19 December 2017

# Modeling Sets

Part 02 of the "Bunch of Sets" series.

This is part 02 of a three part series that starts here.

Equipped with some basic understanding of our problem domain established in part 01 of this series, let us begin to develop the main ingredients for a Ruby model of sets that encompasses the types of sets we have discussed (as well as potentially other ones). We start by discussing a SetMap class that captures the commonalities of classical sets, fuzzy sets, and multisets, while allowing us to easily define each of these specific types via inheritance.

### Hash tables

What do the three types of sets have in common? At first glance, it seems that their internal structure is pretty different: multisets and fuzzy sets have been presented above as consisting of key-value pairs, while classical sets simply consist of a bunch of keys. However, this is merely a matter of representation. In fact, it is rather common to represent a classical set by means of a characteristic function, which maps the members of the set to 1, while all other objects from a given domain are mapped to 0. Taking a cue from this, we extend our key-value notation to classical sets, writing the set { 0, 1, 2 }, for example, as

{ 0: 1, 1: 1, 2: 1 }.

From this perspective, it becomes obvious that the membership information for a set—be it a fuzzy set, a classical set, or a multiset—may be stored in a hash table:

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{ 'a' => 1, 'b' => 1, 'c' => 1 }     # 'classical set'
{ 'a' => 1, 'b' => 1, 'c' => 2 }     # 'multiset'
{ 'a' => 0.3, 'b' => 1, 'c' => 0.6 } # 'fuzzy set'


Hash tables will form the basis of our representation of sets. Of course, we would not want to directly expose such a table to the user of our set class. The user need not even be aware that we are using a hash table to store her set. Rather, the hash instance that stores the set keys and their associated scores will be a collaborator object to our set object. The initialize method of our SetMap class sets the stage for this:

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class SetMap
def initialize
@hash = {}
@size = 0
end
end


Besides the @hash instance variable, we also decide to maintain an instance variable @size, in the interest of being able to look up the size of our set in constant time. The size of a set is commonly defined as the sum of the scores of its keys, and the idea is that @size will always store this value in an up-to-date fashion. We also make available a getter method size that returns the current value of @size, omitted here.

### Valid scores

What distinguishes the types of sets we have seen above from each other? It is primarily what counts as a valid score according to each type:

• A classical set either contains or does not contain a particular key, so the only valid scores are 0 and 1.
• A multiset may contain a given key n times, where n is a non-negative integer.
• A fuzzy set scores a given key to a degree in the unit interval from 0 to 1.

So in each case, we have to be able to express a range of possible values. We set up a bunch of class methods and class instance variables for this purpose:

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class SetMap
def self.score_type
@score_type || raise(SetError, '@score_type not initialized')
end

def self.min_score
@min_score || raise(SetError, '@min_score not initialized')
end

def self.max_score
@max_score || raise(SetError, '@max_score not initialized')
end

def self.valid_score?(val)
val.is_a?(score_type) && (min_score..max_score).cover?(val)
end
end


The first three class methods defined above are getters (on the level of the class object) for the class instance variables @score_type (the kind of object we may use as a score for a key), @min_score (the smallest value that may be used as a score), and @max_score (the largest value that maye be used as a score). The fourth method uses these getters and describes what constitutes a valid score as a predicate.

As the second disjunct of each of the above getter methods tells us loud and clear, we are missing something so far: our class instance variables have not been set to any value! Initializing those class instance variables is precisely the job description of our target classes.

### Target classes

SetMap is meant to be subclassed, with each subclass defining a particular set type by specifying a range of legal scores via the class instance variables @score_type, @min_score and @max_score. Here is the code for classical sets:

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class ClassicalSet < SetMap
@score_type = Integer
@min_score = 0
@max_score = 1
end


In other words, the only valid scores for the keys of classical sets are the integers 0 and 1. For fuzzy sets, we write:

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class FuzzySet < SetMap
@score_type = Numeric
@min_score = 0
@max_score = 1
end


So any Numeric instance in the closed interval [0, 1] is a valid score for a fuzzy set key (we choose Numeric so as to allow both floats and integers). Finally, for multisets:

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class MultiSet < SetMap
@score_type = Integer
@min_score = 0
@max_score = Float::INFINITY
end


The Float::INFINITY constant has the property that x < Float::INFINITY for any numeric x. Setting @max_score to Float::INFINITY is thus a way of saying that, for multisets, there is no maximal score: any non-negative integer is allowed.

