Announcements: Project 5Everything we have been learning thus far will

enable us to solve interesting problems

Project 5 will focus on applying the skills we have learned on a problem from biology, specifically computational biology

Email your group / group requests by Friday

Project 5 will be released tomorrow late afternoon

AnnouncementsMidterm 2: Same curve as Midterm 1

Pre Lab 15 will be released next Friday

Stale version of week 12 slides was accidently uploaded to the wiki (correct slides were presented in class) – this has been fixed, please re download

O(1) Exampledef isOdd(list): return (len(list)%2 == 1)

>>> isOdd([0])True>>> isOdd([0,1])False

Clicker Questiondef getFirst(list): if len(list) == 0: return -1 return (list[0])

A: O(n) B: O(n2)C: O(1)

>>> getFirst([])-1>>> getFirst([0,1,2,3])0>>> getFirst(["a", "b", "c"])'a’

Building the IntuitionLogic Puzzle:

You have 9 marbles. 8 marbles weigh 1 ounce each, & one marble weighs 1.5 ounces. You are unable to determine which is the heavier marble by looking at them. How do you find the marble which weighs more?

Solution 1: Weigh one marble vs another

What is the complexity of this solution?

Finding the complexityStep 1: What is our input?

The marbles

Step 2: How much work do we do per marble?We weight each marble once (except one)

Step 3: What is the total work we did?8 measurementsWhat if we had 100 marbles or 1000?

Clicker Question: What is the complexity of this

algorithm?

A: O(n) B: O(n2)C: O(1) D: O(log n)

We can do better!Lets pull some intuition from our search

algorithm that was O(log n)We want a way to eliminated ½ (or more) of the

marbles with each measurement

How might we do this?What about weighing multiple marbles at once?

The Optimal SolutionSplit the marbles into three groups

We can then weigh two of the groups

Finding the complexity of the optimal solution

Step 1: What is our input? The marbles

Step 2: How much work do we do per marble? Logarithmic

Step 3: What is the total work we did?2 measurementsWhat if we had 100 marbles or 1000?

What happens at each step?

We eliminated 2/3rds of the marbles

Clicker Question: What is the complexity of this

algorithm?

A: O(n) B: O(n2)C: O(1)D: O(log n)

SortingMotivation

We can answer questions like min/max very efficiently

We can search very efficientlyWhat if we need to search many many times

Many algorithms require their input to be sorted

Intuition behind bubble sort

Background reading (homework):

http://en.wikipedia.org/wiki/Bubble_sort

Bubble sort takes a list and returns a sorted listCompare each pair of adjacent items and swap

them if the one to the right is smaller Assumption: we want our list sorted from smallest to

greatest

Intuition behind bubble sort

[5, 7, 9, 0, 3, 5, 6]

[5, 7, 9, 0, 3, 5, 6]

[5, 7, 9, 0, 3, 5, 6]

[5, 7, 9, 0, 3, 5, 6]

[5, 7, 0, 9, 3, 5, 6]

[5, 7, 0, 3, 9, 5, 6]

[5, 7, 0, 3, 5, 9, 6]

[5, 7, 0, 3, 5, 6, 9]

Intuition behind bubble sort

[5, 7, 0, 3, 5, 6, 9]

[5, 7, 0, 3, 5, 6, 9]

[5, 7, 0, 3, 5, 6, 9]

[5, 0, 7, 3, 5, 6, 9]

[5, 0, 3, 7, 5, 6, 9]

[5, 0, 3, 5, 7, 6, 9]

[5, 0, 3, 5, 6, 7, 9]

[5, 0, 3, 5, 6, 7, 9]

Intuition behind bubble sort

How many passes do we have to do before we are guaranteed the list is sorted?n passes, where n is the length of the list

In each pass we do how much work?n-1 comparisons

What is the total work? Complexity?

