Functional linked lists in Python
Linked lists are fundamental data structures that every programmer should know. This article explains how to implement a simple linked list data type in Python using a functional programming style.
The excellent book Programming in Scala inspired me to play with functional programming concepts in Python. I ended up implementing a basic linked list data structure using a Lisp-like functional style that I want to share with you.
I wrote most of this using Pythonista on my iPad. Pythonista is a Python IDE-slash-scratchpad and surprisingly fun to work with. It’s great when you’re stuck without a laptop and want to explore some CS fundamentals :)
So without further ado, let’s dig into the implementation.
Constructing linked lists
Our linked list data structure consists of two fundamental building blocks:
Nil represents the empty list and serves as a sentinel for longer lists. The
cons operation extends a list at the front by inserting a new value.
The lists we construct using this method consist of nested 2-tuples. For example, the list
[1, 2, 3] is represented by the expression
cons(1, cons(2, cons(3, Nil))) which evaluates to the nested tuples
(1, (2, (3, Nil))).
Nil = None def cons(x, xs=Nil): return (x, xs) assert cons(0) == (0, Nil) assert cons(0, (1, (2, Nil))) == (0, (1, (2, Nil)))
Why should we use this structure?
First, the cons operation is deeply rooted in the history of functional programming. From Lisp’s cons cells to ML’s and Scala’s
:: operator, cons is everywhere – you can even use it as a verb.
Second, tuples are a convenient way to define simple data structures. For something as simple as our list building blocks, we don’t necessarily have to define a proper class. Also, it keeps this introduction short and sweet.
Third, tuples are immutable in Python which means their state cannot be modified after creation. Immutability is often a desired property because it helps you write simpler and more thread-safe code. I like this article by John Carmack where he shares his views on functional programming and immutability.
Abstracting away the tuple construction using the
cons function gives us a lot of flexibility on how lists are represented internally as Python objects. For example, instead of using 2-tuples we could store our elements in a chain of anonymous functions with Python’s
def cons(x, xs=Nil): return lambda i: x if i == 0 else xs
To write simpler tests for more complex list operations we’ll introduce the helper function
lst. It allows us to define list instances using a more convenient syntax and without deeply nested
def lst(*xs): if not xs: return Nil else: return cons(xs, lst(*xs[1:])) assert lst() == Nil assert lst(1) == (1, Nil) assert lst(1, 2, 3, 4) == (1, (2, (3, (4, Nil))))
All operations on linked lists can be expressed in terms of the three fundamental operations
headreturns the first element of a list.
tailreturns a list containing all elements except the first.
Trueif the list contains zero elements.
You’ll see later that these three operations are enough to implement a simple sorting algorithm like insertion sort.
def head(xs): return xs assert head(lst(1, 2, 3)) == 1
def tail(xs): return xs assert tail(lst(1, 2, 3, 4)) == lst(2, 3, 4)
def is_empty(xs): return xs is Nil assert is_empty(Nil) assert not is_empty(lst(1, 2, 3))
Length and concatenation
length operation returns the number of elements in a given list. To find the length of a list we need to scan all of its n elements. Therefore this operation has a time complexity of O(n).
def length(xs): if is_empty(xs): return 0 else: return 1 + length(tail(xs)) assert length(lst(1, 2, 3, 4)) == 4 assert length(Nil) == 0
concat takes two lists as arguments and concatenates them. The result of
concat(xs, ys) is a new list that contains all elements in
xs followed by all elements in
ys. We implement the function with a simple divide and conquer algorithm.
def concat(xs, ys): if is_empty(xs): return ys else: return cons(head(xs), concat(tail(xs), ys)) assert concat(lst(1, 2), lst(3, 4)) == lst(1, 2, 3, 4)
Last, init, and list reversal
The basic operations
tail have corresponding operations
last returns the last element of a non-empty list and
init returns all elements except the last one (the initial elements).
def last(xs): if is_empty(tail(xs)): return head(xs) else: return last(tail(xs)) assert last(lst(1, 3, 3, 4)) == 4
def init(xs): if is_empty(tail(tail(xs))): return cons(head(xs)) else: return cons(head(xs), init(tail(xs))) assert init(lst(1, 2, 3, 4)) == lst(1, 2, 3)
Both operations need O(n) time to compute their result. Therefore it’s a good idea to reverse a list if you frequently use
init to access its elements. The
reverse function below implements list reversal, but in a slow way that takes O(n²) time.
def reverse(xs): if is_empty(xs): return xs else: return concat(reverse(tail(xs)), cons(head(xs), Nil)) assert reverse(Nil) == Nil assert reverse(cons(0, Nil)) == (0, Nil) assert reverse(lst(1, 2, 3, 4)) == lst(4, 3, 2, 1) assert reverse(reverse(lst(1, 2, 3, 4))) == lst(1, 2, 3, 4)
Prefixes and suffixes
The following operations
tail by returning arbitrary prefixes and suffixes of a list. For example,
take(2, xs) returns the first two elements of the list
drop(3, xs) returns everything except the last three elements in
def take(n, xs): if n == 0: return Nil else: return cons(head(xs), take(n-1, tail(xs))) assert take(2, lst(1, 2, 3, 4)) == lst(1, 2)
def drop(n, xs): if n == 0: return xs else: return drop(n-1, tail(xs)) assert drop(1, lst(1, 2, 3)) == lst(2, 3) assert drop(2, lst(1, 2, 3, 4)) == lst(3, 4)
Random element selection on linked lists doesn’t really make sense in terms of time complexity – accessing an element at index n requires O(n) time. However, the element access operation
apply is simple to implement using
def apply(i, xs): return head(drop(i, xs)) assert apply(0, lst(1, 2, 3, 4)) == 1 assert apply(2, lst(1, 2, 3, 4)) == 3
More complex examples
The three basic operations
is_empty are all we need to implement a simple (and slow) sorting algorithm like insertion sort.
def insert(x, xs): if is_empty(xs) or x <= head(xs): return cons(x, xs) else: return cons(head(xs), insert(x, tail(xs))) assert insert(0, lst(1, 2, 3, 4)) == lst(0, 1, 2, 3, 4) assert insert(99, lst(1, 2, 3, 4)) == lst(1, 2, 3, 4, 99) assert insert(3, lst(1, 2, 4)) == lst(1, 2, 3, 4) def isort(xs): if is_empty(xs): return xs else: return insert(head(xs), isort(tail(xs))) assert isort(lst(1, 2, 3, 4)) == lst(1, 2, 3, 4) assert isort(lst(3, 1, 2, 4)) == lst(1, 2, 3, 4)
to_string operation flattens the recursive structure of a given list and returns a Python-style string representation of its elements. This is useful for debugging and makes for a nice little programming exercise.
def to_string(xs, prefix="[", sep=", ", postfix="]"): def _to_string(xs): if is_empty(xs): return "" elif is_empty(tail(xs)): return str(head(xs)) else: return str(head(xs)) + sep + _to_string(tail(xs)) return prefix + _to_string(xs) + postfix assert to_string(lst(1, 2, 3, 4)) == "[1, 2, 3, 4]"
Where to go from here
This article is more of a thought experiment than a guide on how to implement a useful linked list in Python. Keep in mind that the above code has severe restrictions and is not fit for real life use. For example, if you use this linked list implementation with larger example lists you’ll quickly hit recursion depth limits (CPython doesn’t optimize tail recursion).
I spent a few fun hours playing with functional programming concepts in Python and I hope I inspired you to do the same. If you want to explore functional programming in ‘real world’ Python check out the following resources: