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Ruby Cookbook by Leonard Richardson, Lucas Carlson

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Chapter 4. Arrays

Like all high-level languages, Ruby has built-in support for arrays, objects that contain ordered lists of other objects. You can use arrays (often in conjunction with hashes) to build and use complex data structures without having to define any custom classes.

An array in Ruby is an ordered list of elements. Each element is a reference to some object, the way a Ruby variable is a reference to some object. For convenience, throughout this book we usually talk about arrays as though the array elements were the actual objects, not references to the objects. Since Ruby (unlike languages like C) gives no way of manipulating object references directly, the distinction rarely matters.

The simplest way to create a new array is to put a comma-separated list of object references between square brackets. The object references can be predefined variables (my_var), anonymous objects created on the spot ('my string', 4.7, or MyClass.new), or expressions (a+b, object.method). A single array can contain references to objects of many different types:

	a1 = []                              # => []
	a2 = [1, 2, 3]                       # => [1, 2, 3]
	a3 = [1, 2, 3, 'a', 'b', 'c', nil]   # => [1, 2, 3, "a", "b", "c", nil]

	n1 = 4
	n2 = 6
	sum_and_difference = [n1, n2, n1+n2, n1-n2]
	# => [4, 6, 10, -2]

If your array contains only strings, you may find it simpler to build your array by enclosing the strings in the w{} syntax, separated by whitespace. This saves you from having to write all those quotes and comma:

	%w{1 2 3}                            # => ["1", "2", "3"]
	%w{The rat sat
	   on the mat}
	# => ["The", "rat", "sat", "on", "the", "mat"]

The << operator is the simplest way to add a value to an array. Ruby dynamically resizes arrays as elements are added and removed.

	a = [1, 2, 3]                  # => [1, 2, 3]
	a << 4.0                       # => [1, 2, 3, 4.0]
	a << 'five'                    # => [1, 2, 3, 4.0, "five"]

An array element can be any object reference, including a reference to another array. An array can even contain a reference to itself, though this is usually a bad idea, since it can send your code into infinite loops.

	a = [1,2,3]                      # => [1, 2, 3]
	a << [4, 5, 6]                   # => [1, 2, 3, [4, 5, 6]]
	a << a                           # => [1, 2, 3, [4, 5, 6], […]]

As in most other programming languages, the elements of an array are numbered with indexes starting from zero. An array element can be looked up by passing its index into the array index operator []. The first element of an array can be accessed with a[0], the second with a[1], and so on.

Negative indexes count from the end of the array: the last element of an array can be accessed with a[-1], the second-to-last with a[-2], and so on. See Recipe 4.13 for more ways of using the array indexing operator.

The size of an array is available through the Array#size method. Because the index numbering starts from zero, the index of the last element of an array is the size of the array, minus one.

	a = [1, 2, 3, [4, 5, 6]]
	a.size                               # => 4
	a << a                               # => [1, 2, 3, [4, 5, 6], […]]
	a.size                               # => 5

	a[0]                                 # => 1
	a[3]                                 # => [4, 5, 6]
	a[3][0]                              # => 4
	a[3].size                            # => 3

	a[-2]                                # => [4, 5, 6]
	a[-1]                                # => [1, 2, 3, [4, 5, 6], […]]
	a[a.size-1]                          # => [1, 2, 3, [4, 5, 6], […]]

	a[-1][-1]                            # => [1, 2, 3, [4, 5, 6], […]]
	a[-1][-1][-1]                        # => [1, 2, 3, [4, 5, 6], […]]

All languages with arrays have constructs for iterating over them (even if it’s just a for loop). Languages like Java and Python have general iterator methods similar to Ruby’s, but they’re usually used for iterating over arrays. In Ruby, iterators are the standard way of traversing all data structures: array iterators are just their simplest manifestation.

Ruby’s array iterators deserve special study because they’re Ruby’s simplest and most accessible iterator methods. If you come to Ruby from another language, you’ll probably start off thinking of iterator methods as letting you treat aspects of a data structure “like an array.” Recipe 4.1 covers the basic array iterator methods, including ones in the Enumerable module that you’ll encounter over and over again in different contexts.

The Set class, included in Ruby’s standard library, is a useful alternative to the Array class for many basic algorithms. A Ruby set models a mathematical set: sets are not ordered, and cannot contain more than one reference to the same object. For more about sets, see Recipes 4.14 and 4.15.

4.1. Iterating Over an Array


You want to perform some operation on each item in an array.


Iterate over the array with Enumerable#each. Put into a block the code you want to execute for each item in the array.

	[1, 2, 3, 4].each { |x| puts x }
	# 1
	# 2
	# 3
	# 4

If you want to produce a new array based on a transformation of some other array, use Enumerable#collect along with a block that takes one element and transforms it:

	[1, 2, 3, 4].collect { |x| x ** 2 }             # => [1, 4, 9, 16]


Ruby supports for loops and the other iteration constructs found in most modern programming languages, but its prefered idiom is a code block fed to an method like each or collect.

Methods like each and collect are called generators or iterators: they iterate over a data structure, yielding one element at a time to whatever code block you’ve attached. Once your code block completes, they continue the iteration and yield the next item in the data structure (according to whatever definition of “next” the generator supports). These methods are covered in detail in Chapter 7.

In a method like each, the return value of the code block, if any, is ignored. Methods like collect take a more active role. After they yield an element of a data structure to a code block, they use the return value in some way. The collect method uses the return value of its attached block as an element in a new array.

Although commonly used in arrays, the collect method is actually defined in the Enumerable module, which the Array class includes. Many other Ruby classes (Hash and Range are just two) include the Enumerable methods; it’s a sort of baseline for Ruby objects that provide iterators. Though Enumerable does not define the each method, it must be defined by any class that includes Enumerable, so you’ll see that method a lot, too. This is covered in Recipe 9.4.

If you need to have the array indexes along with the array elements, use Enumerable#each_with_index.

	['a', 'b', 'c'].each_with_index do |item, index|
	  puts "At position #{index}: #{item}"
	# At position 0: a
	# At position 1: b
	# At position 2: c

Ruby’s Array class also defines several generators not seen in Enumerable . For instance , to iterate over a list in reverse order, use the reverse_each method:

	[1, 2, 3, 4]. 
reverse_each { |x| puts x }
	# 4
	# 3
	# 2
	# 1

Enumerable#collect has a destructive equivalent: Array# collect!, also known as Arary#map! (a helpful alias for Python programmers). This method acts just like collect, but instead of creating a new array to hold the return values of its calls to the code block, it replaces each item in the old array with the corresponding value from the code block. This saves memory and time, but it destroys the old array:

	array = ['a', 'b', 'c']
	array.collect! { |x| x.upcase }
	array                                # => ["A", "B", "C"]
	array.map! { |x| x.downcase }
	array                                # => ["a", "b", "c"]

If you need to skip certain elements of an array, you can use the iterator methods Range#step and Integer#upto instead of Array#each. These methods generate a sequence of numbers that you can use as successive indexes into an array.

	array = ['junk', 'junk', 'junk', 'val1', 'val2']
	3.upto(array.length-1) { |i| puts "Value #{array[i]}" }
	# Value val1
	# Value val2

	array = ['1', 'a', '2', 'b', '3', 'c']
	(0..array.length-1).step(2) do |i|
	  puts "Letter #{array[i]} is #{array[i+1]}"
	# Letter 1 is a
	# Letter 2 is b
	# Letter 3 is c

Like most other programming languages, Ruby lets you define for, while, and until loops—but you shouldn’t need them very often. The for construct is equivalent to each, whether it’s applied to an array or a range:

	for element in ['a', 'b', 'c']
	  puts element
	# a
	# b
	# c

	for element in (1..3)
	  puts element
	# 1
	# 2
	# 3

The while and until constructs take a boolean expression and execute the loop while the expression is true (while)or until it becomes true (until). All three of the following code snippets generate the same output:

	array = ['cherry', 'strawberry', 'orange']

	for index in (0…array.length)
	  puts "At position #{index}: #{array[index]}"

	index = 0
	while index < array.length
	  puts "At position #{index}: #{array[index]}"
	  index += 1

	index = 0
	until index == array.length
	  puts "At position #{index}: #{array[index]}"
	  index += 1

	# At position 0: cherry
	# At position 1: strawberry
	# At position 2: orange

These constructs don’t make for very idiomatic Ruby. You should only need to use them when you’re iterating over a data structure in a way that doesn’t already have an iterator method (for instance, if you’re traversing a custom tree structure). Even then, it’s more idiomatic if you only use them to define your own iterator methods.

