AnnouncementsNo Labs / Recitation this weekOn Friday we will talk about Project 3

Release late afternoon / evening tomorrowCryptography

Midterm Curve 0-35 D 36-39 C- 40-50 C 51-55 C+ 56-59 B- 60-68 B 69-74 B+ 75-79 A- 80-100 A

Data StructuresMore List MethodsOur first encoding

Matrix

CQ:Are these programs equivalent?

b = [‘h’,’e’,’l’,’l’,’o’]b.insert(len(b), “w”)print(b)

b = [‘h’,’e’,’l’,’l’,’o’]b.append(“w”)print(b)

21

A: yesB: no

Advanced List OperationsL = [0, 1, 2, 0]L.reverse()

print(L) will print: [0, 2, 1, 0]L.remove(0)

print(L) will print: [2, 1, 0]L.remove(0)

print(L) will print: [2, 1]print (L.index(2)) will print 0

Why are Lists useful?They provide a mechanism for creating a

collection of items

def doubleList(b): i = 0 while i < len(b): b[i] = 2 * b[i] i = i +1 return (b)

print(doubleList([1,2,3]))

Using Lists to Create Our Own Encodings

Python provides a number of encodings for us:Binary, ASCII, Unicode, RGB, Pictures etc.

We know the basic building blocks, but we are still missing something …We need to be able to create our own encodings

What if I want to write a program that operates on proteins?

Under the hood: what is a matrix?

A matrix is not “pre defined” in python We “construct” a way to encode a matrix

through the use of listsWe will see how to do so using a single list (not

ideal)We will see how to do so using a list of lists

Two different ways Row by Row Column by Column 1 2

3 4( )

ListsSuppose we wanted to extract the value 3

The first set of [] get the array in position 1 of y. The second [] is selecting the element in position 2 of that array. This is equiv. to:

y = [“ABCD”, [1,2,3] , ”CD”, ”D”]y[1][2]

z = y[1]z[2]

Lets make it more concrete!

Lets revisit encoding a matrix

Lets try something simple:A = [1, -1, 0, 2]B = [1, 0, 0, 0.5, 3, 4, -1, -3, 6]

Does this work?We lose a bit of information in this encoding

Which numbers correspond to which rowWe can explicitly keep track of rows through a

row length variable

B = [1, 0, 0, 0.5, 3, 4, -1, -3, 6]rowLength = 3B[rowLength*y +x]

Lets convince ourselves

x = 0y = 0B[3*0 + 0]

B = [1, 0, 0, 0.5, 3, 4, -1, -3, 6]rowLength = 3B[rowLength*y +x]

x = 1y = 1B[3*1 + 1]

x = 2y = 1B[3*1 + 2]

Can we encode it another way?

We can encode column by column, but we lose some information againWhich numbers correspond to which column

We can explicitly keep track of columns through a column length variable

B = [1, 0.5, -1, 0, 3, -3, 0, 4, 6]columnLength = 3B[columnLength*x + y]

Lets convince ourselves

x = 0y = 0B[3*0 + 0]

x = 1y = 1B[3*1 + 1]

x = 2y = 1B[3*2 + 1]

B = [1, 0.5, -1, 0, 3, -3, 0, 4, 6]columnLength = 3B[columnLength*x + y]

Lists of ListsRecall that when we had a string in our list

B = [“ABCD”, 0, 1, 3]We could utilize the bracket syntax multiple

timesprint B[0][1] would print B

Lists can store other ListsB = [[0, 1, 3], 4, 5, 6]

Another way to encode a Matrix

Lets take a look at our example matrix

What about this?B= [[1, 0, 0], [0.5, 3, 4], [-1, -3, 6]]

Why is this important?We can now write code that more closely

resembles mathematical notation i.e., we can use x and y to index into our matrix

B = [[1, 0, 0], [0.5, 3, 4], [-1, -3, 6]]for x in range(3): for y in range(3): print (B[x][y])

…but first some more notation

We can use the “*” to create a multi element sequence6 * [0] results in a sequence of 6 0’s[0, 0, 0, 0, 0, 0]3 * [0, 0] results in a sequence of 6 0’s[0, 0, 0, 0, 0, 0]10 * [0, 1, 2] results in what?

What is going on under the hood?

Lets leverage some algebraic properties3 * [0, 0] is another way to write[0, 0] + [0, 0] + [0, 0]

We know that “+” concatenates two sequences together

What about lists of lists?We have another syntax for creating lists

[ X for i in range(y)]This creates a list with y elements of X

Example: [ 0 for i in range(6)] ≡ [0]*6[0, 0 ,0 ,0 ,0 ,0]

Example: [[0, 0] for i in range(3)] ≡ [[0,0]]*3[[0, 0], [0, 0], [0, 0]]

What does this does: [2*[0] for i in range(3)]?

