!

# Quicksort

• Divide and conquer

Given a list $L$, we divide the sorting problem into two sub-problems.

1. Sort the elements in $L$ smaller than $x$: $L_1$

2. Sort the elements in $L$ larger than $x$: $L_2$

• Combine

$\mathrm{sorted}(L) = \mathrm{sorted}(L_1) \oplus \{x\} \oplus \mathrm{sorted}(L_2)$

# Partition

Given a list $L$ and an element $x$ in $L$, partitioning $L[p \dots r]$ by the pivot element $x$ does the following:

Rearrange the elements in $L[p\dots r]$ so that there exists a position $q$ such that after rearrangment:

1. $L[q] = x$
2. $\forall i\in [p, q-1],\ L[i] \leq x$
3. $\forall i\in [q+1, r],\ L[i] \geq x$
4. !

!

## Version 0.1: simple, but takes more memory

Let’s implement partition(L, p, r) with the minimal effort:

• Pick $x = L[r]$
• Filter the list for smaller elements: $L_1 = \{y\in L[p\dots r-1]: y\leq x\}$
• Filter the list for larger elements: $L_2 = \{y\in L[p\dots r-1]: y > x\}$
• Reconstruct: $L[p \dots r] = L_1 \oplus \{x\} \oplus L_2$
• !
def partition(L, p, r):
x = L[r]
L1 = [y for y in L[p:r] if y <= x]
L2 = [y for y in L[p:r] if y > x]
L[p:r+1] = L1 + [x] + L2


!

## Version 0.2: more involved, but takes no additional memory

!

def partition(L, p, r):
x = L[r]
i = p - 1
for j in range(p, r):
if L[j] <= x:
i += 1
L[i], L[j] = L[j], L[i]
L[i+1], L[r] = L[r], L[i+1]


!

It only involves element swapping, so it requires zero additional memory.

But its correctness is much less obvious.

# Analysis of partition

!

def partition(L, p, r):
x = L[r]
i = p - 1
for j in range(p, r):
if L[j] <= x:
i += 1
L[i], L[j] = L[j], L[i]
L[i+1], L[r] = L[r], L[i+1]


!

Bookkeeping is the key:

• !
• $p$ and $r$ are the boundaries of the sublist to be partitioned.
• $r$ also holds the value of the pivot element.
• Elements in $[p\dots j]$ are processed.
• $i$ is a marker of the boundary between smaller and larger elements.

# Analysis of partition

!

def partition(L, p, r):
x = L[r]
i = p - 1
for j in range(p, r):
if L[j] <= x:
i += 1
L[i], L[j] = L[j], L[i]
L[i+1], L[r] = L[r], L[i+1]


!

Loop invariance:

After each iteration, $L[p, j]$ is properly partitioned:

• $L[p, i] \leq x$
• $L[i+1, j] > x$
• !

# Quicksort

def quicksort(L, p, r):
if p < r:
k = partition(L, p, r)
quicksort(L, p, k-1)
quicksort(L, k+1, r)


# Performance Analysis

!

We will only give a terse outline of the analysis. The full version is deferred to later lectures.

# Performance Analysis

Let there be $n$ elements in the array.

Key obsevation:

• partition(L, p, r) takes $(r - p)$ iterations. Each iteration has a fixed number of instructions. So, $T_\mathrm{partition} = c_1 (r-p) \leq c\cdot n$.

• At each invocation of quicksort(L, p, k-1) and quicksort(L, k+1, r), we need to determine the number of elements: $k-1-p$, and $r-k-1$.

# Performance Analysis: Worst-case

• The worst case for $T_\mathrm{partition} = n$
• The worst case for $k-1-p = n-1$
• The worst case for $r-k-1 = n-1$

We have enough to set-up an equation to estimate the number of instructions of quicksort(L, a, b):

Let $|L[a\dots b]| = n$.

$$\begin{eqnarray} T_\mathrm{quicksort}(n) &=& n + 2\cdot T_\mathrm{quicksort}(n-1) \end{eqnarray}$$

!

This leads to a (poor) performance characteristics of: $\approx n^2$

!

We will show how to solve such equations (known as recurrence equations) in the next section of this course.

# Performance Analysis: average case

Quicksort as the name suggests is actually quite quick (most of the time…)

Revisiting k = partition(L, p, q).

The pivot value should be somewhere in the middle of the subarray. Namely: $k\approx p+q/2$, and so $k-1-p \approx n/2$, and $q-k-1\approx n/2$.

Revisiting the recurrence equation:

$$\begin{eqnarray} T_\mathrm{quicksort}(n) &=& n + 2\cdot T_\mathrm{quicksort}(n/2) \end{eqnarray}$$

This leads to a satisfying performance characteristics of: $\approx n\log(n)$

# Summary

!

• Partition: a simple version

• Partition: a zero-memory version

• Quicksort: a recursive sorting algorithm

# Challenge

Is QUICKSORT a stable sorting algorithm?

Recall:

Stable sorting requires an additional condition on the permutation:

$$\forall i,j\in [0, \mathrm{length}(x)],\ i < j\ \mathrm{and}\ f(x[i], x[j]) = 0 \implies \pi(i) < \pi(j)$$