The following code is a hybrid of each and each_reverse. It switches back and forth between iterating from the beginning of an array and iterating from its end.

	array = [1,2,3,4,5]
	new_array = []
	front_index = 0

	back_index = array.length-1
	while front_index <= back_index
	  new_array << array[front_index]
	  front_index += 1
	  if front_index <= back_index
	   new_array << array[back_index]
	    back_index -= 1
	new_array                            # => [1, 5, 2, 4, 3]

That code works, but it becomes reusable when defined as an iterator. Put it into the Array class, and it becomes a universally accessible way of doing iteration, the colleague of each and reverse_each:

	class Array
	 def each_from_both_sides
	    front_index = 0
	    back_index = self.length-1
	    while front_index <= back_index
	      yield self[front_index]
	      front_index += 1
	      if front_index <= back_index
	    yield self[back_index]
	        back_index -= 1

	new_array = []
	[1,2,3,4,5].each_from_both_sides { |x| new_array << x }
	new_array                            # => [1, 5, 2, 4, 3]

This “burning the candle at both ends” behavior can also be defined as a collecttype method: one which constructs a new array out of multiple calls to the attached code block. The implementation below delegates the actual iteration to the each_ from_both_sides method defined above:

	class Array
	  def collect_from_both_sides
	    new_array = []
	    each_from_both_sides { |x| new_array << yield(x) }
	    return new_array

	["ham", "eggs", "and"].collect_from_both_sides { |x| x.capitalize }
	# => ["Ham", "And", "Eggs"]

See Also

  • Chapter 7, especially Recipe 7.5, “Writing an Iterator Over a Data Structure,” and Recipe 7.9, “Looping Through Multiple Iterables in Parallel”

4.2. Rearranging Values Without Using Temporary Variables


You want to rearrange a number of variables, or assign the elements of an array to individual variables.


Use a single assignment statement. Put the destination variables on the left-hand side, and line each one up with a variable (or expression) on the right side.

A simple swap:

	a = 1
	b = 2
	a, b = b, a
	a                                  # => 2
	b                                  # => 1

A more complex rearrangement:

	a, b, c = :red, :green, :blue
	c, a, b = a, b, c
	a                                  # => :green
	b                                  # => :blue
	c                                  # => :red

You can split out an array into its components:

	array = [:red, :green, :blue]
	c, a, b = array
	a                                  # => :green
	b                                  # => :blue
	c                                  # => :red

You can even use the splat operator to extract items from the front of the array:

	a, b, *c = [12, 14, 178, 89, 90]
	a                                   # => 12
	b                                   # => 14
	c                                   # => [178, 89, 90]


Ruby assignment statements are very versatile. When you put a comma-separated list of variables on the left-hand side of an assignment statement, it’s equivalent to assigning each variable in the list the corresponding right-hand value. Not only does this make your code more compact and readable, it frees you from having to keep track of temporary variables when you swap variables.

Ruby works behind the scenes to allocate temporary storage space for variables that would otherwise be overwritten, so you don’t have to do it yourself. You don’t have to write this kind of code in Ruby:

	a, b = 1, 2
	x = a
	a = b
	b = x

The right-hand side of the assignment statement can get almost arbitrarily complicated:

	a, b = 5, 10
	a, b = b/a, a-1                      # => [2, 4]

	a, b, c = 'A', 'B', 'C'
	a, b, c = [a, b], { b => c }, a
	a                                    # => ["A", "B"]
	b                                    # => {"B"=>"C"}
	c                                    # => "A"

If there are more variables on the left side of the equal sign than on the right side, the extra variables on the left side get assigned nil. This is usually an unwanted side effect.

	a, b = 1, 2
	a, b = b
	a                                    # => 2
	b                                    # => nil

One final nugget of code that is interesting enough to mention even though it has no legitimate use in Ruby: it doesn’t save enough memory to be useful, and it’s slower than doing a swap with an assignment. It’s possible to swap two integer variables using bitwise XOR, without using any additional storage space at all (not even implicitly):

	a, b = rand(1000), rand(1000)        # => [595, 742]
	a = a ^ b                            # => 181
	b = b ^ a                            # => 595
	a = a ^ b                            # => 742

	[a, b]                               # => [742, 595]

In terms of the cookbook metaphor, this final snippet is a dessert—no nutritional value, but it sure is tasty.

4.3. Stripping Duplicate Elements from an Array


You want to strip all duplicate elements from an array, or prevent duplicate elements from being added in the first place.


Use Array#uniq to create a new array, based on an existing array but with no duplicate elements. Array#uniq! strips duplicate elements from an existing array.

	survey_results = [1, 2, 7, 1, 1, 5, 2, 5, 1]
	distinct_answers = survey_results.uniq        # => [1, 2, 7, 5]
	survey_results                                # => [1, 2, 7, 5]

To ensure that duplicate values never get into your list, use a Set instead of an array. If you try to add a duplicate element to a Set, nothing will happen.

	require 'set'
	survey_results = [1, 2, 7, 1, 1, 5, 2, 5, 1]
	distinct_answers = survey_results.to_set
	# => #<Set: {5, 1, 7, 2}>

	games = [["Alice", "Bob"], ["Carol", "Ted"],
	         ["Alice", "Mallory"], ["Ted", "Bob"]]
	players = games.inject(Set.new) { |set, game| game.each { |p| set << p }; set }
	# => #<Set: {"Alice", "Mallory", "Ted", "Carol", "Bob"}>

	players << "Ted"
	# => #<Set: {"Alice", "Mallory", "Ted", "Carol", "Bob"}>


The common element between these two solutions is the hash (see Chapter 5). Array#uniq iterates over an array, using each element as a key in a hash that it always checks to see if it encountered an element earlier in the iteration. A Set keeps the same kind of hash from the beginning, and rejects elements already in the hash. You see something that acts like an array, but it won’t accept duplicates. In either case, two objects are considered “duplicates” if they have the same result for ==.

The return value of Array#uniq is itself an array, and nothing prevents you from adding duplicate elements to it later on. If you want to start enforcing uniqueness in perpetuity, you should turn the array into a Set instead of calling uniq. Requiring the set library will define a new method Enumerable#to_set, which does this.

