Fibonacci

This blog post looks at a few algorithms to generate Fibonacci numbers. For a much better treatment of these algorithms, I recommend From Mathematics to Generic Programming. The implementations are provided in poorly written Rust, as I’m just learning the language.

Learning Rust and going through The Rust Programming Language, I got to the end of chapter 3, where there are a few simple exercises. One of them is Generate the nth Fibonacci number.

The Fibonacci Sequence

The Fibonacci sequence is defined as the sequence \(F_n = F_{n-1} + F_{n-2}\) with the seed values \(F_0 = 0\) and \(F_1 = 1\).

The first few values of the sequence are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ….

A Naïve Algorithm

Directly translating the definition above into an algorithm to compute the n-th Fibonacci number yields:

fn fib(n: i32) -> i32 {
    if n == 0 { return 0; }
    if n == 1 { return 1; }

    fib(n - 1) + fib(n - 2)
}

This algorithm works for very small n values but it is extremely inefficient as it has exponential time complexity and linear space complexity (based on stack depth). Since fib(n - 1) = fib(n - 2) + fib(n - 3), a call like fib(n - 1) + fib(n - 2) is equivalent to (fib(n - 2) + fib(n - 3)) + fib(n - 2), so the same elements of the sequence end up being computed over and over again.

Bottom-Up Approach

A better way to generate the nth Fibonacci number is to build it bottom-up, starting from \(F_0\) and \(F_1\):

fn fib2(n: i32) -> i32 {
    if n == 0 { return 0; }

    let mut a = 0;
    let mut b = 1;

    for _ in 1..n {
        let c = a + b;
        a = b;
        b = c;
    }

    b
}

Here we start with \(a = F_0, b = F_1\) and with each iteration of the loop we advance from \(a = F_k, b = F_{k+1}\) to \(a = F_{k+1}, b = F_{k+2}\) until b becomes \(F_n\).

Compared to the first algorithm, this is highly efficient, as it has linear complexity and requires constant space. There are faster ways to compute the nth Fibonacci number though.

Matrix Form

The Fibonacci sequence can also be described in matrix form as follows:

\[\begin{split}\begin{bmatrix} F_{k+2} \\ F_{k+1} \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix} * \begin{bmatrix} F_{k+1} \\ F_k \end{bmatrix}\end{split}\]

Note that the next pair of numbers in the sequence, \(F_{k+3}, F_{k+2}\) can be expressed as:

\[\begin{split}\begin{bmatrix} F_{k+3} \\ F_{k+2} \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix} * \begin{bmatrix} F_{k+2} \\ F_{k+1} \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix} * \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix} * \begin{bmatrix} F_{k+1} \\ F_k \end{bmatrix}\end{split}\]

Thus we have the formula:

\[\begin{split}\begin{bmatrix} F_n \\ F_n - 1 \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}^{n-1} * \begin{bmatrix} F_1 \\ F_0 \end{bmatrix}\end{split}\]

Since \(F_0 = 0\) and \(F_1 = 1\), \(F_n\) is the element at index \((0, 0)\) in:

\[\begin{split}\begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}^{n-1}\end{split}\]

Assuming we have a function for exponentiating 2x2 matrices exp2x2, we can implement an algorithm to compute the nth Fibonacci number like this:

fn fib3(n: i32) -> i32 {
    if n == 0 { return 0; }
    if n == 1 { return 1; }

    let a = [
        [1, 1],
        [1, 0],
    ];

    exp2x2(a, n - 1)[0][0]
}

The complexity of this algorithm is given by the complexity of exp2x2. A simple implementation of matrix exponentiation given a matrix multiplication function mul2x2 is:

fn exp2x2(a: [[i32; 2]; 2], n: i32) -> [[i32; 2]; 2] {
    let mut result = [
        [1, 0],
        [0, 1],
    ];

    for _ in 0..n {
        result = mul2x2(result, a);
    }

    result
}

This function computes a^n by starting with the identity matrix and multiplying it with a n times. The function for multiplying two 2x2 matrices is trivial:

fn mul2x2(a: [[i32; 2]; 2], b: [[i32; 2]; 2]) -> [[i32; 2]; 2] {
    [
        [a[0][0] * b[0][0] + a[1][0] * b[0][1], a[0][0] * b[1][0] + a[1][0] * b[1][1]],
        [a[0][1] * b[0][0] + a[1][1] * b[0][1], a[0][1] * b[1][0] + a[1][1] * b[1][1]],
    ]
}

With mul2x2 and exp2x2, our fib3 algorithm has linear complexity, which is determined by the number of times we call mul2x2 in our exponentiation function. There is a faster way to do exponentiation though: observe that \(x^7 = x^4 * x^2 * x\). In general, any number n and can be decomposed as a series of powers of two. So we can implement a fast_exp2x2 which works as follows:

if n == 1, return a
if n is even, return fast_exp2x2(a * a, n / 2)
if n is odd, return fast_exp2x2(a * a, (n - 1) / 2) * a

