Why is insertion sort Θ(n^2) in the average case?

Question

Insertion sort has a runtime that is Ω(n) (when the input is sorted) and O(n²) (when the input is reverse sorted). On average, it runs in Θ(n²

Answer 1

To answer this question, let's first determine how we can evaluate the runtime of insertion sort. If we can find a nice mathematical expression for the runtime, we can then manipulate that expression to determine the average runtime.

The key observation we need to have is that the runtime of insertion sort is closely related to the number of inversions in the input array. An inversion in an array is a pair of elements A[i] and A[j] that are in the wrong relative order - that is, i < j, but A[j] < A[i]. For example, in this array:

0 1 3 2 4 5

There is one inversion: the 3 and 2 should be switched. In this array:

4 1 0 3 2

There are 6 inversions:

• 4 and 1
• 4 and 0
• 4 and 3
• 4 and 2
• 1 and 0
• 3 and 2

One important property of inversions is that a sorted array has no inversions in it, since every element should be smaller than everything coming after it and larger than everything coming before it.

The reason this is significant is that there is a direct link between the amount of work done in insertion sort and the number of inversions in the original array. To see this, let's review some quick pseudocode for insertion sort:

• For i = 2 .. n: (Assuming 1-indexing)
  • Set j = i - 1.
  • While A[j] > A[j + 1]:
    • Swap A[j] and A[j + 1].
    • Set j = j - 1.

Normally, when determining the total amount of work done by a function like this, we could determine the maximum amount of work done by the inner loop, then multiply it by the number of iterations of the outer loop. This will give an upper bound, but not necessarily a tight bound. A better way to account for the total work done is to recognize that there are two different sources of work:

• The outer loop, which counts 2, 3, ..., n, and
• The inner loop, which performs swaps.

That outer loop always does Θ(n) work. The inner loop, however, does an amount of work that's proportional to the total number of swaps made across the entire runtime of the algorithm. To see how much work that loop will do, we will need to determine how many total swaps are made across all iterations of the algorithm.

This is where inversions come in. Notice that when insertion sort runs, it always swaps adjacent elements in the array, and it only swaps the two elements if they form an inversion. So what happens to the total number of inversions in the array after we perform a swap? Well, graphically, we have this:

 [---- X ----] A[j] A[j+1] [---- Y ----]

Here, X is the part of the array coming before the swapped pair and Y is the part of the array coming after the swapped pair.

Let's suppose that we swap A[j] and A[j+1]. What happens to the number of inversions? Well, let's consider some arbitrary inversion between two elements. There are 6 possibilities:

• Both elements are in X, or both elements are in Y, or one element is in X and one element is in Y. Then the inversion is still there, since we didn't move any of those elements.
• One element is in X or Y and the other is either A[j] or A[j+1]. Then the inversion is still there, since the relative orderings of the elements haven't changed, even though their absolute positions might have.
• One element is A[j] and the other A[j+1]. Then the inversion is removed after the swap.

This means that after performing a swap, we decrease the number of inversions by exactly one, because only the inversion of the adjacent pair has disappeared. This is hugely important for the following reason: If we start off with I inversions, each swap will decrease the number by exactly one. Once no inversions are left, no more swaps are performed. Therefore, the number of swaps equals the number of inversions!

Given this, we can accurately express the runtime of insertion sort as Θ(n + I), where I is the number of inversions of the original array. This matches our original runtime bounds - in a sorted array, there are 0 inversions, and the runtime is Θ(n + 0) = Θ(n), and in a reverse-sorted array, there are n(n - 1)/2 inversions, and the runtime is Θ(n + n(n-1)/2) = Θ(n²). Nifty!

So now we have a super precise way of analyzing the runtime of insertion sort given a particular array. Let's see how we can analyze its average runtime. To do this, we'll need to make an assumption about the distribution of the inputs. Since insertion sort is a comparison-based sorting algorithm, the actual values of the input array don't actually matter; only their relative ordering actually matters. In what follows, I'm going to assume that all the array elements are distinct, though if this isn't the case the analysis doesn't change all that much. I'll point out where things go off-script when we get there.

To solve this problem, we're going to introduce a bunch of indicator variables of the form X_ij, where X_ij is a random variable that is 1 if A[i] and A[j] form an inversion and 0 otherwise. There will be n(n - 1)/2 of these variables, one for each distinct pair of elements. Note that these variables account for each possible inversion in the array.

Given these X's, we can define a new random variable I that is equal to the total number of inversions in the array. This will be given by the sum of the X's:

I = Σ X_ij

We're interested in E[I], the expected number of inversions in the array. Using linearity of expectation, this is

E[I] = E[Σ X_ij] = Σ E[X_ij]

So now if we can get the value of E[X_ij], we can determine the expected number of inversions and, therefore, the expected runtime!

Fortunately, since all the X_ij's are binary indicator variables, we have that

E[X_ij] = Pr[X_ij = 1] = Pr[A[i] and A[j] are an inversion]

So what's the probability, given a random input array with no duplicates, that A[i] and A[j] are an inversion? Well, half the time, A[i] will be less than A[j], and the other half of the time A[i] will be greater than A[j]. (If duplicates are allowed, there's a sneaky extra term to handle duplicates, but we'll ignore that for now). Consequently, the probability that there's an inversion between A[i] and A[j] is 1 / 2. Therefore:

E[I] = ΣE[X_ij] = Σ (1 / 2)

Since there are n(n - 1)/2 terms in the sum, this works out to

E[I] = n(n - 1) / 4 = Θ(n²)

And so, on expectation, there will be Θ(n²) inversions, so on expectation the runtime will be Θ(n² + n) = Θ(n²). This explains why the average-case behavior of insertion sort is Θ(n²).

Hope this helps!