Array#uniq preserves the original order of the array (that is, the first instance of an object remains in its original location), but a Set has no order, because its internal implementation is a hash. To get array-like order in a Set, combine this recipe with Recipe 5.8 and subclass Set to use an OrderedHash:

	class OrderedSet < Set
	  def initialize
	    @hash ||= OrderedHash.new

Needing to strip all instances of a particular value from an array is a problem that often comes up. Ruby provides Array#delete for this task, and Array#compact for the special case of removing nil values.

	a = [1, 2, nil, 3, 3, nil, nil, nil, 5]
	a.compact                                # => [1, 2, 3, 3, 5]

	a                                        # => [1, 2, nil, nil, nil, nil, 5]

4.4. Reversing an Array


Your array is the wrong way around: the last item should be first and the first should be last.


Use reverse to create a new array with the items reversed. Internal subarrays will not themselves be reversed.

	[1,2,3].reverse                           # => [3, 2, 1]
	[1,[2,3,4],5].reverse                     # => [5, [2, 3, 4], 1]


Like many operations on basic Ruby types, reverse has a corresponding method, reverse!, which reverses an array in place:

	a = [1,2,3]
	a                                       # => [3, 2, 1]

Don’t reverse an array if you just need to iterate over it backwards. Don’t use a for loop either; the reverse_each iterator is more idiomatic.

See Also

  • Recipe 1.4, " Reversing a String by Words or Characters”

  • Recipe 4.1, “Iterating Over an Array,” talks about using Array#reverse_each to iterate over an array in reverse order

  • Recipe 4.2, “Rearranging Values Without Using Temporary Variables”

4.5. Sorting an Array


You want to sort an array of objects, possibly according to some custom notion of what “sorting” means.


Homogeneous arrays of common data types, like strings or numbers, can be sorted “naturally” by just calling Array#sort:

	[5.01, -5, 0, 5].sort                                # => [-5, 0, 5, 5.01] 
	["Utahraptor", "Ankylosaur", "Maiasaur"].sort
	# => ["Ankylosaur", "Maiasaur", "Utahraptor"]

To sort objects based on one of their data members, or by the results of a method call, use Array#sort_by. This code sorts an array of arrays by size, regardless of their contents:

	arrays = [[1,2,3], [100], [10,20]]  
	arrays.sort_by { |x| x.size }                    # => [[100], [10, 20], [1, 2, 3]]

To do a more general sort, create a code block that compares the relevant aspect of any two given objects. Pass this block into the sort method of the array you want to sort.

This code sorts an array of numbers in ascending numeric order, except that the number 42 will always be at the end of the list:

	[1, 100, 42, 23, 26, 10000].sort do |x, y|
	  x == 42 ? 1 : x <=> y
	# => [1, 23, 26, 100, 10000, 42]


If there is one “canonical” way to sort a particular class of object, then you can have that class implement the <=> comparison operator. This is how Ruby automatically knows how to sort numbers in ascending order and strings in ascending ASCII order: Numeric and String both implement the comparison operator.

The sort_by method sorts an array using a Schwartzian transform (see Recipe 4.6 for an in-depth discussion). This is the most useful customized sort, because it’s fast and easy to define. In this example, we use sort_by to sort on any one of an object’s fields.

	class Animal
	 attr_reader :name, :eyes, :appendages

	 def initialize(name, eyes, appendages)
	  @name, @eyes, @appendages = name, eyes, appendages

	 def inspect

	animals = [Animal.new("octopus", 2, 8),
	           Animal.new("spider", 6, 8),
	           Animal.new("bee", 5, 6),
	           Animal.new("elephant", 2, 4),
	           Animal.new("crab", 2, 10)]

	animals.sort_by { |x| x.eyes }
	# => [octopus, elephant, crab, bee, spider]

	animals.sort_by { |x| x.appendages }
	# => [elephant, bee, octopus, spider, crab]

If you pass a block into sort, Ruby calls the block to make comparisons instead of using the comparison operator. This is the most general possible sort, and it’s useful for cases where sort_by won’t work.

The comparison operator and a sort code block both take one argument: an object against which to compare self. A call to <=> (or a sort code block) should return–1 if self is “less than” the given object (and should therefore show up before it in a sorted list). It should return 1 if self is “greater than” the given object (and should show up after it in a sorted list), and 0 if the objects are "equal" (and it doesn’t matter which one shows up first). You can usually avoid remembering this by delegating the return value to some other object’s <=> implementation.

See Also

  • Recipe 4.6, “Ignoring Case When Sorting Strings,” covers the workings of the Schwartzian Transform

  • Recipe 4.7, “Making Sure a Sorted Array Stays Sorted”

  • Recipe 4.10, “Shuffling an Array”

  • If you need to find the minimum or maximum item in a list according to some criteria, don’t sort it just to save writing some code; see Recipe 4.11, “Getting the N Smallest Items of an Array,” for other options

4.6. Ignoring Case When Sorting Strings


When you sort a list of strings, the strings beginning with uppercase letters sort before the strings beginning with lowercase letters.

	list = ["Albania", "anteater", "zorilla", "Zaire"]
	# => ["Albania", "Zaire", "anteater", "zorilla"]

You want an alphabetical sort, regardless of case.


Use Array#sort_by. This is both the fastest and the shortest solution.

	list.sort_by { |x| x.downcase }
	# => ["Albania", "anteater", "Zaire", "zorilla"]


The Array#sort_by method was introduced in Recipe 4.5, but it’s worth discussing in detail because it’s so useful. It uses a technique called a Schwartzian Transform. This common technique is like writing the following Ruby code (but it’s a lot faster, because it’s implemented in C):

	list.collect { |s| [s.downcase, s] }. 
sort.collect { |subarray| subarray[1] }

It works like this: Ruby creates a new array containing two-element subarrays. Each subarray contains a value of String#downcase, along with the original string. This new array is sorted, and then the original strings (now sorted by their values for String#downcase) are recovered from the subarrays. String#downcase is called only once for each string.

A sort is the most common occurance of this pattern, but it shows up whenever an algorithm calls a particular method on the same objects over and over again. If you’re not sorting, you can’t use Ruby’s internal Schwartzian Transform, but you can save time by caching, or memoizing, the results of each distinct method call.

If you need to implement a Schwartzian Transform in Ruby, it’s faster to use a hash than an array:

	m = {}  
	list.sort { |x,y| (m[x] ||= x.downcase) <=> (m[y] ||= y.downcase) }

This technique is especially important if the method you need to call has side effects. You certainly don’t want to call such methods more than once!

See Also

  • The Ruby FAQ, question 9.15

  • Recipe 4.5, “Sorting an Array”

4.7. Making Sure a Sorted Array Stays Sorted


You want to make sure an array stays sorted, even as you replace its elements or add new elements to it.


Subclass Array and override the methods that add items to the array. The new implementations add every new item to a position that maintains the sortedness of the array.

As you can see below, there are a lot of these methods. If you can guarantee that a particular method will never be called, you can get away with not overriding it.

SortedArray < Array

	  def initialize(*args, &sort_by)
	    @sort_by = sort_by || Proc.new { |x,y| x <=> y }
	    sort! &sort_by

	  def insert(i, v)
	    # The next line could be further optimized to perform a
	    # binary search.
	    insert_before = index(find { |x| @sort_by.call(x, v) == 1 })
	    super(insert_before ? insert_before : -1, v)

	  def <<(v)
	    insert(0, v)

	  alias push <<
	  alias unshift <<

Some methods, like collect!, can modify the items in an array, taking them out of sort order. Some methods, like flatten!, can add new elements to strange places in an array. Rather than figuring out a way to implement these methods in a way that preserves the sortedness of the array, we’ll just let them run and then re-sort the array.[1]

	  ["collect!", "flatten!", "[]="].each do |method_name|
	    module_eval %{
	      def #{method_name}(*args)
	        sort! &@sort_by

	  def reverse!
	    #Do nothing; reversing the array would disorder it.

