def is_prime?(num)
(2...num).each do |divisor|
return false if num % divisor == 0
end
true
end
def circular_prime(num)
circular_primes = []
if num < 2
return false
end
(2..num-1).each do |number|
is_prime?(number)
result = [number.to_s]
(0..number.to_s.size-2).each do |x|
var = result.last.split('')
result << var.unshift(var.pop).join if var.uniq.size != 1
end
circular_primes << result if result.all?{ |x| is_prime? x.to_i }
end
end
When I tested it with 10,100,1000, and 10_000, the script executed very fast, but when I tested with 100_000, the shell displayed an Interrupt error. Where is the weak point, and how can I fix it?
Your is_prime? method is too slow. Some minor improvement could be:
You don't have to test divisor all the way upto num, the square root of
num is enough.
You can skip all even numbers since 2 is the only even prime
number.
However, this is still not good enough because the algorithm is slow. Since you need to generate prime numbers among consecutive integers, consider using Sieve of Eratosthenes.
Of course there's prime standard library but I assume you want to do this manually.
This script's time complexity is O(n^2), so it's expected that it will take long time to finish for big input. Try using https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes to find primes and remembering which number is a prime in some array.
I have this code that seems to be working for number 6-8 digits, in a normal time period.
When I enter bigger values the amount get ridiculously bug. It takes more than 4 hours to complete.
Here is my code.
#Fermat's factorization method
def get_largest_prime n
t=(Math.sqrt(n)+1).floor
k=0
prime_numbers=[]
while (t+k)<n
element = (t+k)**2-n
if is_integer? Math.sqrt(element)
#store prime numbers
prime_numbers << t+k+Math.sqrt(element)
prime_numbers << t+k-Math.sqrt(element)
#puts "Prime Factors of #{n} are: #{t+k+Math.sqrt(element)} and #{t+k-Math.sqrt(element)}"
end
k+=1
end
puts "Prime Factors: "+prime_numbers.to_s
end
#making sure 450.0 is 450, for example.
def is_integer? number
number.to_i == number ? true : false
end
get_largest_prime 600851475143
Running this will take more than 4 hours.
But running it for value ' 600851' for example or ' 60085167' does not take a lot of time. Any help ?
First note that Fermat factorisation doesn't give you prime factors in general.
Then, you run it until t+k >= n, that means you run the while loop n - t times, since t is roughly sqrt(n), that is an O(n) algorithm. For a largish n like 600851475143 (about 6*10^11), that is bound to take long.
You need to change the algorithm. When you have found a pair of divisors (both larger than 1), factorise them both recursively. If the smaller of the found factors is 1, that is a prime factor.
Doing that (forgive the bad style, I barely know ruby):
#Fermat's factorization method
def get_largest_prime n
t=(Math.sqrt(n)+1).floor
k=0
prime_numbers=[]
while (t+k)<n
element = (t+k)**2-n
if is_integer? Math.sqrt(element)
#store prime numbers
a = t+k+Math.sqrt(element)
b = t+k-Math.sqrt(element)
if b == 1
prime_numbers << a
break
end
prime_numbers += get_largest_prime a
prime_numbers += get_largest_prime b
break
#puts "Prime Factors of #{n} are: #{t+k+Math.sqrt(element)} and #{t+k-Math.sqrt(element)}"
end
k+=1
end
return prime_numbers
end
#making sure 450.0 is 450, for example.
def is_integer? number
number.to_i == number ? true : false
end
a = get_largest_prime 600851475143
puts "Prime Factors: "+a.to_s
solves the given problem quickly.
However, it will still take a long time for numbers that have no divisors close to the square root.
The standard factorisation by trial division has much better worst-case behaviour (O(sqrt(n) worst case). A mixed approach can be slightly faster than pure trial division, though.
Two effects here:
1) When an integer gets larger than 2**31 in Ruby, it uses a different, and slower, representation
2) There are no known factorisation algorithms that don't eventually perform badly once the number gets large enough - technically they all get slower worse than any polynomial of the (number of digits of) the number you want to factorise.