We stop when our exponent is 1 and return a. If n is even, we square the base and halve the exponent (for example \(x^8 = (x*x)^4\)). If n is odd, we do the same but multiply by the base (for example \(x^9 = (x*x)^4 * x\)). This is a recursive algorithm which halves n at each step, so we have logarithmic time and space complexity.

fn fast_exp2x2(a: [[i32; 2]; 2], n: i32) -> [[i32; 2]; 2] {
    if n == 1 { return a; }

    let mut result = fast_exp2x2(mul2x2(a, a), n >> 1);

    if n & 1 == 1 {
        result = mul2x2(a, result);
    }

    result
}

This is a very efficient way to compute the nth Fibonacci number. But where does this fast exponentiation algorithm come from?

Ancient Egyptian Multiplication

The ancient Egyptian multiplication algorithm comes from the Rhind Papyrus from around 1500 BC. The idea is very similar to our fast exponentiation algorithm: we can implement a fast multiplication algorithm by relying on addition and doubling (eg. \(x * 7 = x * 4 + x * 2 + x\)). The steps or our egyptian_mul algorithm are:

if n == 1, return a
if n is even, return egyptian_mul(a + a, n / 2)
if n is odd, return egyptian_mul(a + a, (n - 1) / 2) + a

An implementation in Rust is:

fn egyptian_mul(a: i32, n: i32) -> i32 {
    if n == 1 { return a; }

    let mut result = egyptian_mul(a + a, n >> 1);

    if n & 1 == 1 {
        result += a;
    }

    result
}

This multiplication algorithm relies only on addition, and multiplies a by n in \(log(n)\) steps.

egyptian_mul and fast_exp2x2 algorithms have the same structure since they are fundamentally the same: they provide an efficient way to implement an operation defined as applying another operation n times. Multiplication is, by definition, repeated addition. Similarly, exponentiation is, by definition, repeated multiplication. We can generalize these to an algorithm that given an initial value a of any type T, an operation op(T, T) -> T, and n, the number of times to apply op, provides an efficient computation using doubling and halving:

fn op_n_times<T, Op>(a: T, op: &Op, n: i32) -> T
    where T: Copy,
          Op: Fn(T, T) -> T {
    if n == 1 { return a; }

    let mut result = op_n_times::<T, Op>(op(a, a), &op, n >> 1);
    if n & 1 == 1 {
        result = op(a, result);
    }

    result
}

We can express Egyptian multiplication (egyptian_mul) as addition applied n times:

fn egyptian_mul(a: i32, n: i32) -> i32 {
    op_n_times(a, &std::ops::Add::add, n)
}

Similarly, we can express fast matrix exponentiation (fast_exp2x2) as matrix multiplication applied n times:

fn fast_exp2x2(a: [[i32; 2]; 2], n: i32) -> [[i32; 2]; 2] {
    op_n_times(a, &mul2x2, n)
}

Industrial Strength Fibonacci

I wanted to benchmark the two interesting implementations: fib2 and fib4. The first exponential complexity implementation is highly inefficient and even for small values of N (eg. N = 50) it takes a long time to complete. fib3 has linear complexity like fib2, but while fib2 just performs additions and assignments on each iteration, fib3 performs matrix multiplication, which is more expensive. So fib2 and fib4 are more interesting to look at.

Turns out that the Fibonacci sequence grows quite fast, the 100th Fibonacci number is 354224848179261915075, which does not fit in an i32. So let’s update the implementations to use num::BigUint, an arbitrary precision number. First is fib2:

extern crate num;

use num::bigint::{BigUint, ToBigUint};

fn fib2(n: i32) -> BigUint {
    if n == 0 { return 0.to_biguint().unwrap(); }

    let mut a = 0.to_biguint().unwrap();
    let mut b = 1.to_biguint().unwrap();

    for _ in 1..n {
        let c = &a + &b;
        a = b;
        b = c;
    }

    b
}

For fib4, we need to update mul2x2 to work with BigUint array references, so we don’t copy BigUint arrays:

fn mul2x2(a: &[[BigUint; 2]; 2], b: &[[BigUint; 2]; 2]) -> [[BigUint; 2]; 2] {
    [
        [&a[0][0] * &b[0][0] + &a[1][0] * &b[0][1], &a[0][0] * &b[1][0] + &a[1][0] * &b[1][1]],
        [&a[0][1] * &b[0][0] + &a[1][1] * &b[0][1], &a[0][1] * &b[1][0] + &a[1][1] * &b[1][1]],
    ]
}