A SortedArray created from an unsorted array will end up sorted:

	a = SortedArray.new([3,2,1])         # => [1, 2, 3]


Many methods of Array are much faster on sorted arrays, so it’s often useful to expend some overhead on keeping an array sorted over time. Removing items from a sorted array won’t unsort it, but adding or modifying items can. Keeping a sorted array sorted means intercepting and reimplementing every sneaky way of putting objects into the array.

The SortedArray constructor accepts any code block you can pass into Array#sort, and keeps the array sorted according to that code block. The default code block uses the comparison operator (<=>) used by sort.

	unsorted= ["b", "aa", "a", "cccc", "1", "zzzzz", "k", "z"]
	strings_by_alpha = SortedArray.new(unsorted)
	# => ["1", "a", "aa", "b", "cccc", "k", "z", "zzzzz"]
	strings_by_length = SortedArray.new(unsorted) do |x,y|
	  x.length <=> y.length
	# => ["b", "z", "a", "k", "1", "aa", "cccc", "zzzzz"]

The methods that add elements to an array specify where in the array they operate: push operates on the end of the array, and insert operates on a specified spot. SortedArray responds to these methods but it ignores the caller’s request to put elements in a certain place. Every new element is inserted into a position that keeps the array sorted.

	a << -1                            # => [-1, 1, 2, 3]
	a << 1.5                           # => [-1, 1, 1.5, 2, 3]
	a.push(2.5)                        # => [-1, 1, 1.5, 2, 2.5, 3]
	a.unshift(1.6)                     # => [-1, 1, 1.5, 1.6, 2, 2.5, 3]

For methods like collect! and array assignment ([]=)that allow complex changes to an array, the simplest solution is to allow the changes to go through and then re-sort:

	a = SortedArray.new([10, 6, 4, -4, 200, 100])  
	# => [-4, 4, 6, 10, 100, 200]
	a.collect! { |x| x * -1 }          # => [-200, -100, -10, -6, -4, 4]

	a[3] = 25
	a                                  # => [-200, -100, -10, -4, 4, 25]
	# That is, -6 has been replaced by 25 and the array has been re-sorted.

	a[1..2] = [6000, 10, 600, 6]
	a                                  # => [-200, -4, 4, 6, 10, 25, 600, 6000]
	# That is, -100 and -10 have been replaced by 6000, 10, 600, and 6,
	# and the array has been re-sorted.

But with a little more work, we can write a more efficient implementation of array assignment that gives the same behavior. What happens when you run a command like a[0]= 10 on a SortedArray? The first element in the SortedArray is replaced by 10, and the SortedArray is re-sorted. This is equivalent to removing the first element in the array, then adding the value 10 to a place in the array that keeps it sorted.

Array#[]= implements three different types of array assignment, but all three can be modeled as a series of removals followed by a series of insertions. We can use this fact to implement a more efficient version of SortedArray#[]=:.

	  def []=(*args)
	    if args.size == 3
	      #e.g. "a[6,3] = [1,2,3]"
	      start, length, value = args
	      slice! Range.new(start, start+length, true)
	      (value.respond_to? :each) ? value.each { |x| self << x } : self << value
	    elsif args.size == 2
	      index, value = args
	      if index.is_a? Numeric
	        #e.g. "a[0] = 10" (the most common form of array assignment)
	        self << value
	      elsif index.is_a? Range
	        #e.g. "a[0..3] = [1,2,3]"
	        slice! index
	        (value.respond_to? :each) ? value.each { |x| self << x } : self << value
	        #Not supported. Delegate to superclass; will probably give an error.
	      #Not supported. Delegate to superclass; will probably give an error.

Just as before, the sort will be maintained even when you use array assignment to replace some of a SortedArray’s elements with other objects. But this implementation doesn’t have to re-sort the array every time.

	a = SortedArray.new([1,2,3,4,5,6])
	a[0] = 10
	a                                     # => [2, 3, 4, 5, 6, 10]

	a[0, 2] = [100, 200]
	a                                     # => [4, 5, 6, 10, 100, 200]

	a[1..2] = [-4, 6]
	a                                     # => [-4, 4, 6, 10, 100, 200]

It’s possible to subvert the sortedness of a SortedArray by modifying an object in place in a way that changes its sort order. Since the SortedArray never hears about the change to this object, it has no way of updating itself to move that object to its new sort position:[2]

	stripes = 
SortedArray.new(["aardwolf", "zebrafish"])
	stripes                                 # => ["aardwolf", "ZEBRAFISH"]
	stripes.sort!                           # => ["ZEBRAFISH", "aardwolf"]

If this bothers you, you can make a SortedArray keep frozen copies of objects instead of the objects themselves. This solution hurts performance and uses more memory, but it will also prevent objects from being modified after being put into the SortedArray. This code adds a convenience method to Object that makes a frozen copy of the object:

	class Object
	  def to_frozen
	    f = self
	    unless frozen?
	         f = dup.freeze
	       rescue TypeError  
	         #This object can't be duped (e.g. Fixnum); fortunately,
	         #it usually can't be modified either 
	    return f

The FrozenCopySortedArray stores frozen copies of objects instead of the objects themselves:

	class FrozenCopySortedArray < SortedArray
	  def insert(i, v)  
	    insert_before = index(find { |x| x > v })
	    super(insert_before ? insert_before : -1, v.to_frozen)

	  ["initialize", "collect!", "flatten!"].each do |method_name|
	    define_method(method_name) do 
	      each_with_index { |x, i| self[i] = x.to_frozen }
	      # No need to sort; by doing an assignment to every element
	      # in the array, we've made #insert keep the array sorted.

	stripes = SortedArray.new(["aardwolf", "zebrafish"])
	# TypeError: can't modify frozen string

Unlike a regular array, which can have elements of arbitrarily different data classes, all the elements of a SortedArray must be mutually comparable. For instance, you can mix integers and floating-point numbers within a SortedArray, but you can’t mix integers and strings. Any data set that would cause Array# sort to fail makes an invalid SortedArray:

	[1, "string"].sort
	# ArgumentError: comparison of Fixnum with String failed

	a = SortedArray.new([1]) 
	a << "string" 
	# ArgumentError: comparison of Fixnum with String failed

One other pitfall: operations that create a new object, such as |=, +=, and to_a will turn an SortedArray into a (possibly unsorted) array.

	a = SortedArray.new([3, 2, 1])       # => [1, 2, 3]
	a += [1, -10]                        # => [1, 2, 3, 1, -10]
	a.class                              # => Array

The simplest way to avoid this is to override these methods to transform the resulting array back into a SortedArray:

	class SortedArray  
	  def + (other_array) 

See Also

4.8. Summing the Items of an Array


You want to add together many objects in an array.


There are two good ways to accomplish this in Ruby. Plain vanilla iteration is a simple way to approach the problem:

	collection = [1, 2, 3, 4, 5]
	sum = 0 
	collection.each {|i| sum += i} 
	sum                                       # => 15

However this is such a common action that Ruby has a special iterator method called inject, which saves a little code:

	collection = [1, 2, 3, 4, 5]
inject(0) {|sum, i| sum + i}      # => 15


Notice that in the inject solution, we didn’t need to define the variable total variable outside the scope of iteration. Instead, its scope moved into the iteration. In the example above, the initial value for total is the first argument to inject. We changed the += to + because the block given to inject is evaluated on each value of the collection, and the total variable is set to its output every time.

