# Sort

### Introduction

Sorting is the process of placing elements from a collection in some kind of order. This suggests that sorting is an important area of study in computer science and sorting also provides instructive examples of how algorithms and data structures work together, and of how the correct choice of data structure can make dramatic improvements in execution speed and memory usage.

An understanding of recursion allows us to consider more sophisticated procedures for sorting, notably quick sort, one of the fastest sorting algorithms. An understanding of recursive algorithms also leads also to the recognition that data structures themselves can be recursive. This insight is illustrated through an analysis of the heap sort algorithm, in which a recursive data structure known as a heap is used for the purposes of sorting.

### 1. What is sorting?

#### Definition

Suppose we have a collection, $C$, of items of data, (and we limit our discussion to dealing mostly with linear structures). Sorting $C$ means rearranging (reordering) its data items, based on an ordering property of each item, into ascending or descending order (and a reordering of a sequence has exactly the same items as the original sequence, but in a different order).

If it is sorted into ascending order then if item $A$ comes before item $B$ it must also be true that $A \leq B$, according to the particular ordering property we are using. If $C$ is sorted in descending order then we would have $A \geq B$.

A formal, mathematical statement of sorting in, say, ascending order would run as follows:

Given a sequence of $n$ items $(x1, x2, …, xn)$, find a reordering $(x'_1, x'_2, ..., x'_n)$ such that $x'_1 \leq x'_2 \leq ... \leq x'_n$.

This leads us into writing our specification for ascending sort:

Name: Sort A sequence of elements $C = \{c_1, c_2, c_3, ..., c_n\}$ A re-ordering of $C$: $C' = \{c'_1, c'_2, c'_3, ..., c'_n\}$ All $c_i$ in $C$ must have the same ordering property $c'_1\leq c'_2\leq c'_3\leq ... \leq c'_n$

and we continue, in general terms, but conduct our discussion in terms of ascending sort for simplicity.

#### Complexity

To sort a small number of items, a complex sorting method may be more trouble than it is worth and the overhead may be too high – a simple, rough-and-ready, unsystematic method may suffice. We often refer to such simple strategies as naive sorting or straight sorting methods.

Sorting a large number of items can take a substantial amount of computing resources and the efficiency of a sorting algorithm becomes material, and every significant improvement can matter. Two different units of computation are commonly considered when measuring the complexity and evaluating the overall efficiency of a sorting algorithm:

1. Comparison, the most commonly used unit, where we determine if one item is larger or smaller than another; and
2. Swap, if, during sorting, two values are found to be out of order, exchanging the positions of the two items is a swap. This exchange is costly.

### 2. Naive Sorting

#### Bubble Sort

This algorithm iterates several times over the list. On the first pass it ‘bubbles’ the largest item up to its correct place, through a series of swaps. On the next pass it does the same with the next largest item, and so on, but the length of the section of the sequence over which further comparisons and swaps are made are reduced by 1 containing comparison to the unsorted portion. In this way, the items at the end, which are in their correct positions, no long need to be compared (as they have been sorted).



In [23]:

def bubbleSort(alist):
for passnum in range(len(alist)-1,0,-1):
for i in range(passnum):
if alist[i]>alist[i+1]:
temp = alist[i]
alist[i] = alist[i+1]
alist[i+1] = temp
# print(alist) TO SEE EACH STEP

# Note the use of three-step assignment rather than simulatenous assignment, i.e. a, b = b, a.

alist = [54,26,93,17,77,31,44,55,20]
bubbleSort(alist)
print(alist)




[17, 20, 26, 31, 44, 54, 55, 77, 93]



To analyze the bubble sort, we note that regardless of how the items are arranged in the initial list, $n−1$ passes will be made to sort a list of size $n$. Hence, the total number of comparisons is the sum of the first $n−1$ integers. The sum $n$ integers is $\frac{1}{2}n^2 + \frac{1}{2}n$ and thus the sum $n-1$ integers is $\frac{1}{2}n^2 + \frac{1}{2}n - n = \frac{1}{2}n^2 - \frac{1}{2}n$. Hence, there are $O(n^2)$ comparisons. In the best case, if the list is already ordered, no exchanges will be made. However, in the worst case, every comparison will cause an exchange. On average, we exchange half of the time.

A bubble sort is often considered the most inefficient sorting method since it must exchange items before the final location is known. These “wasted” exchange operations are very costly. However, because the bubble sort makes passes through the entire unsorted portion of the list, it has the capability to do something most sorting algorithms cannot. In particular, if during a pass there are no exchanges, then we know that the list must be sorted. A bubble sort can be modified to stop early if it finds that the list has become sorted. This means that for lists that require just a few passes, a bubble sort may have an advantage in that it will recognize the sorted list and stop.

The next piece of code shows this modification, which is often referred to as the short bubble.



In [24]:

def shortBubbleSort(alist):
exchanges = True
passnum = len(alist)-1
while passnum > 0 and exchanges:
exchanges = False
for i in range(passnum):
if alist[i]>alist[i+1]:
exchanges = True
temp = alist[i]
alist[i] = alist[i+1]
alist[i+1] = temp
# print(alist)
passnum = passnum-1

alist=[20,30,40,90,50,60,70,80,100,110]
shortBubbleSort(alist)
print(alist)




[20, 30, 40, 50, 60, 70, 80, 90, 100, 110]



If no exchanges are made during an inner loop, this means the list must already be sorted. No further processing is necessary, and so the short bubble sort algorithm stops at this point, rather than continue with unnecessary iterations. The Boolean variable exchanges changes from false to true as soon as an exchange is made. If it has not changed to true during an iteration of the inner loop, the algorithm terminates.

Both variants of bubble sort are afflicted with one particular potential problem, known as the problem of the hares and the tortoises:

The item 99 (a hare), located at the beginning of the list, makes its way speedily up to its place at the end. By contrast, 1 (a tortoise) is situated at the end of the list and crawls down to its correct place at the beginning with tortuous slowness.

In bubble sort, once the largest item in the sequence is encountered, it moves to its place at the end in a single pass. Larger items near the beginning of the sequence are the hares and are quickly swapped up to the end in this way. Smaller items near the end of the sequence are the tortoises, which trudge down to their places at the beginning only after all n − 1 iterations have occurred.

As with any algorithm, the actual input data that bubble sort is working on may seriously affect its performance. A worst-case scenario for bubble sort is that in which the input list it is given is already sorted, but in reverse order. In such a reverse-sorted list, tortoises will predominate.

Can anything be done about this? Many have tried, and the bubble sort algorithm has been subjected to many tweaks over the years; but analysis shows that the bubble sort and its minor improvements are inferior to both the insertion and the selection sorts; and in fact the bubblesort has hardly anything to recommend it except its catchy name.



In [ ]:

#### Bubble Sort