python has a built-in sort method for lists:

```
>>> L = range(10)
>>> print L
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
>>> from random import *
>>> shuffle(L)
>>> print L
[3, 1, 7, 4, 5, 9, 0, 6, 2, 8]
>>> L.sort()
>>> print L
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
```

How does `L.sort()`

work, and is it a *good* sort to use?

How would *you* sort numbers in a list? If you had to write a function to sort a given list, how would you do it? Try it out using some playing cards -- see if you can come up with a sorting algorithm. Once you have an idea, demo it for me or a ninja. Make sure your function:

- works with the playing cards initially face down
- only uses compare and swap

Once you have shown us a working sort algorithm, see if you can code it up in a file called `sorts.py`

. There are many different ways to sort a list. We will add 3 or 4 different sorting functions to this file. Then we can just add `from sorts import *`

in our other programs that need a sorting function.

Make sure you add a testing section to `sorts.py`

. Here is a small example (assuming you called your sort function `mysort()`

:

```
if __name__ == "__main__":
from random import shuffle
N = 100
L = range(N)
shuffle(L)
mysort(L)
assert L == range(N)
```

In most languages, to swap two items in a list, you would do this (assuming `i`

and `j`

are valid indecies):

```
tmp = L[i] # save ith value to temporary location
L[i] = L[j] # copy jth value to ith position
L[j] = tmp # copy from tmp to jth position
```

Since this is a common operation, python provides a shortcut:

`L[j],L[i] = L[i],L[j]`

This is the *selection sort* algorithm:

- given a list
`L`

- find the smallest number in the list, swap it to position 0
- find the next smallest number in the list, swap it to position 1
- find the next smallest number in the list, swap it to position 2
- and so on....

It is called *selection sort* because each time you are selecting the smallest number from the remaining unsorted elements.

I assume you understand the "find the smallest number in a list" part, but here it is:

```
ismall = 0 # index of first element in list
for i in range(len(L)):
if L[i] < L[ismall]:
ismall = i # found a smaller item, so save position
# after the for loop, ismall should be the index of the
# smallest item in the list
print "smallest is ", L[ismall]
```

The selection sort uses the above code, but not just for position 0. A second *outer* `for loop`

is needed to run the above code for all values from 0 to the last index of the list.

Here is a video of the selection sort algorithm (click to play):

See if you can add `selectionSort()`

to our `sorts.py`

file of sorts. Include some test code at the bottom of `sorts.py`

to make sure your `selectionSort()`

function is working properly.

This is the *bubble sort* algorithm:

- given a list
`L`

- for every item in the list, compare to the item just to the right, swap if needed
- keep doing the above until you go from one end of the list to the other and don't make any swaps!

Here's a video of the bubble sort algorithm (click to play):

See if you can add `bubbleSort()`

to our `sorts.py`

file. Include some test code at the bottom of `sorts.py`

to make sure your `bubbleSort()`

function is working properly.

This is the *insertion sort* algorithm:

- assume item at index 0 is already "sorted"
- starting with item at index 1, check all items to the left and swap positions if needed
- do the same for item at index 2, where now items at index 0 and 1 should be in order
- do the same for item at index 3, where now items at index 0, 1, and 2 are in order ...and so on

Notice that, for each index, all items to the left are *in order*, and you are *inserting* the next item into the correct spot.

Here is a video of the insertion sort algorithm (click to play):

See if you can add `insertionSort()`

to our `sorts.py`

file. Include some test code at the bottom of `sorts.py`

!

Each of the sorting algorithms we considered last time (selection, bubble, and insertion sort) has *nested loops*, where each loop's length depends on the size on the list being sorted (*N*). This means the number of steps required for each sort, or the runtime, depends on the size of the list *squared*. These are called *quadratic* or N-squared sorts, because if the size of the list doubles, the number of steps **quadruples**. For example, if the runtime of the bubble sort was 10 seconds for a list of size `N`

, it would take approximately 40 seconds to sort a list of size `2N`

.

Let's see if we can show this by actually timing a few sorts. Here is the code we will use to time a particular sort:

```
from time import time
t1 = time()
selectionSort(L)
t2 = time()
print "selection sort time: %8.4f" % (t2-t1)
```

If we do the above for lists of size `N`

, `2N`

, and `4N`

, we should see the quadratic behavior.

```
$ python timesorts.py
N: 2000
selection sort time: 0.1416
bubble sort time: 0.4503
insertion sort time: 0.2378
python sort time: 0.0003
$ python timesorts.py
N: 4000
selection sort time: 0.5836
bubble sort time: 1.7433
insertion sort time: 0.9037
python sort time: 0.0007
$ python timesorts.py
N: 8000
selection sort time: 2.2732
bubble sort time: 7.1467
insertion sort time: 4.2049
python sort time: 0.0015
```

Notice three things:

- the quadratic nature of the first three sorts
- the built-in python sort appears to be much faster
- the built-in python sort does not appear to be quadratic

How does the python `L.sort()`

method work? Are there other, better sorting algorithms?

Here is the basic merge sort algorithm:

- if the list has two or more items in it, split it into two lists
- sort each of the two split lists
- merge the two sorted lists back together into one sorted list

The merge step is actually not that hard, given two sorted lists. If you know each list is already sorted, you can just compare the first item in each list, and take the smaller of the two.

A more interesting question is, how do we sort each of the split lists? Do we use selection sort, bubble sort, or some other sort?

Since we are talking about merge sort, can we use merge sort to sort each of the split lists? This is called *recursion*: we are writing a function (`mergeSort()`

), and somewhere in that function we call *that function* (or a copy of that function) to help solve the problem. This may seem odd, but it is perfectly fine. If you think about stack diagrams, calling any function just puts another function frame on the stack. Assuming there is an end somewhere, it is perfectly fine for a function to *call itself* and put another copy on the stack.

So here is a version of the `mergeSort()`

function, assuming we have a valid `merge()`

function, which is not hard to write (we will write it next week):

```
def mergeSort(L):
"""use merge sort to sort L from low to high"""
if len(L) > 1:
# split into two lists
half = len(L)/2
L1 = L[0:half]
L2 = L[half:]
# sort each list
mergeSort(L1)
mergeSort(L2)
# merge them back into one sorted list
merge(L1,L2,L)
```

We will write this and test it next week, after we learn all about recursion...

CS21 Topics