You can think of the inject example as equivalent to the following code:

	collection = [1, 2, 3, 4, 5]
	sum = 0
	sum = sum + 1
	sum = sum + 2
	sum = sum + 3
	sum = sum + 4
	sum = sum + 5

Although inject is the preferred way of summing over a collection, inject is generally a few times slower than each. The speed difference does not grow exponentially, so you don’t need to always be worrying about it as you write code. But after the fact, it’s a good idea to look for inject calls in crucial spots that you can change to use faster iteration methods like each.

Nothing stops you from using other kinds of operators in your inject code blocks. For example, you could multiply:

	collection = [1, 2, 3, 4, 5]
	collection.inject(1) {|total, i| total * i}  # => 120

Many of the other recipes in this book use inject to build data structures or run calculations on them.

See Also

4.9. Sorting an Array by Frequency of Appearance


You want to sort an array so that its least-frequently-appearing items come first.


Build a histogram of the frequencies of the objects in the array, then use it as a lookup table in conjunction with the sort_ by method.

The following method puts the least frequently-appearing objects first. Objects that have the same frequency are sorted normally, with the comparison operator.

	module Enumerable
	    histogram = inject(Hash.new(0)) { |hash, x| hash[x] += 1; hash}
	    sort_by { |x| [histogram[x], x] }

	# => [3, 8, 9, 16, 2, 2, 1, 1, 1, 4, 4, 4]


The sort_by_frequency method uses sort_by, a method introduced in Recipe 4.5 and described in detail in Recipe 4.6. The technique here is a little different from other uses of sort_by, because it sorts by two different criteria. We want to first compare the relative frequencies of two items. If the relative frequencies are equal, we want to compare the items themselves. That way, all the instances of a given item will show up together in the sorted list.

The block you pass to Enumerable#sort_by can return only a single sort key for each object, but that sort key can be an array. Ruby compares two arrays by comparing their corresponding elements, one at a time. As soon as an element of one array is different from an element of another, the comparison stops, returning the comparison of the two different elements. If one of the arrays runs out of elements, the longer one sorts first. Here are some quick examples:

	[1,2] <=> [0,2]                        # => 1
	[1,2] <=> [1,2]                        # => 0
	[1,2] <=> [2,2]                        # => -1
	[1,2] <=> [1,1]                        # => 1
	[1,2] <=> [1,3]                        # => -1
	[1,2] <=> [1]                          # => 1
	[1,2] <=> [3]                          # => -1
	[1,2] <=> [0,1,2]                      # => 1
	[1,2] <=> []                           # => 1

In our case, all the arrays contain two elements: the relative frequency of an object in the array, and the object itself. If two objects have different frequencies, the first elements of their arrays will differ, and the items will be sorted based on their frequencies. If two items have the same frequency, the first element of each array will be the same. The comparison method will move on to the second array element, which means the two objects will be sorted based on their values.

If you don’t mind elements with the same frequency showing up in an unsorted order, you can speed up the sort a little by comparing only the histogram frequencies:

	module Enumerable
	  def sort_by_frequency_faster
	    histogram = inject(Hash.new(0)) { |hash, x| hash[x] += 1; hash}
	    sort_by { |x| histogram[x] }

	# => [16, 8, 3, 9, 2, 2, 4, 1, 1, 4, 4, 1]

To sort the list so that the most-frequently-appearing items show up first, either invert the result of sort_by_frequency, or multiply the histogram values by–1 when passing them into sort_by:

	module Enumerable
	  def sort_by_frequency_descending
	    histogram = inject(Hash.new(0)) { |hash, x| hash[x] += 1; hash}
	    sort_by { |x| [histogram[x] * -1, x]}

	# => [1, 1, 1, 4, 4, 4, 2, 2, 3, 8, 9, 16]

If you want to sort a list by the frequency of its elements, but not have repeated elements actually show up in the sorted list, you can run the list through Array#uniq after sorting it. However, since the keys of the histogram are just the distinct elements of the array, it’s more efficient to sort the keys of the histogram and return those:

	module Enumerable
	  def sort_distinct_by_frequency
	    histogram = inject(Hash.new(0)) { |hash, x| hash[x] += 1; hash }
	    histogram.keys.sort_by { |x| [histogram[x], x] }

	# => [3, 8, 9, 16, 2, 1, 4]

See Also

4.10. Shuffling an Array


You want to put the elements of an array in random order.


The simplest way to shuffle an array (in Ruby 1.8 and above) is to sort it randomly:

	[1,2,3].sort_by { rand } # => [1, 3, 2]

This is not the fastest way, though.


It’s hard to beat a random sort for brevity of code, but it does a lot of extra work. Like any general sort, a random sort will do about n log n variable swaps. But to shuffle a list, it suffices to put a randomly selected element in each position of the list. This can be done with only n variable swaps.

	class Array
	  def shuffle!  
	    each_index do |i| 
	      j = rand(length-i) + i
	      self[j], self[i] = self[i], self[j]  

	  def shuffle  

If you’re shuffling a very large list, either Array#shuffle or Array#shuffle! will be significantly faster than a random sort. Here’s a real-world example of shuffling using Array#shuffle:

	class Card
	  def initialize(suit, rank)
	    @suit = suit  
	    @rank = rank  

	  def to_s
	    "#{@suit} of #{@rank}"

	class Deck < Array
	  attr_reader :cards  
	  @@suits = %w{Spades Hearts Clubs Diamonds}
	  @@ranks = %w{Ace 2 3 4 5 6 7 8 9 10 Jack Queen King} 

	  def initialize
	    @@suits.each { |suit| @@ranks.each { |rank| self << Card.new(rank, suit) } }

	deck = Deck.new
	deck.collect { |card| card.to_s }
	# => ["Ace of Spades", "2 of Spades", "3 of Spades", "4 of Spades",…]
	deck.collect { |card| card.to_s }
	# => ["6 of Clubs", "8 of Diamonds", "2 of Hearts", "5 of Clubs",…]

See Also

  • Recipe 2.5, “Generating Random Numbers”

  • The Facets Core library provides implementations of Array#shuffle and Array#shuffle!

4.11. Getting the N Smallest Items of an Array


You want to find the smallest few items in an array, or the largest, or the most extreme according to some other measure.


If you only need to find the single smallest item according to some measure, use Enumerable#min. By default, it uses the <=> method to see whether one item is “smaller” than another, but you can override this by passing in a code block.

	[3, 5, 11, 16].min 
	# => 3
	["three", "five", "eleven", "sixteen"].min
	# => "eleven"
	["three", "five", "eleven", "sixteen"].min { |x,y| x.size <=> y.size }
	# => "five"

Similarly, if you need to find the single largest item, use Enumerable#max.

	[3, 5, 11, 16].max 
	# => 16
	["three", "five", "eleven", "sixteen"].max
	# => "three"
	["three", "five", "eleven", "sixteen"].max { |x,y| x.size <=> y.size }
	# => "sixteen"

By default, arrays are sorted by their natural order: numbers are sorted by value, strings by their position in the ASCII collating sequence (basically alphabetical order, but all lowercase characters precede all uppercase characters). Hence, in the previous examples, “three” is the largest string, and “eleven” the smallest.