You could speed things up by using
Math.sqrt(element)
less. Assign result of it to a variable, before all the tests. Note this will not "fix" your problem. Ultimately it won't run fast enough above a certain number - even if you transferred everything to C (although you might squeeze out a couple of extra digits before C got slow)
Possibly, you can try the code below.
def prime_factor limit
(2..Math.sqrt(limit).to_i).inject([]) do |memo, var|
memo << var if limit % var == 0 and !memo.any? {|d| var % d == 0}
memo
end
end
prime_result = prime_factor 600851475143
puts prime_result.max
The less loops you use, the faster your code will run ;-) (less cpu cycles). Try to do everything recursively like here on this program that finds the largest prime factor
I am trying to create a program that will test whether a value is prime, but I don't know how. This is my code:
class DetermineIfPrime
def initialize (nth_value)
#nth_value = nth_value
primetest
end
def primetest
if Prime.prime?(#nth_value)
puts ("#{#nth_value} is prime")
else
puts ("This is not a prime number.")
end
rescue Exception
puts ("#{$!.class}")
puts ("#{$!}")
end
end
And every time I run that it returns this.
NameError
uninitialized constant DetermineIfPrime::Prime
I tried other ways to do the job, but I think this is the closest I can get.
I also tried this:
class DetermineIfPrime
def initialize (nth_value)
#nth_value = nth_value
primetest
end
def primetest
for test_value in [2, 3, 5, 7, 9, 11, 13] do
if (#nth_value % test_value) == 0
puts ("#{#nth_value} is not divisible by #{test_value}")
else
puts ("This is not a prime number since this is divisible by #{test_value}")
break
end
end
end
end
Or am I just doing something wrong?
Ruby has built in method to check if number is prime or not.
require 'prime'
Prime.prime?(2) #=> true
Prime.prime?(4) #=> false
def is_prime?(num)
return false if num <= 1
Math.sqrt(num).to_i.downto(2).each {|i| return false if num % i == 0}
true
end
First, we check for 0 and 1, as they're not prime. Then we basically just check every number less than num to see if it divides. However, as explained here, for every factor greater than the square root of num, there's one that's less, so we only look between 2 and the square root.
Update
def is_prime?(num)
return if num <= 1
(2..Math.sqrt(num)).none? { |i| (num % i).zero? }
end
The error you are getting is because you haven't required Primein your code, You need to do require Prime in your file.
One cool way I found here, to check whether a number is prime or not is following:
class Fixnum
def prime?
('1' * self) !~ /^1?$|^(11+?)\1+$/
end
end
10.prime?
From an algorithmic standpoint, checking if a number is prime can be done by checking all numbers up to and including (rounding down to previous integer) said number's square root.
For example, checking if 100 is prime involves checking everything up to 10.
Checking 99 means only going to 9.
** Another way to think about it **
Each factor has a pair (3 is a factor of 36, and 3's pair is 12).
The pair is on the other side of the square root (square root of 6 is 36, 3 < 6, 12 > 6).
So by checking everything until the square root (and not going over) ensures you check all possible factors.
You can make it quicker by having a list of prime numbers to compare, as you are doing. If you have a maximum limit that's reasonably small, you could just have a list of primes and do a direct lookup to see if that number is prime.
def is_prime?(num)
Math.sqrt(num).floor.downto(2).each {|i| return false if num % i == 0}
true
end
lol sorry for resurrecting a super old questions, but it's the first one that came up in google.
Basically, it loops through possible divisors, using the square root as the max number to check to save time on very large numbers.
In response to your question, while you can approach the problem by using Ruby's Prime I am going to write code to answer it on its own.
Consider that all you need to do is determine a factor that is smaller than the integer's square root. Any number larger than the integer's square root as a factor requires a second factor to render the number as the product. (e.g. square root of 15 is approx 3.8 so if you find 5 as a factor it is only a factor with the factor pair 3 and 5!!)
def isPrime?(num)
(2..Math.sqrt(num)).each { |i| return false if num % i == 0}
true
end
Hope that helps!!
(To first answer the question: yes, you are doing something wrong. As BLUEPIXY mentions, you need to put require 'prime' somewhere above the line that calls Prime.prime?. Typically on line 1.)
Now, a lot of answers have been given that don't use Prime.prime?, and I thought it might be interesting to benchmark some of them, along with a possible improvement of my own that I had in mind.
###TL;DR
I benchmarked several solutions, including a couple of my own; using a while loop and skipping even numbers performs best.
Methods tested
Here are the methods I used from the answers:
require 'prime'
def prime1?(num)
return if num <= 1
(2..Math.sqrt(num)).none? { |i| (num % i).zero? }
end
def prime2?(num)
return false if num <= 1
Math.sqrt(num).to_i.downto(2) {|i| return false if num % i == 0}
true
end
def prime3?(num)
Prime.prime?(num)
end
def prime4?(num)
('1' * num) !~ /^1?$|^(11+?)\1+$/
end
prime1? is AndreiMotinga's updated version. prime2? is his original version (with the superfluous each method removed). prime3? is Reboot's, using prime library. prime4? is Saurabh's regex version (minus the Fixnum monkey-patch).
A couple more methods to test
The improvement I had in mind was to leverage the fact that even numbers can't be prime, and leave them out of the iteration loop. So, this method uses the #step method to iterate over only odd numbers, starting with 3:
def prime5?(num)
return true if num == 2
return false if num <= 1 || num.even?
3.step(Math.sqrt(num).floor, 2) { |i| return false if (num % i).zero? }
true
end
I thought as well that it might be interesting to see how a "primitive" implementation of the same algorithm, using a while loop, might perform. So, here's one:
def prime6?(num)
return true if num == 2
return false if num <= 1 || num.even?
i = 3
top = Math.sqrt(num).floor
loop do
return false if (num % i).zero?
i += 2
break if i > top
end
true
end
Benchmarks
I did a simple benchmark on each of these, timing a call to each method with the prime number 67,280,421,310,721. For example:
start = Time.now
prime1? 67280421310721
puts "prime1? #{Time.now - start}"
start = Time.now
prime2? 67280421310721
puts "prime2? #{Time.now - start}"
# etc.
As I suspected I would have to do, I canceled prime4? after about 60 seconds. Presumably, it takes quite a bit longer than 60 seconds to assign north of 6.7 trillion '1''s to memory, and then apply a regex filter to the result — assuming it's possible on a given machine to allocate the necessary memory in the first place. (On mine, it would seem that there isn't: I went into irb, put in '1' * 67280421310721, made and ate dinner, came back to the computer, and found Killed: 9 as the response. That looks like a SignalException raised when the process got killed.)
The other results are:
prime1? 3.085434
prime2? 1.149405
prime3? 1.236517
prime5? 0.748564
prime6? 0.377235
Some (tentative) conclusions
I suppose that isn't really surprising that the primitive solution with the while loop is fastest, since it's probably closer than the others to what's going on under the hood. It is a bit surprising that it's three times faster than Prime.prime?, though. (After looking at the source code in the doc it is less so. There are lots of bells and whistles in the Prime object.)
AndreiMotinga's updated version is nearly three times as slow as his original, which suggests that the #none? method isn't much of a performer, at least in this context.
Finally, the regex version might be cool, but it certainly doesn't appear to have much practical value, and using it in a monkey-patch of a core class looks like something to avoid entirely.
If you are going to use any Prime functions you must include the Prime library. This problem can be solved without the use of the prime library however.
def isPrime?(num)
(2..Math.sqrt(num)).each { |i|
if num % i == 0 && i < num
return false
end
}
true
end
Something like this would work.
Try this
def prime?(num)
2.upto(Math.sqrt(num).ceil) do |i|
break if num%i==0
return true if i==Math.sqrt(num).ceil
end
return false
end
So most of the answers here are doing the same thing in slightly different ways which is one of the cool things about Ruby, but I'm a pretty new student (which is why I was looking this up in the first place) and so here's my version with comment explanations in the code:
def isprime n # starting with 2 because testing for a prime means you don't want to test division by 1
2.upto(Math.sqrt(n)) do |x| # testing up to the square root of the number because going past there is excessive
if n % x == 0
# n is the number being called from the program
# x is the number we're dividing by, counting from 2 up to the square root of the number
return false # this means the number is not prime
else
return true # this means the number is prime
end
end
end