Pré-visualização50 páginas

constant into a Roman number; this lets you issue a statement like XI + IX, and get XX as the result (http://www.rubyquiz.com/ quiz22.html) 2.15 Generating a Sequence of Numbers Problem You want to iterate over a (possibly infinite) sequence of numbers the way you can iterate over an array or a range. Solution Write a generator function that yields each number in the sequence. def fibonacci(limit = nil) seed1 = 0 seed2 = 1 while not limit or seed2 <= limit yield seed2 seed1, seed2 = seed2, seed1 + seed2 2.15 Generating a Sequence of Numbers | 79 end end fibonacci(3) { |x| puts x } # 1 # 1 # 2 # 3 fibonacci(1) { |x| puts x } # 1 # 1 fibonacci { |x| break if x > 20; puts x } # 1 # 1 # 2 # 3 # 5 # 8 # 13 Discussion A generator for a sequence of numbers works just like one that iterates over an array or other data structure. The main difference is that iterations over a data structure usually have a natural stopping point, whereas most common number sequences are infinite. One strategy is to implement a method called each that yields the entire sequence. This works especially well if the sequence is finite. If not, it\u2019s the responsibility of the code block that consumes the sequence to stop the iteration with the break keyword. Range#each is an example of an iterator over a finite sequence, while Prime#each enu- merates the infinite set of prime numbers. Range#each is implemented in C, but here\u2019s a (much slower) pure Ruby implementation for study. This code uses self.begin and self.end to call Range#begin and Range#end, because begin and end are reserved words in Ruby. class Range def each_slow x = self.begin while x <= self.end yield x x = x.succ end end end (1..3).each_slow {|x| puts x} # 1 # 2 # 3 80 | Chapter 2: Numbers The other kind of sequence generator iterates over a finite portion of an infinite sequence. These are methods like Fixnum#upto and Fixnum#step: they take a start and/ or an end point as input, and generate a finite sequence within those boundaries. class Fixnum def double_upto(stop) x = self until x > stop yield x x = x * 2 end end end 10.double_upto(50) { |x| puts x } # 10 # 20 # 40 Most sequences move monotonically up or down, but it doesn\u2019t have to be that way: def oscillator x = 1 while true yield x x *= -2 end end oscillator { |x| puts x; break if x.abs > 50; } # 1 # -2 # 4 # -8 # 16 # -32 # 64 Though integer sequences are the most common, any type of number can be used in a sequence. For instance, Float#step works just like Integer#step: 1.5.step(2.0, 0.25) { |x| puts x } # => 1.5 # => 1.75 # => 2.0 Float objects don\u2019t have the resolution to represent every real number. Very small differences between numbers are lost. This means that some Float sequences you might think would go on forever will eventually end: def zeno(start, stop) distance = stop - start travelled = start while travelled < stop and distance > 0 yield travelled distance = distance / 2.0 travelled += distance 2.16 Generating Prime Numbers | 81 end end steps = 0 zeno(0, 1) { steps += 1 } steps # => 54 See Also \u2022 Recipe 1.16, \u201cGenerating a Succession of Strings\u201d \u2022 Recipe 2.16, \u201cGenerating Prime Numbers,\u201d shows optimizations for generating a very well-studied number sequence \u2022 Recipe 4.1, \u201cIterating Over an Array\u201d \u2022 Chapter 7 has more on this kind of generator method 2.16 Generating Prime Numbers Problem You want to generate a sequence of prime numbers, or find all prime numbers below a certain threshold. Solution Instantiate the Prime class to create a prime number generator. Call Prime#succ to get the next prime number in the sequence. require 'mathn' primes = Prime.new primes.succ # => 2 primes.succ # => 3 Use Prime#each to iterate over the prime numbers: primes.each { |x| puts x; break if x > 15; } # 5 # 7 # 11 # 13 # 17 primes.succ # => 19 Discussion Because prime numbers are both mathematically interesting and useful in crypto- graphic applications, a lot of study has been lavished on them. Many algorithms have been devised for generating prime numbers and determining whether a number is prime. The code in this recipe walks a line between efficiency and ease of implementation. 82 | Chapter 2: Numbers The best-known prime number algorithm is the Sieve of Eratosthenes, which finds all primes in a certain range by iterating over that range multiple times. On the first pass, it eliminates every even number greater than 2, on the second pass every third num- ber after 3, on the third pass every fifth number after 5, and so on. This implementa- tion of the Sieve is based on a sample program packaged with the Ruby distribution: def sieve(max=100) sieve = [] (2..max).each { |i| sieve[i] = i } (2..Math.sqrt(max)).each do |i| (i*i).step(max, i) { |j| sieve[j] = nil } if sieve[i] end sieve.compact end sieve(10) # => [2, 3, 5, 7] sieve(100000).size # => 9592 The Sieve is a fast way to find the primes smaller than a certain number, but it\u2019s memory-inefficient and it\u2019s not suitable for generating an infinite sequence of prime numbers. It\u2019s also not very compatible with the Ruby idiom of generator methods. This is where the Prime class comes in. A Prime object stores the current state of one iteration over the set of primes. It con- tains all information necessary to calculate the next prime number in the sequence. Prime#each repeatedly calls Prime#succ and yields it up to whatever code block was passed in. Ruby 1.9 has an efficient implementation of Prime#each, but Ruby 1.8 has a very slow implementation. The following code is based on the 1.9 implementation, and it illustrates many of the simple tricks that drastically speed up algorithms that find or use primes. You can use this code, or just paste the code from Ruby 1.9\u2019s mathn.rb into your 1.8 program. The first trick is to share a single list of primes between all Prime objects by making it a class variable. This makes it much faster to iterate over multiple Prime instances, but it also uses more memory because the list of primes will never be garbage-collected. We initialize the list with the first few prime numbers. This helps early performance a little bit, but it\u2019s mainly to get rid of edge cases. The class variable @@check_next tracks the next number we think might be prime. require 'mathn' class Prime @@primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101] @@check_next = 103 end 2.16 Generating Prime Numbers | 83 A number is prime if it has no factors: more precisely, if it has no prime factors between 2 and its square root. This code uses the list of prime numbers not only as a cache, but as a data structure to help find larger prime numbers. Instead of checking all the possible factors of a number, we only need to check some of the prime factors. To avoid calculating square roots, we have @@limit track the largest prime number less than the square root of @@check_next. We can decide when to increment it by calculating squares instead of square roots: class Prime # @@primes[3] < sqrt(@@check_next) < @@primes[4] @@limit = 3 # sqrt(121) == @@primes[4] @@increment_limit_at = 121 end Now we need a new implementation of Prime#succ. Starting from @@check_next, the new implementation iterates over numbers until it finds one that\u2019s prime, then returns the prime number. But it doesn\u2019t iterate over the numbers one at a time: we can do better than that. It skips even numbers and numbers divisible by