It gets more complicated when you need to get a number of the smallest or largest elements according to some measurement: say, the top 5 or the bottom 10. The simplest solution is to sort the list and skim the items you want off of the top or bottom.

	l = [1, 60, 21, 100, -5, 20, 60, 22, 85, 91, 4, 66]
	sorted = l.sort

	#The top 5
	# => [60, 66, 85, 91, 100]  

	#The bottom 5 
	# => [-5, 1, 4, 20, 21]

Despite the simplicity of this technique, it’s inefficient to sort the entire list unless the number of items you want to extract approaches the size of the list.


The min and max methods work by picking the first element of the array as a “champion,” then iterating over the rest of the list trying to find an element that can beat the current champion on the appropriate metric. When it finds one, that element becomes the new champion. An element that can beat the old champion can also beat any of the other contenders seen up to that point, so one run through the list suffices to find the maximum or minimum.

The naive solution to finding more than one smallest item is to repeat this process multiple times. Iterate over the Array once to find the smallest item, then iterate over it again to find the next-smallest item, and so on. This is naive for the same reason a bubble sort is naive: you’re repeating many of your comparisons more times than necessary. Indeed, if you run this algorithm once for every item in the array (trying to find the n smallest items in an array of n items), you get a bubble sort.

Sorting the list beforehand is better when you need to find more than a small fraction of the items in the list, but it’s possible to do better. After all, you don’t really want to sort the whole list: you just want to sort the bottom of the list to find the smallest items. You don’t care if the other elements are unsorted because you’re not interested in those elements anyway.

To sort only the smallest elements, you can keep a sorted “stable” of champions, and kick the largest champion out of the stable whenever you find an element that’s smaller. If you encounter a number that’s too large to enter the stable, you can ignore it from that point on. This process rapidly cuts down on the number of elements you must consider, making this approach faster than doing a sort.

The SortedList class from Recipe 4.7 is useful for this task. The min_n method below creates a SortedList “stable” that keeps its elements sorted based on the same block being used to find the minimum. It keeps the stable at a certain size by kicking out the largest item in the stable whenever a smaller item is found. The max_n method works similarly, but the comparisons are reversed, and the smallest element in the stable is kicked out when a larger element is found.

	module Enumerable
	  def min_n(n, &block)
	    block = Proc.new { |x,y| x <=> y } if block == nil
	    stable = SortedArray.new(&block) 
	    each do |x|
	      stable << x if stable.size < n or block.call(x, stable[-1]) == -1
	      stable.pop until stable.size <= n
	    return stable 

	  def max_n(n, &block)
	    block = Proc.new { |x,y| x <=> y } if block == nil
	    stable = SortedArray.new(&block) 
	    each do |x|
	      stable << x if stable.size < n or block.call(x, stable[0]) == 1
	      stable.shift until stable.size <= n
	    return stable 
	l = [1, 60, 21, 100, -5, 20, 60, 22, 85, 91, 4, 66]
	# => [60, 66, 85, 91, 100]  
	# => [-5, 1, 4, 20, 21]
	l.min_n(5) { |x,y| x.abs <=> y.abs }
	# => [1, 4, -5, 20, 21]

See Also

  • Recipe 4.7, “Making Sure a Sorted Array Stays Sorted”

4.12. Building Up a Hash Using Injection


You want to create a hash from the values in an array.


As seen in Recipe 4.8, the most straightforward way to solve this kind of problem is to use Enumerable#inject. The inject method takes one parameter (the object to build up, in this case a hash), and a block specifying the action to take on each item. The block takes two parameters: the object being built up (the hash), and one of the items from the array.

Here’s a straightforward use of inject to build a hash out of an array of key-value pairs:

	collection = [ [1, 'one'], [2, 'two'], [3, 'three'],
	               [4, 'four'], [5, 'five'] 
	collection.inject({}) do |hash, value|
	  hash[value.first] = value.last 
	# => {5=>"five", 1=>"one", 2=>"two", 3=>"three", 4=>"four"}


Why is there that somewhat incongrous expression hash at the end of the inject block above? Because the next time it calls the block, inject uses the value it got from the block the last time it called the block. When you’re using inject to build a data structure, the last line of code in the block should evaluate to the object you’re building up: in this case, our hash.

This is probably the most common inject-related gotcha. Here’s some code that doesn’t work:

	collection.dup.inject({}) { |hash, value| hash[value.first] = value.last }
	# IndexError: index 3 out of string

Why doesn’t this work? Because hash assignment returns the assigned value, not the hash.

	Hash.new["key"] = "some value"       # => "some value"

In the broken example above, when inject calls the code block for the second and subsequent times, it does not pass the hash as the code block’s first argument. It passes in the last value to be assigned to the hash. In this case, that’s a string (maybe “one” or “four”). The hash has been lost forever, and the inject block crashes when it tries to treat a string as a hash.

Hash#update can be used like hash assignment, except it returns the hash instead of the assigned value (and it’s slower). So this code will work:

	collection.inject({}) do |hash, value|
	  hash.update value.first => value.last 
	# => {5=>"five", 1=>"ontwo", 2=>"two", 3=>"three", 4=>"four"}

Ryan Carver came up with a more sophisticated way of building a hash out of an array: define a general method for all arrays called to_h.

	class Array
	  def to_h(default=nil)
	    Hash[ *inject([]) { |a, value| a.push value, default || yield(value) } ]

The magic of this method is that you can provide a code block to customize how keys in the array are mapped to values.

	a = [1, 2, 3]

	# => {1=>true, 2=>true, 3=>true}

	a.to_h { |value| [value * -1, value * 2] }
	# => {1=>[-1, 2], 2=>[-2, 4], 3=>[-3, 6]}


4.13. Extracting Portions of Arrays


Given an array, you want to retrieve the elements of the array that occupy certain positions or have certain properties. You might to do this in a way that removes the matching elements from the original array.


To gather a chunk of an array without modifying it, use the array retrieval operator Array#[], or its alias Array#slice.

The array retrieval operator has three forms, which are the same as the corresponding forms for substring accesses. The simplest and most common form is array[index].It takes a number as input, treats it as an index into the array, and returns the element at that index. If the input is negative, it counts from the end of the array. If the array is smaller than the index, it returns nil. If performance is a big consideration for you, Array#at will do the same thing, and it’s a little faster than Array#[]:

	a = ("a".."h").to_a                # => ["a", "b", "c", "d", "e", "f", "g", "h"]

	a[0]                               # => "a"
	a[1]                               # => "b"

	a.at(1)                            # => "b" 
	a.slice(1)                         # => "b"
	a[-1]                              # => "h"
	a[-2]                              # => "g"
	a[1000]                            # => nil 
	a[-1000]                           # => nil

The second form is array[range]. This form retrieves every element identified by an index in the given range, and returns those elements as a new array.

