# Fibonacci in Rust

This post goes through different approaches to generating the Fibonacci sequence in Rust. It compares the speed of these approaches using the benchmarking crate criterion.

You’ve probably already heard of the Fibonacci sequence. It is a sequence of numbers named after the mathematician Leonardo of Pisa, that’s generated using a simple rule, yet pops up in many unexpected places in math and nature, often in relation to the golden ratio.

To generate the sequence, start with the sequence 1, 1. Then, generate the next element in the sequence by adding up the last two elements:

• 1, 1
• 1, 1, 2
• 1, 1, 2, 3

Mathematically, this can elegantly be expressed using recursion. In this formula, n represents a given position in the sequence starting with 0.

$fib(n) = \begin{cases} 1 &\text{if } n \leq 1 \\ fib(n-2)+fib(n-1) &\text{if } n \gt 1 \end{cases}$

## Implementation

We will be implementing the Fibonacci logic using Rust. Rust is a programming language initially developed at Mozilla that can guarantee memory safety, while at the same time being super fast. We’ll try out different approaches and see which one runs fastest.

If you want to follow along on your computer, you can find simple installation instructions for Rust and cargo, the Rust package manager, for just about any OS, on the Rust homepage: https://www.rust-lang.org/tools/install.

Alternatively, you can also code along online on the Rust playgroundwithout having to install anything!

### Setup

If you are not coding along on your machine, feel free to skip this section.

First, let’s create a new Rust library project from the command-line using cargo:

cargo new --lib rust_fibonaccicd rust_fibonacci/

The file src/lib.rs is where we’ll be writing source code later on. You can delete it’s pregenerated contents for now, as we won’t go through writing tests for such a simple program.

Now, the only aspect missing is the benchmarking, so that we can compare the different approaches to calculating the Fibonacci sequence. To do this, we can use the criterion crate, which allows us to write benchmarks and run them using cargo. Add the following code to the Cargo.toml file, just below the [dependencies] section at the bottom:

[dev-dependencies]criterion = "0.3"[[bench]]name = "fibonacci_benchmark"harness = false

Create a directory called benches in the base directory and place an empty file called fibonacci_benchmark.rs into it.

Once you’re done, the rust_fibonacci/ project directory content should look something like this:

.
├── benches
│   └── fibonacci_benchmark.rs
├── Cargo.lock
├── Cargo.toml
└── src
└── lib.rs


## #1: Standard

INFO
If you're using the Rust playground, just add all the code examples above the main function. The playground just allows you to use one single file to write your code in.

Now, let’s get coding! Probably the most straightforward way to implement the Fibonacci sequence, would be to just start with two variables a and b, that keep track of the last two elements of the sequence and build from there. Let’s do that! Add this code to src/lib.rs:

pub fn fib_standard(n: usize) -> usize {    let mut a = 1;    let mut b = 1;    for _ in 1..n {        let old = a;        a = b;        b += old;    }    b}

Notice the pub keyword: We use it, so that we can import the code from other files. This will be useful when benchmarking the functions later on.

Here, we declare a and b as mut, i.e. mutable, to allow us to mutate or change their values. In Rust, all variables are imutable by default.

We start our loop at 1, because the first two values are already defined and the range is non-inclusive for its end.

## #2: Recursion

Another approach to implementing the Fibonacci algorithm would be to just translate the recursive mathematical definition from the introduction into Rust code. Add the following function in src/lib.rs:

pub fn fib_recursive(n: usize) -> usize {    match n {        0 | 1 => 1,        _ => fib_recursive(n-2) + fib_recursive(n-1),    }}

In this case, the match operator comes in really handy. It works just like the conditional function definition in math! It is short, clean and concise. This means we don’t have to write endless if { ... } else if { ... } else { ... } clauses.

One last thing to note, is the implicit return. We did not have to use the return statement, because the last expression is automatically returned. However, it is important not to end with a semi-colon, which would make the function return nothing (or (), to be more precise).

When we take a close look at this function, it might become clear that it is pretty inefficient. When calculating fib_recursive(n), we end up calculating the Fibonacci sequence twice every step down from n, although it would be enough to calculate the sequence once. This is where memoizationcomes in.

Runtime complexity

The recursive approach has the runtime complexity $O(2^n)$.

This is because the time complexity of fib_recursive(n) approximately doubles for every n, because it computes fib_recursive(n-1) and fib_recursive(n-2).

If we want to be more exact about the statement “it approximately doubles”, we can say the following about this factor $a$:

\begin{aligned} a^n &= a^{(n-1)} &+ &a^{(n-2)} \quad | : a^{(n-2)} \\ a^2 &= a &+ &1 \\ a &= \frac{1 \pm \sqrt{5}}{2} \end{aligned}

We can safely ignore the second solution $\frac{1 - \sqrt{5}}{2}$, which is negative. This leaves us with $a = \frac{1 + \sqrt{5}}{2}$, the golden ratio. What a coincidence! The asymptotically tight bound on the running time of fib_recursive is thus $\Theta(a^n)$, where $a$ is the golden ratio.

You can find out more about asymptotic notation in computer science on Khan Academy.

## #3: Memoization

We will use a std::collections::HashMap, which is similar to a dict in Python or an Object in JavaScript, to keep track of all the values we’ve already calculated. Then, we can quickly check, whether a given value has already been encountered and can return this, before wasting time on a redundant calculation. Add this code to your lib.rs file:

use std::collections::HashMap;pub fn fib_memoization(n: usize, memo: &mut HashMap<usize, usize>) -> usize {    if let Some(v) = memo.get(&n) {        return *v;    }    let v = match n {        0 | 1 => 1,        _ => fib_memoization(n-2, memo) + fib_memoization(n-1, memo),    };    memo.insert(n, v);    v}

We first check, whether the current n is in the HashMap, by checking whether the value at n is Some. If no value has jet been recorded, memo.get(&n) will return None and the pattern won’t match.

