15 May 2017

# Sorting in Ruby

Why does Ruby have two sorting methods, rather than one?

Ruby has two built-in methods for sorting collections: sort and sort_by. Both are contained in the Enumerable module, which any Ruby class can include as long as it implements a an each method for iterating over instances of the class. The elements of a collection also need to implement a three-way comparison method <=> (the “spaceship operator”) if we want to invoke sort or sort_by on that collection. My question for this post is this: why would there be two sort methods rather than just one?

## Exploring sort_by

Let’s first try to get a clearer picture how sort_by works. Start with an example. Suppose we would like to sort the following array by the numerical values of its string elements:

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arr = ['0', '10', '3']


This can be achieved using sort_by as follows:

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arr.sort_by { |string| string.to_i }
# => ['0', '3', '10']


Or using shorthand:

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arr.sort_by(&:to_i)
# => ['0', '3', '10']


Now the question is: how does sort_by do its magic? My initial hunch when exploring this topic was that sort_by seems fairly closely related to the functionality provided by the map method, also contained in Enumerable. There is a hint in the Ruby Docs that points in the same direction: “The current implementation of sort_by generates an array of tuples containing the original collection element and the mapped value.”

Based on this, it looks like what sort_by must be doing is something like this:

1. Transform the given array to an array of pairs (making use of the block that was passed).
2. Sort the array of pairs by accessing the second component of each pair (relying on the <=> method defined for the second component).
3. Project each pair to its first component.

The result of step 3 is your sorted array.

After some Googling, I found out that this is actually a pretty well-known technique, often called Schwartzian transform among Perl programmers. So it does look like sort_by works just in this way.

Let’s make things more concrete by implementing a toy version of sort_by ourselves. This will come in handy in the second part of the post, when we want to compare sort and sort_by. Plus it’s a nice exercise – when learning a language, it can be useful to re-implement methods of interest to deepen understanding, I have been told at Launch School.

First, we observe that we can actually express the above three steps in Ruby code fairly easily – this is where the map method comes into play. For our running example, observe that the sorted array can be obtained as follows:

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arr.map { |elem| [elem, elem.to_i] } # step (1)
.sort { |pair1, pair2| pair1.last <=> pair2.last } # step (2)
.map { |pair| pair.first } # step (3)
# => ['0', '3', '10']


Notice that we have replaced the invocation of sort_by with calls to map, sort and <=>. This is obviously a lot more cumbersome than using sort_by itself – but it makes it fairly clear how our re-implementation of sort_by should look like. Here it is:

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module Enumerable
def my_sort_by
self.map { |elem| [elem, yield(elem)] }
.sort { |pair1, pair2| pair1.last <=> pair2.last }
.map { |pair| pair.first }
end
end


Looking at the Rubinius code for sort_by – a hint I got from one of the instructors at Launch School when sharing a draft of this post – we see just this pattern: a map invocation followed by a sort invocation followed by a map invocation. Rubinius even has a special class for representing tuples of the required kind. It is called SortedElement and comes with a <=> method that compares instances based on the value of the second element of the tuple.

For our running example, my_sort_by yields the desired return value:

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arr.my_sort_by { |elem| elem.to_i } # => ['0', '3', '10']


Or, using shorthand:

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arr.my_sort_by(&:to_i) # => ['0', '3', '10']


Let’s now return to our original question: why would Ruby have two sort methods rather than just one?

## The cost of transformation

It turns out that there is a reason for favouring sort_by over sort, at least in certain scenarios: efficiency.

Both sort and sort_by are based on comparisons of elements of the collection we want to sort. Comparison-based sorting has a lower bound of $O(n \log n)$, which is to say that it is not possible to come up with a (comparison-based) sorting algorithm that performs better in a worst-case scenario. This is because, in the worst case, $n \log n$ comparisons of elements have to be made in order to determine the correct sort order. Ruby uses quicksort for sorting, an algorithm that has a worst case complexity of $O(n^2)$, but runs in $O(n \log n)$ on average (as it turns out, $O(n \log n)$ is the lower bound for the average case as well, so quicksort is optimal for average cases).

Since quicksort is the algorithm powering both sort and sort_by, wouldn’t it be reasonable to think that both methods should have the same performance? The answer is no, and the reason is that the overall picture is complicated by the fact that we often do not wish to sort a given collection as is (i.e., relying on the <=> operator provided for its elements), but rather relying on some special “property” of its elements, i.e., a sort criterion, or sort key. For example, we may want to sort user profiles by users’ last names, or available moves in a game by their expected utility – or strings by integer value, as in our running example above. The last name, the expected utility, the integer value – those would be our sort keys.

Now unless the sort keys are simply given to us along with the values we want to sort, we will have to compute those keys ourselves. This takes time above and beyond the actual sorting. And this is where sort and sort_by differ.

Both of these methods allow us to do key-based sorting by passing a block with the method invocation. Suppose we have a method key that transforms the elements of a collection list to sort keys of the required kind (in our running example, the relevant transformation is simply to convert strings to integers, because numeric values are what we want to sort by). Then we can sort our list with the sort method as follows:

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list.sort { |elem1, elem2| key(elem1) <=> key(elem2) }


Using sort_by, as we have seen above, we would do it like this:

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list.sort_by { |elem| key(elem) }


It looks like using sort_by saves us a little bit of typing. But we are also saving a lot of computation steps, potentially. Remember from above: what is happening “under the hood” when we invoke sort_by in the way just described is something like this:

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list.map { |elem| [elem, key(elem)] }
.sort { |pair1, pair2| pair1.last <=> pair2.last }
.map { |pair| pair.first }


Clearly, this procedure involves $n$ calls to the key method, assuming that the length of list is $n$: we call key once for each element elem of list, storing the pair [elem, key(elem)] in our intermediate array. However, when using sort, the number of calls to the key method may be quite a bit larger. As observed earlier, sorting our list involves making $O(n\log n)$ comparisons on average, and even $O(n^2)$ comparisons in the worst case (since quicksort is Ruby’s search algorithm of choice). If we use sort, each such comparison will require two on the fly calls to the key method. In cases where key itself is a time-consuming transformation, having to perform it $O(n\log n)$ times (or even $O(n^2)$, in the worst case) rather than merely $O(n)$ times will make a big difference indeed.

So computing the keys ahead of time, and saving them for later use – as sort_by does, a technique known as memoization – may come with a significant performance gain over computing the keys on an as-needed basis, as sort does. If, on the other hand, the transformation is trivial, sort may still be faster than sort_by – the time saved by avoiding calls to the key method may then be more than offset by the effort of calling map twice. The Ruby Docs for sort_by give an example of this. So, as usual, there are trade-offs involved.

Thanks to Pete Hanson from Launch School for valuable information on the topic of this post.

# Ben Rodenhäuser

Notes on programming