And this is really all there is to it! Specializing the capabilities of SetMap to a particular target class boils down to providing appropriate values for a bunch of class instance variables.

Of course, we have not yet demonstrated what the interface for SetMap actually looks like. But from now on, we will write methods that work equally well for all three set types under consideration: classical sets, fuzzy sets, and multisets.

### Key insertion

The most basic part of the interface of any set class is arguably the capability of inserting scores for particular keys. Here is the SetMap#insert method:

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def insert(key, val = 1)
raise(SetError, 'Illegal value') unless self.class.valid_score?(val)
old_score = self[key]
@hash[key] = [self[key] + val, self.class.max_score].min
@size = @size + (self[key] - old_score)
self
end


The general idea of this method is to increment the score of key by val. As per line 2 of the snippet, this will work only if val is a valid score according to the implementation of valid_score? (which in turn depends on the values of the class instance variables @score_type, @min_score and @max_score). If that is the case, we use what is called a bounded sum to add val to self[key], capping off the sum at @max_score (line 4).

Let’s try this out using our target classes:

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fuzzy_set = FuzzySet.new
fuzzy_set.insert('a', 0.5)
fuzzy_set.insert('a', 0.3)
fuzzy_set #=> #<FuzzySet: {"a": 0.8}>

multi_set = MultiSet.new
multi_set.insert('a')
multi_set.insert('a', 2)
multi_set #=> #<MultiSet: {"a": 3}>

classical_set = ClassicalSet.new
classical_set.insert('a')
classical_set.insert('a')
classical_set #=> #<ClassicalSet: {"a": 1}>


These are the desired results (assuming a—standard—inspect method which we have not shown). Notice that the score range we have specified for classical sets in tandem with the bounded sum ensures that inserting the same key twice has the same effect as inserting it once: the sum of 1 and 1 bounded by 1 is again 1.

Returning to the earlier snippet, we also need to keep track of the size of our set (line 5). Here, we also neutralize rounding errors that might occur for types of sets that allow floating point numbers as scores.

### Key retrieval

Next, consider SetMap#retrieve:

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def retrieve(key)
@hash[key] ? @hash[key] : 0
end
alias [] retrieve


The retrieve method (which we alias as []) wraps the element reference method of our internal hash. If the hash does not contain a certain key, @hash[key] will return nil. In that case, retrieve(key) (or, equivalently as per our alias, self[key]) will return 0. Alternatively, we could haver set a default value for @hash, but the current way seems slightly more explicit.

Observe that key retrieval is fast: accessing a hash key takes constant time on average (disregarding some fine-print), i.e., as the number of keys in a hash increases, the average time necessary to recover the value for a key does not increase. This is one of the main reasons why using hash tables to model sets is an attractive choice.

### Key removal

While it would be possible to tweak our approach and express removal of a key as insertion with a negative score, we prefer to keep things simple here:

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def remove(key, val = 1)
raise(SetError, 'Illegal value') unless self.class.valid_score?(val)
old_score = self[key]
@hash[key] = [self[key] - val, self.class.min_score].max
@size = @size - (old_score - self[key])
self
end


SetMap#remove is perfectly symmetric to the earlier insert method, using a bounded difference instead of a bounded sum. For our three target classes, this ensures that negative scores cannot occur.

### Enumeration

The below SetMap#each_pair method enumerates set keys and their associated scores in a straightforward manner, piggybacking on Hash#each_pair. Notice that we only yield key-value pairs for which the value is non-zero, since a key scored with value 0 is not considered part of our set.

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def each_pair

@hash.each_pair do |key, val|
yield([key, val]) if val != 0
end

self
end
alias each each_pair


As we will see in the next post, each_pair forms the basis for all our methods that iterate over sets. This includes pretty much all the interesting operations on sets—union, intersection, and the like. Since each_pair is aliassed as each, it also allows us to include the Enumerable module, which any respectable Ruby collection class should have access to.

Continue to part 03 of the series where we discuss operations on sets.

# Ben Rodenhäuser

Notes on programming