Next, we compute the sequence value just as when using plain recursion. The only difference is, that we save the value to our memo before returning it.

Notice how we write &mut HashMap<usize, usize> in the function definition. This is part of Rust’s borrow checker, that ensure memory safety for our program. By declaring the memo as mutable, Rust ensures that only one part of the program has write access at a time and that no other part of the program can read from the memo while we have write access to it and might be modifying it unexpectedly.

## #4: Iterator

One last way to implement the Fibonacci sequence that this post will cover is using Rust iterators. You might be familiar with this concept, especially if you’ve already used Generators in JavaScriptor Iterators in Python.

Rust iterators implement the Iterator traitand expose a next function, which returns the next element of the iterator or None, if the iteration is over.

Lets implement this iterator principle using a struct. The struct will save the last two elements a and b of the sequence, starting at 1. It will then generate the next value just as in the standard approach:

pub struct FibIterator {    a: usize,    b: usize}impl Default for FibIterator {    fn default() -> Self {        FibIterator { a: 1, b: 1 }    }}impl Iterator for FibIterator {    type Item = usize;    fn next(&mut self) -> Option<Self::Item> {        let curr = self.a;        self.a = self.b;        self.b = curr + self.a;        Some(curr)    }}

A few things to notice here. First, notice how we all methods of the struct are wrapped in impl blocks. This separates the struct definition from it’s methods, helping your code stay clean.

Also, we write a default method that takes no arguments and returns an initialized FibIterator. As this method is not associated to a struct instance, i.e. an initialized FibIterator with concrete values for a and b, it is called an associated function. We can call these types of functions using ::. In this case, we would call FibIterator::default() to construct a new instance.

The Iterator and Default traits are implemented using the impl Foo for Bar statement. In the impl block of the Iterator trait, we define a next function that just returns the sum of the two last elements in the sequence. This way, the iterator can just keep generating new integers of the sequence on demand. Because iterators in Rust are lazy, these integers are only generated when needed.

INFO

A trait is a set of common functions all structs must implement, to have this trait. In the case of Iterator, this is solely the next function.

Traits are useful, because they allow other functions to accept different types, while making sure that all of these different types share a common interface.

When implementing Iterator, this trait unlocks a whole set of other useful methods such as skip, take, filter, and many more, that all rely on the next method we implemented. These all come built-in with the trait and we don’t need any additional work to implement these.

Iterators are an important part of Rust, as they allow to write code in a concise functional style, while incurring no additional performance. When compiling the code, Rust will optimize the operations away and turn the iterators into classical for loops in the background. That means you don’t have to choose between writing fast and clean code, you can do both!

## Bechmarking

Finally, we will compare the different approaches by benchmarking the different functions. Add the following code to the benches/fibonacci_benchmark.rs file. Note that benchmarking does not seem possible on the online Rust playground.

use criterion::{criterion_group, criterion_main, BenchmarkId, Criterion};use rust_fibonacci::*;use std::collections::HashMap;fn bench_fibs(c: &mut Criterion) {    let mut group = c.benchmark_group("Fibonacci");    for i in [20, 21].iter() {        group.bench_with_input(BenchmarkId::new("Standard", i), i, |b, i| {            b.iter(|| fib_standard(*i))        });        group.bench_with_input(BenchmarkId::new("Recursion", i), i, |b, i| {            b.iter(|| fib_recursive(*i))        });        group.bench_with_input(BenchmarkId::new("Memoization", i), i, |b, i| {            b.iter(|| {                let mut memo = HashMap::new();                fib_memoization(*i, &mut memo);            })        });        group.bench_with_input(BenchmarkId::new("Iterator", i), i, |b, i| {            b.iter(|| {                FibIterator::new().nth(*i).unwrap();            })        });    }    group.finish();}criterion_group!(benches, bench_fibs);criterion_main!(benches);

This code creates a test group called Fibonacci and benchmarks the four different approaches using the same input. Run the benchmark in your terminal:

cargo bench

Once the benchmarks are done, you can view a nice HTML report in your browser by opening target/criterion/Fibonacci/report/index.html. Running on my machine gave me the following stats:

You can clearly see, that the naive recursive solution is the least performant approach, as its execution time increases (exponentially, but not visible with 2 inputs) with the workload. The memoized version, in contrast, shows a great improvement, but it still incurs the performance overhead of initializing and managing the memo, making it less performant than the two last approaches.

The iterator and standard seem to be indistinguishable. On my machine, the execution of the iterator takes ~34ns for both inputs, the standard approach around ~4.5ns for both inputs.

You can find more detailed graphs and charts for every function in the corresponding target/criterion/Fibonacci/<APPROACH>/report/index.html folder.

## Conclusion

We’ve implemented and benchmarked four different approaches to generating the Fibonacci sequence.

Although the recursive solution is short and concise, it is by far the least performant and can become too slow to calculate for larger inputs. The memoized solution is interesting, in that it combines the conciseness of the recursive approach with a greater speed. The standard approach, on the other hand, seems to be the fastest, but it is arguably the least elegant.

Finally, the iterator solution appears to be by far the most versatile while at the same time being very fast. Additionally, it allows the user to work with the sequence in a very convenient way, e.g. by filtering, mapping, etc.

Overall, it becomes clear that iterators are a very versatile and performant aspect of Rust, that are also worth considering in other languages such as Python or JavaScript.

The final code of this project is open source and available here: https://github.com/umcconnell/rust_fibonacci

Feedback, questions, comments or improvements are welcome!