Changing our Intuition into Code

def BubbleSort(myList): swapped = True while swapped: swapped = False for i in range(len(myList)-1): if myList[i] > myList[i+1]: temp = myList[i] myList[i] = myList[i+1] myList[i+1] = temp swapped = True return myList

The main loop

Keep executing theloop IF we do a swap

Changing our Intuition into Code

def BubbleSort(myList): swapped = True while swapped: swapped = False for i in range(len(myList)-1): if myList[i] > myList[i+1]: temp = myList[i] myList[i] = myList[i+1] myList[i+1] = temp swapped = True return myList

The loop that executes the swaps

Changing our Intuition into Code

def BubbleSort(myList): swapped = True while swapped: swapped = False for i in range(len(myList)-1): if myList[i] > myList[i+1]: temp = myList[i] myList[i] = myList[i+1] myList[i+1] = temp swapped = True return myList

Check if the twonumbersshould be swapped

Changing our Intuition into Code

def BubbleSort(myList): swapped = True while swapped: swapped = False for i in range(len(myList)-1): if myList[i] > myList[i+1]: temp = myList[i] myList[i] = myList[i+1] myList[i+1] = temp swapped = True return myList

Swap!

Fast Swapping of Two Variables

Python provides us the ability to perform the swap in a much more efficient manner

>>> a = 5>>> b = 7>>> a, b = b, a>>> print a7>>> print b5

variable1, variable 2 = variable2, variable1

Changing our Intuition into Code

def BubbleSort(myList): swapped = True while swapped: swapped = False for i in range(len(myList)-1): if myList[i] > myList[i+1]: myList[i], myList[i+1] =

myList[i+1], myList[i] swapped = True return myList

Swap!

HomeworkReach Chapter 11 from the text book

Can we sort faster?Bubble sort certainly will sort our data for us

Unfortunately it simply is not fast enough

We can sort faster!There are algorithms which sort in O(n log n) or log

linear time

Lets reason why this is the case

Observation 1: We can merge two sorted lists in

linear timeWhat is in the input?

Both the lists, n = total amount of elements

Why is the complexity linear?We must examine each element in each of the lists Its linear in the total amount of elements

O(len(list1) + len(list2)) = O(n)

Observation 1: We can merge two sorted lists in

linear time[5,9,10, 100, 555]

[3,4,12, 88, 535]

[3]

Observation 1: We can merge two sorted lists in

linear time[5,9,10, 100, 555]

[3,4,12, 88, 535]

[3, 4]

Observation 1: We can merge two sorted lists in

linear time[5,9,10, 100, 555]

[3,4,12, 88, 535]

[3,4,5]

Observation 1: We can merge two sorted lists in

linear time

[5,9,10, 100, 555]

[3,4,12, 88, 535]

[3,4,5,9]

Observation 1: We can merge two sorted lists in

linear time

[5,9,10, 100, 555]

[3,4,12, 88, 535]

[3,4,5,9,10]

Observation 1: We can merge two sorted lists in

linear time[5,9,10, 100, 555]

[3,4,12, 88, 535]

[3,4,5,9,10,12]

Observation 1: We can merge two sorted lists in

linear time[5,9,10, 100, 555]

[3,4,12, 88, 535]

[3,4,5,9,10,12, 88]

Observation 2Notice that merging two lists of length one ends

up producing a sorted list of length two

[5]

[3]

[3]

[5]

[3]

[3,5]

[5]

[3]

[3,5]

Lets build the intuition for Merge-Sort

We know we can merge sorted lists in linear time

We know that merging two lists of length one results in a sorted list of length two

Lets split our unsorted list into a bunch of lists of length one and merge them into progressively bigger lists!We split a list into two smaller lists of equal partsKeep splitting until we have lists of length one

Visual Representation

log(n)n elementsmerged

Putting it all togetherWe know that there are log(n) splits

At each “level” we split each list in two

We know that we need to merge a total of n elements at each “level”n * log(n) thus O(n log n)

SynopsisTook a look at the code for bubble sort

We learned an efficient way to swap the contents of two variables (or list locations)

We built the intuition as to why we can sort faster than quadratic time Introduced the concept of merge sort

HomeworkStart working on Project 5

Play around with the concepts presented in recitation as well as the pre labThey will help both with the lab AND project 5

Review the project 4 solutionMany of the same concepts will be used in project

5