As a recent effort to improve my Computer Science fundamentals I decided to learn more about algorithms, how they work, how to prove their correctness and how to measure their efficiency. I’ve used different types of reference material for this and have left links where appropriate.
Notes on the implementation
To explore this algorithm I’ll be writing an implementation in Rust, but I will initially be implement some built in functions myself in order to fully understand what the algorithm is doing, but I will compare my implementation to a more idiomatic solution as well. Additionally, Rust’s borrow checker gives me some nice guard rails to both understand how the code is (not) memory safe and to me from doing stupid things. It’s also a nice exercise to simply learn more about the language, and I’ll nerd out about some implementation details whilst examining the algorithm.
Insertion sort
The insertion sort is “an efficient algorithm for sorting a small number of elements”^{1}. A common way to describe it would be how you would sort playing cards if you started with a pile of mixed card, and an empty hand. Every time you pick up a card you would insert it into the correct poistion so that the cards you’re holding are always sorted. While this is a nice example (it goes well with the name of the algorithm), it didn’t help me really visualize how the algorithm works. But let’s start with defining what is suppoed to happen in the first place:
$\begin{align*} \text{Input: }\quad & \langle a_1, a_2, \cdots a_n \rangle \\ \text{Output: }\quad & \langle a'_1, a'_2, \cdots a'_n \rangle \end{align*}$where the output is a permutation (read: reordering) such that every element in the array is in order, from smallest to largest:
$a'_1 \leq a'_2 \leq \cdots a'_n$Arguably there are actually 2 inputs, the first one being the sequence, the second one, $n$, being the length of the sequence, but because in all programming languages I know this can be trivially calculated from the array/list/vector/slice itself.
I asked ChatGPT to generate some pseudocode for the algorithm because I was too lazy to come up with a good balance of aestetic and readability myself. I think it did a pretty good job:
Procedure InsertionSort(A[0..n1])
Input: An array A[0..n1] of n elements
Output: Array A[0..n1] sorted in nondecreasing order
for i ← 1 to n1 do
key ← A[i]
// Insert A[i] into the sorted sequence A[0..i1]
j ← i  1
while j ≥ 0 and A[j] > key do
A[j + 1] ← A[j] // Move elements of A[0..i1], that are
// greater than key, to one position ahead
// of their current position
j ← j  1
A[j + 1] ← key
end for
end Procedure
The interesting point to note here is that we make use of a subarray, or sorted squence as it’s called in the pseudocode. In the common playing cards example, this subarray represents our hand. In the example we would naturally start off with an empty hand and fill it as we pick up cards, but when we write the algorithm we already start off with the first element of the array, in the pseudocode we can see this because we start the iteration at the second index (1 rather than 0).
In this example the subarray is not intuitive to track but it’s an implicite slice of the input,
where i
acts as a marker to increase it’s upper bound. After every iteration of the loop i
increments
which pushed the bound of the subarray from left to right (from smaller indices to larger ones).
The value at this marker gets assigned to a variable key
with the goal of finding the correct position
to insert it into the array (I guess that’s where the name comes from).
In order to find this position we follow some rules (sometimes also called loop invariants^{2} ^{3}):
 We ensure that we don’t underflow the array by checking
j
is not negative.  We check that the previous element,
A[j]
is greater thankey
, if it isn’t then we are already at a valid position where we can insertkey
If we fall withing these rules we moving elements to the right (we are swapping elments upwards),
and decrementing j
. Once we no longer meet the conditions for the inner loop, we have arrived
at a suitble insertion point and can bind the key
to that point in the array.
Implementation
Let’s write a very naive implementation that follows the pseudocode as closely as possible:
fn insertion_sort<T>(arr: &mut [T])
where
T: Ord + Copy,
{
let n = arr.len();
for i in 1..n {
let key = arr[i];
let sub_arr = &mut arr[0..=i];
let mut j = i as isize  1;
while j >= 0 && sub_arr[j as usize] > key {
sub_arr[(j + 1) as usize] = sub_arr[j as usize];
j = 1;
}
sub_arr[(j + 1) as usize] = key;
}
}
Now, I know what you’re thinking: “This looks like terrible code! Too much type casting! There are many better ways to do this!” And you’re right, this is pretty bad to look at and there is a better way to write this. But before we refactor this, let’s examine how the Rust codes works. I’ll start at the beginning, with the function signature.