A range in which both numbers are negative will retrieve elements counting from the end of the array. You can mix positive and negative indices where that makes sense:

	a[2..5]                                # => ["c", "d", "e", "f"]
	a[2…5]                               # => ["c", "d", "e"]
	a[0..0]                                # => ["a"]
	a[1..-4]                               # => ["b", "c", "d", "e"]
	a[5..1000]                             # => ["f", "g", "h"]

	a[2..0]                                # => []
	a[0…0]                               # => []

	a[-3..2]                               # => []

The third form is array[start_index, length]. This is equivalent to array[range. new(start_index…start_index+length)].

	a[2, 4]                                # => ["c", "d", "e", "f"]
	a[2, 3]                                # => ["c", "d", "e"]
	a[0, 1]                                # => ["a"]
	a[1, 2]                                # => ["b", "c"]  
	a[-4, 2]                               # => ["e", "f"] 
	a[5, 1000]                             # => ["f", "g", "h"]

To remove a slice from the array, use Array#slice!. This method takes the same arguments and returns the same results as Array#slice, but as a side effect, the objects it retrieves are removed from the array.

	a.slice!(2..5)                       # => ["c", "d", "e", "f"]
	a                                    # => ["a", "b", "g", "h"]

	a.slice!(0)                          # => "a"
	a                                    # => ["b", "g", "h"]

	a.slice!(1,2)                        # => ["g", "h"]
	a                                    # => ["b"]


The Array methods [], slice, and slice! work well if you need to extract one particular elements, or a set of adjacent elements. There are two other main possibilities: you might need to retrieve the elements at an arbitrary set of indexes, or (a catch-all) you might need to retrieve all elements with a certain property that can be determined with a code block.

To nondestructively gather the elements at particular indexes in an array, pass in any number of indices to Array#values_at. Results will be returned in a new array, in the same order they were requested.

	a = ("a".."h").to_a             # => ["a", "b", "c", "d", "e", "f", "g", "h"]
	a.values_at(0)                  # => ["a"]
	a.values_at(1, 0, -2)           # => ["b", "a", "g"]
	a.values_at(4, 6, 6, 7, 4, 0, 3)# => ["e", "g", "g", "h", "e", "a", "d"]

Enumerable#find_all finds all elements in an array (or other class with Enumerable mixed in)for which the specified code block returns true. Enumerable#reject will find all elements for which the specified code block returns false.

	a.find_all { |x| x < "e" }       # => ["a", "b", "c", "d"]
	a.reject { |x| x < "e" }         # => ["e", "f", "g", "h"]

To find all elements in an array that match a regular expression, you can use Enumerable#grep instead of defining a block that does the regular expression match:

	a.grep /[aeiou]/                    # => ["a", "e"]
	a.grep /[^g]/                       # => ["a", "b", "c", "d", "e", "f", "h"]

It’s a little tricky to implement a destructive version of Array#values_at, because removing one element from an array changes the indexes of all subsequent elements. We can let Ruby do the work, though, by replacing each element we want to remove with a dummy object that we know cannot already be present in the array. We can then use the C-backed method Array#delete to remove all instances of the dummy object from the array. This is much faster than using Array#slice! to remove elements one at a time, because each call to Array#slice! forces Ruby to rearrange the array to be contiguous.

If you know that your array contains no nil values, you can set your undesired values to nil, then use use Array#compress! to remove them. The solution below is more general.

	class Array
	  def strip_values_at!(*args)
	      #For each mentioned index, replace its value with a dummy object.
	      values = []
	      dummy = Object.new  
	      args.each do |i|
	        if i < size
	         values << self[i]  
	          self[i] = dummy
	      #Strip out the dummy object.
	      return values 

	a = ("a".."h").to_a
	a.strip_values_at!(1, 0, -2)         # => ["b", "a", "g"]
	a                                    # => ["c", "d", "e", "f", "h"]

	a.strip_values_at!(1000)             # => [] 
	a                                    # => ["c", "d", "e", "f", "h"]

Array#reject! removes all items from an array that match a code block, but it doesn’t return the removed items, so it won’t do for a destructive equivalent of Enumerable#find_all. This implementation of a method called extract! picks up where Array#reject! leaves off:

	class Array
	  def extract! 
	    ary = self.dup
	    self.reject! { |x| yield x }
	    ary - self

	a = ("a".."h").to_a
	a.extract! { |x| x < "e" && x != "b" }   # => ["a", "c", "d"]
	a                                        # => ["b", "e", "f", "g", "h"]

Finally, a convenience method called grep_extract! provides a method that destructively approximates the behavior of Enumerable#grep.

	class Array
	   def grep_extract!(re)
	   extract! { |x| re.match(x) }

	a = ("a".."h").to_a
	a.grep_extract!(/[aeiou]/)           # => ["a", "e"]
	a                                    # => ["b", "c", "d", "f", "g", "h"]

See Also

  • Strings support the array lookup operator, slice, slice!, and all the methods of Enumerable, so you can treat them like arrays in many respects; see Recipe 1.13, “Getting the Parts of a String You Want”

4.14. Computing Set Operations on Arrays


You want to find the union, intersection, difference, or Cartesian product of two arrays, or the complement of a single array with respect to some universe.


Array objects have overloaded arithmetic and logical operators to provide the three simplest set operations:

	[1,2,3] | [1,4,5]                    # => [1, 2, 3, 4, 5] 

	[1,2,3] & [1,4,5]                    # => [1]

	[1,2,3] - [1,4,5]                    # => [2, 3]

Set objects overload the same operators, as well as the exclusive-or operator (^).If you already have Arrays, though, it’s more efficient to deconstruct the XOR operation into its three component operations.

	require 'set' 
	a = [1,2,3]
	b = [3,4,5]
	a.to_set ^ b.to_set                  # => #<Set: {5, 1, 2, 4}>  
	(a | b) - (a & b)                    # => [1, 2, 4, 5]


Set objects are intended to model mathematical sets: where arrays are ordered and can contain duplicate entries, Sets model an unordered collection of unique items. Set not only overrides operators for set operations, it provides English-language aliases for the three most common operators: Set#union, Set#intersection, and Set#difference. An array can only perform a set operation on another array, but a Set can perform a set operation on any Enumerable.

	array = [1,2,3]
	set = [3,4,5].to_s  
	array & set                      # => TypeError: can't convert Set into Array
	set & array                      # => #<Set: {3}>

You might think that Set objects would be optimized for set operations, but they’re actually optimized for constant-time membership checks (internally, a Set is based on a hash). Set union is faster when the left-hand object is a Set object, but intersection and difference are significantly faster when both objects are arrays. It’s not worth it to convert arrays into Sets just so you can say you performed set operations on Set objects.

The union and intersection set operations remove duplicate entries from arrays. The difference operation does not remove duplicate entries from an array except as part of a subtraction.

	[3,3] & [3,3]                        # => [3]
	[3,3] | [3,3]                        # => [3]
	[1,2,3,3] - [1]                      # => [2, 3, 3]
	[1,2,3,3] - [3]                      # => [1, 2]
	[1,2,3,3] - [2,2,3]                  # => [1]


If you want the complement of an array with respect to some small universe, create that universe and use the difference operation:

	u = [:red, :orange, :yellow, :green, :blue, :indigo, :violet]
	a = [:red, :blue]
	u - a                    # => [:orange, :yellow, :green, :indigo, :violet]

More often, the relevant universe is infinite (the set of natural numbers)or extremely large (the set of three-letter strings). The best strategy here is to define a generator and use it to iterate through the complement. Be sure to break when you’re done; you don’t want to iterate over an infinite set.

	def natural_numbers_except(exclude) 
	  exclude_map = {}
	  exclude.each { |x| exclude_map[x] = true }
	  x = 1
	  while true
	    yield x unless exclude_map[x] 
	    x = x.succ

	natural_numbers_except([2,3,6,7]) do |x|  
	  break if x > 10  
	  puts x  
	# 1
	# 4
	# 5
	# 8
	# 9
	# 10

Cartesian product

To get the Cartesian product of two arrays, write a nested iteration over both lists and append each pair of items to a new array. This code is attached to Enumerable so you can also use it with Sets or any other Enumerable.

	module Enumberable
	  def cartesian(other)
	    res = []
	    each { |x| other.each { |y| res << [x, y] } }  
	    return res

	# => [[1, "a"], [1, 5], [1, 6],
	#    [2, "a"], [2, 5], [2, 6],
	#    [3, "a"], [3, 5], [3, 6]

This version uses Enumerable#inject to make the code more concise; however, the original version is more efficient.

	module Enumerable  
	  def cartesian(other)
	    inject([]) { |res, x| other.inject(res) { |res, y| res << [x,y] } }

See Also

  • See Recipe 2.5, “Generating Random Numbers,” for an example (constructing a deck of cards from suits and ranks)that could benefit from a function to calculate the Cartesian product

  • Recipe 2.10, “Multiplying Matrices”

4.15. Partitioning or Classifying a Set


You want to partition a Set or array based on some attribute of its elements. All elements that go “together” in some code-specific sense should be grouped together in distinct data structures. For instance, if you’re partitioning by color, all the green objects in a Set should be grouped together, separate from the group of all the red objects in the Set.