We also need to update our op_n_times so the operation now takes &T instead of T. Note this version still works with i32 arrays and numbers, but now the operation is expected to take two references instead of two values. On the other hand we no longer require that T has the Copy trait:

fn op_n_times<T, Op>(a: T, op: &Op, n: i32) -> T
    where Op: Fn(&T, &T) -> T {
    if n == 1 { return a; }

    let mut result = op_n_times::<T, Op>(op(&a, &a), &op, n >> 1);
    if n & 1 == 1 {
        result = op(&a, &result);
    }

    result
}

Then we can update our fib4 implementation to use BigUint:

fn fast_exp2x2(a: [[BigUint; 2]; 2], n: i32) -> [[BigUint; 2]; 2] {
    op_n_times(a, &mul2x2, n)
}

fn fib4(n: i32) -> BigUint {
    if n == 0 { return 0.to_biguint().unwrap(); }
    if n == 1 { return 1.to_biguint().unwrap(); }

    let a = [
        [1.to_biguint().unwrap(), 1.to_biguint().unwrap()],
        [1.to_biguint().unwrap(), 0.to_biguint().unwrap()],
    ];

    fast_exp2x2(a, n - 1)[0][0].clone()
}

These two implementations work with arbitrarily large numbers, for example fib4(10_000) is:

33644764876431783266621612005107543310302148460680063906564769974680081442166662368155595513633734025582065332680836159373734790483865268263040892463056431887354544369559827491606602099884183933864652731300088830269235673613135117579297437854413752130520504347701602264758318906527890855154366159582987279682987510631200575428783453215515103870818298969791613127856265033195487140214287532698187962046936097879900350962302291026368131493195275630227837628441540360584402572114334961180023091208287046088923962328835461505776583271252546093591128203925285393434620904245248929403901706233888991085841065183173360437470737908552631764325733993712871937587746897479926305837065742830161637408969178426378624212835258112820516370298089332099905707920064367426202389783111470054074998459250360633560933883831923386783056136435351892133279732908133732642652633989763922723407882928177953580570993691049175470808931841056146322338217465637321248226383092103297701648054726243842374862411453093812206564914032751086643394517512161526545361333111314042436854805106765843493523836959653428071768775328348234345557366719731392746273629108210679280784718035329131176778924659089938635459327894523777674406192240337638674004021330343297496902028328145933418826817683893072003634795623117103101291953169794607632737589253530772552375943788434504067715555779056450443016640119462580972216729758615026968443146952034614932291105970676243268515992834709891284706740862008587135016260312071903172086094081298321581077282076353186624611278245537208532365305775956430072517744315051539600905168603220349163222640885248852433158051534849622434848299380905070483482449327453732624567755879089187190803662058009594743150052402532709746995318770724376825907419939632265984147498193609285223945039707165443156421328157688908058783183404917434556270520223564846495196112460268313970975069382648706613264507665074611512677522748621598642530711298441182622661057163515069260029861704945425047491378115154139941550671256271197133252763631939606902895650288268608362241082050562430701794976171121233066073310059947366875

We can benchmark these implementations using Rust’s built-in benchmarking:

#[bench]
fn fib4_bench(b: &mut Bencher) {
    b.iter(|| fib4(N));
}

#[bench]
fn fib2_bench(b: &mut Bencher) {
    b.iter(|| fib2(N));
}

On my Surface Book, for N = 100, we have:

test tests::fib2_bench ... bench:       7,805 ns/iter (+/- 3,042)
test tests::fib4_bench ... bench:       6,140 ns/iter (+/- 356)

For N = 1_000:

test tests::fib2_bench ... bench:      89,131 ns/iter (+/- 28,346)
test tests::fib4_bench ... bench:      16,307 ns/iter (+/- 2,087)

For N = 10_000:

test tests::fib2_bench ... bench:   2,121,140 ns/iter (+/- 132,448)
test tests::fib4_bench ... bench:     184,625 ns/iter (+/- 12,184)

For N = 100_000:

test tests::fib2_bench ... bench: 128,769,418 ns/iter (+/- 5,198,789)
test tests::fib4_bench ... bench:   7,176,026 ns/iter (+/- 364,400)

While fib2 and fib4 start with about the same performance at N = 100, for N = 100_000, fib4 is significantly faster. The benchmark results don’t seem to reflect the algorithmic complexity of fib2 (linear) and fib4 (logarithmic), I suspect because of the introduction of BigUint and operations on large numbers. Still, the algorithm relying on fast exponentiation performs many times faster on large Ns.

Summary

This blog post covered:

  • Algorithms to generate Fibonacci numbers: naïve recursive (exponential), bottom-up (linear), matrix exponentiation (linear or logarithmic, depending on the matrix exponentiation algorithm).
  • Ancient Egyptian multiplication and fast matrix exponentiation are the same algorithm applied to different operations.
  • A generic algorithm of efficiently applying an operation n times.
  • Algorithms to generate Fibonacci numbers implemented with BigUint for arbitrary precision numbers.
  • Benchmarking the fib2 and fib4 algorithms shows fib4 to be much better as N increases.

My humble conclusion is that generating Fibonacci numbers is more than an introductory exercise.