Function signature
Since insertion sort operates on a sequence in place, I will use a slice, [T]
, since slices seem to be a
good data structure to work with portions of data^{4}.
The Ord
trait ensures that elements of the slice can be ordered in some way (a handy thing if you need
to sort things!).
The Copy
trait is required because we need a way to bind the value at arr[i]
to a variable. This trait
is implemented by default on primitive types, but since the signature is generic over T
we need to be
explicit in telling the compiler that elements of the slice are copyable.
Function body
The function body itself, as mentioned, follows the pseudocode implementation quite closely. Let’s start with the elephant(s) in the room: all those type casts.
I’ll be honest, this kind of confused me and if you’re like me and come from a TypeScript background,
this might be new to you as well. Unlike in many other languages, in Rust indexing into slices and vectors
can only be done with the usize
type (which is unsigned and therefore can never be negative). Easy enough.
However since we also perform some arithmatic on j
, we need to also cast this to a signed version isize
but since we also check that j
can never actually be 0 we cast it back to usize
.
The other thing you may have noticed is that I explicitly added a sub_arr
variable. I did this to make it
very clear that we are operating on a subarray of the input sequence. Both arr
and sub_arr
point to the
same space in memory, which means an mutating operation of one affects the other. It also means that it’s
completely redundant to do it this way, but it helped me understand how things worked.
Now that we understand how the Rust implementation work let’s refactor it slighly to get rid of those annoying type casts:
fn insertion_sort<T>(arr: &mut [T])
where
T: Ord + Copy,
{
for i in 1..arr.len() {
let sub_arr = &mut arr[0..=i];
let mut j = i;
#[allow(clippy::manual_swap)]
while j > 0 && sub_arr[j] < sub_arr[j  1] {
let tmp = sub_arr[j];
sub_arr[j] = sub_arr[j  1];
sub_arr[j  1] = tmp;
j = 1;
}
}
}
In this iteration we take advantage of some of the power of the rust compiler to avoid the various type casts.
The condition of our loop has changed to ensure it is always greater than 0 which means the compiler knows that j
after this point can never be an invalid value for indexing the slice.
As you may have noticed by the supressed clippy warning, we are also reinventing the wheel here by doing a manual swap operation.
Using swap here provides use with a nicer and (in my opinion) more readable way to reason about the code:
if sub_arr[j] < sub_arr[j  1]
then swap them. Keep doing that until the loop condition is no longer valid, in which
case all the items must be sorted.
Since we are using this approach we no longer need the key
variable. We are swaping values
Finally here is the most idiomatic solution I could come up with that implements insertion sort:
fn insertion_sort<T>(arr: &mut [T])
where
T: Ord + Copy,
{
for i in 1..arr.len() {
let mut j = i;
while j > 0 && arr[j] < arr[j  1] {
arr.swap(j, j  1);
j = 1;
}
}
}
Here we got rid of the explicit and redundant sub_array
and are also using the built in method for swapping elements
like normal people would.
Conclusion
Looking at the implementation of this algorithm I can understand why it’s best used on relatively small datasets (I am puerly going on my understanding of how computers work, I did not do any actual benchmarking), since we are iterating over each element, and reassigning elements every time.
In the worst case (where none of the elements are in order), insertion sort it could have a time complexity of: $O(n^2)$ where $n$ is the length of the input sequence.
The algorithm operates on a subarray of the input, which means it has a nested loop which means as the input grows larger, the number of operations increases quadratically.
However, since the algorithm only swaps adjecent elements in memory (which I think has a realtively small overhead), I think it could still be useful for datasets that are almost sorted or simply very small.
References
Footnotes

Cormen, T.H. et al. (2022) Introduction to Algorithms, fourth edition. MIT Press, p.18 ↩

Cormen, T.H. et al. (2022) Introduction to Algorithms, fourth edition. MIT Press, p.20 ↩

Wikipedia. (2020). Loop invariant. [online] Available at: https://en.wikipedia.org/wiki/Loop_invariant. ↩

Code Review Stack Exchange. (n.d.). Insertion sort in Rust. [online] Available at: https://codereview.stackexchange.com/questions/141946/insertionsortinrust [Accessed 13 Mar. 2024]. ↩