Use Set#divide, passing in a code block that returns the partition of the object it’s passed. The result will be a new Set containing a number of partitioned subsets of your original Set.

The code block can accept either a single argument or two arguments.[3] The single-argument version examines each object to see which subset it should go into.

	require 'set' 
	s = Set.new((1..10).collect)
	# => #<Set: {5, 6, 1, 7, 2, 8, 3, 9, 4, 10}>

	# Divide the set into the "true" subset and the "false" subset: that
	# is, the "less than 5" subset and the "not less than 5" subset.
	s.divide { |x| x < 5 }
	# => #<Set: {#<Set: {5, 6, 7, 8, 9, 10}>, #<Set: {1, 2, 3, 4}>}>

	# Divide the set into the "0" subset and the "1" subset: that is, the
	# "even" subset and the "odd" subset.
	s.divide { |x| x % 2 }
	# => #<Set: {#<Set: {6, 2, 8, 4, 10}>, #<Set: {5, 1, 7, 3, 9}>}>

	s = Set.new([1, 2, 3, 'a', 'b', 'c', -1.0, -2.0, -3.0])
	# Divide the set into the "String subset, the "Fixnum" subset, and the
	# "Float" subset.
	s.divide { |x| x.class } 
	# => #<Set: {#<Set: {"a", "b", "c"}>,
	# =>       #<Set: {1, 2, 3}>,
	# =>       #<Set: {-1.0, -3.0, -2.0}>}>

For the two-argument code block version of Set#divide, the code block should return true if both the arguments it has been passed should be put into the same subset.

	s = [1, 2, 3, -1, -2, -4].to_set

	# Divide the set into sets of numbers with the same absolute value.
	s.divide { |x,y| x.abs == y.abs }
	# => #<Set: {#<Set: {-1, 1}>, 
	# =>         #<Set: {2, -2}>, 
	# =>         #<Set: {-4}>,
	# =>         #<Set: {3}>}>

	# Divide the set into sets of adjacent numbers 
	s.divide { |x,y| (x-y).abs == 1 }
	# => #<Set: {#<Set: {1, 2, 3}>,
	# =>         #<Set: {-1}>,
	# =>         #<Set: {-4, -3}>}>

If you want to classify the subsets by the values they have in common, use Set#classify instead of Set#divide. It works like Set#divide, but it returns a hash that maps the names of the subsets to the subsets themselves.

	s.classify { |x| x.class }
	# => {String=>#<Set: {"a", "b", "c"}>,
	# =>  Fixnum=>#<Set: {1, 2, 3}>,
	# =>  Float=>#<Set: {-1.0, -3.0, -2.0}>}


The version of Set#divide that takes a two-argument code block uses the tsort library to turn the Set into a directed graph. The nodes in the graph are the items in the Set. Two nodes x and y in the graph are connected with a vertex (one-way arrow) if the code block returns true when passed |x,y|. For the Set and the two-argument code block given in the example above, the graph looks like Figure 4-1.

The set {1, 2, 3, -1, -2, -4} graphed according to the code block that checks adjacency
Figure 4-1. The set {1, 2, 3, -1, -2, -4} graphed according to the code block that checks adjacency

The Set partitions returned by Set#divide are the strongly connected components of this graph, obtained by iterating over TSort#each_strongly_connected_component. A strongly connected component is a set of nodes such that, starting from any node in the component, you can follow the one-way arrows and get to any other node in the component.

Visually speaking, the strongly connected components are the “clumps” in the graph. 1 and 3 are in the same strongly connected component as 2, because starting from 3 you can follow one-way arrows through 2 and get to 1. Starting from 1, you can follow one-way arrows through 2 and get to 3. This makes 1, 2, and 3 part of the same Set partition, even though there are no direct connections between 1 and 3.

In most real-world scenarios (including all the examples above), the one-way arrows will be symmetrical: if the code returns true for |x,y|, it will also return true for |y,x|. Set#divide will work even if this isn’t true. Consider a Set and a divide code block like the following:

	connections = { 1 => 2, 2 => 3, 3 => 1, 4 => 1 }
	[1,2,3,4].to_set.divide { |x,y| connections[x] == y }
	# => #<Set: {#<Set: {1, 2, 3}>, #<Set: {4}>}>

The corresponding graph looks like Figure 4-2.

The set {1,2,3,4} graphed according to the connection hash
Figure 4-2. The set {1,2,3,4} graphed according to the connection hash

You can get to any other node from 4 by following one-way arrows, but you can’t get to 4 from any of the other nodes. This puts 4 is in a strongly connected component—and a Set partition—all by itself. 1, 2, and 3 form a second strongly connected component—and a second Set partition—because you can get from any of them to any of them by following one-way arrows.

Implementation for arrays

If you’re starting with an array instead of a Set, it’s easy to simulate Set#classify (and the single-argument block form of Set#divide)with a hash. In fact, the code below is almost identical to the current Ruby implementation of Set#classify.

	class Array
	  def classify
	    require 'set'
	    h = {}
	    each do |i|
	      x = yield(i)
	      (h[x] ||= self.class.new) << i

	  def divide(&block)

	[1,1,2,6,6,7,101].divide { |x| x % 2 }
	# => #<Set: {[2, 6, 6], [1, 1, 7, 101]}>

There’s no simple way to implement a version of Array#divide that takes a two-argument block. The TSort class is Set-like, in that it won’t create two different nodes for the same object. The simplest solution is to convert the array into a Set to remove any duplicate values, divide the Set normally, then convert the partitioned subsets into arrays, adding back the duplicate values as you go:

	class Array
	  def divide(&block)
	    if block.arity == 2
	     counts = inject({}) { |h, x| h[x] ||= 0; h[x] += 1; h}
	     to_set.divide(&block).inject([]) do |divided, set|
	       divided << set.inject([]) do |partition, e|
	         counts[e].times { partition << e }  

	[1,1,2,6,6,7,101].divide { |x,y| (x-y).abs == 1 }
	# => [[101], [1, 1, 2], [6, 6, 7]]

Is it worth it? You decide.

[1] We can’t use define_method to define these methods because in Ruby 1.8 you can’t use define_method to create a method that takes a block argument. See Chapter 10 for more on this.

[2] One alternative is to modify SortedArray[] so that when you look up an element of the array, you actually get a delegate object that intercepts all of the element’s method calls, and re-sorts the array whenever the user calls a method that modifies the element in place. This is probably overkill.

[3] This is analogous to the one-argument code block passed into Enumerable#sort_by and the two-argument code block passed into Array#sort.

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