Lets put it all togetherm1 = [ [1, 2, 3, 0], [4, 5, 6, 0], [7, 8, 9, 0] ]m2 = [ [2, 4, 6, 0], [1, 3, 5, 0], [0, -1, -2, 0] ]m3= [ 4*[0] for i in range(3) ]

for x in range(3): for y in range(4): m3[x][y]= m1[x][y]+m2[x][y]

Data structuresWe have constructed our first data structure!

As the name implies, we have given structure to the dataThe data corresponds to the elements in the matrixThe structure is a list of lists

The structure allows us to utilize math-like notation

HomeworkRead through the entire project description

when it becomes available

Announcements: Project 3

Taking a look at cryptography

Message -> Encrypted Message -> Message

Encryption Decryption

Shift CipherHow many possible elements do we have?

Answer: 95 printable charactersWhat happens if we shift something more than

95 positions?Answer: we need to loop around (shifting 99 is the

same as shifting 4)What happens if we shift by a negative shift?

Answer: we can modify a negative shift into a positive shift

Simple ExampleConsider if we had 4 options: “A” “B” “C” “D”Assume encoding for “A” is 0, “B” is 1, “C” is 2,

and “D” is 3Shifting “A” by 2 yields “C” (0+2)Shifting “B” by 2 yields “D” (1+2)Shifting “C” by 2 yields “A” (2+2) ?Shifting “D” by 2 yields “D” (3+2)?

Modular Arithmetic Modular arithmetic allows us to count “in circles” 0 mod 4 = 0 1 mod 4 = 1 2 mod 4 = 2 3 mod 4 = 3 4 mod 4 = 0 5 mod 4 = 1 6 mod 4 = 2 7 mod 4 = 3

Simple ExampleConsider if we had 4 options: “A” “B” “C” “D”Assume encoding for “A” is 0, “B” is 1, “C” is 2,

and “D” is 3Shifting “A” by 2 yields “C” (0+2) mod 4 = 2Shifting “B” by 2 yields “D” (1+2) mod 4 = 3Shifting “C” by 2 yields “A” (2+2) mod 4 = 0Shifting “D” by 2 yields “B” (3+2) mod 4 = 1

Negative Shifts?Lets think about negative shifts … conceptually we

just “go the other way”“D” shifted by -1 is “C” (3-1) mod 4 = 2“C” shifted by -1 is “B” (2-1) mod 4 = 1 Observe that since we are effectively moving in a

circle, a negative shift can be expressed as a positive shift

“D” shifted by 3 is “C” (3 + 3) mod 4 = 2“C” shifted by 3 is “B” (2 + 3) mod 4 = 1

What about large negative shifts?

Observe that shifting -10 is equivalent to shifting -2 Why? We shift 2 times around our “circle” of letters and

then need to shift -2 additional lettersNotice that both shifting -10 and shifting -2 are

equivalent to shifting 2 -10 mod 4 = 2 -2 mod 4 = 2Notice that we can compute a positive shift from a

negative shift by: shift modulo the total number of elements

Clicker Question: did we encode the same matrix

in both programs?

[[1, 2], [3, 4]] [[1, 3], [2, 4]]21

A: yesB: noC: maybe

Lets explore why it depends

1 23 4

1 32 4

[[1, 3], [2, 4]]2

[[1, 2], [3, 4]]1

( () )

Both lists of lists can either be a row by row or acolumn by column encoding

Let us encode something else using lists!

We call this structure a tree

Root

Root

How might we encode such a structure?

What structures do we know of in python?Strings, Ranges, Lists

We know that lists allow us to encode complex structures via sub listsWe used sub list to encode either a row or a

column in the matrixWe used an ‘outer’ list of rows or columns as a

matrix

Can we use the same strategy?

How might we encode a simple tree?

Root

Leaf1 Leaf2 Leaf3

Tree = [‘Leaf1’, ‘Leaf2’, ‘Leaf3’]

Trees can be more complex

Root

Leaf1 Leaf2

Leaf3 Leaf4 Leaf5

Tree = [‘Leaf1’, ‘Leaf2’, [‘Leaf3’, ‘Leaf4’, ‘Leaf5’]]

Trees can be more complex

Root

Leaf1

Leaf2

Leaf3 Leaf4 Leaf5

Tree = [[‘Leaf0’,‘Leaf1’], ‘Leaf2’, [‘Leaf3’, ‘Leaf4’, ‘Leaf5’]]

Leaf0

Trees can be more complex

Root

Leaf1

Leaf2

Leaf3 Leaf4

Leaf6Tree = [[‘Leaf0’,‘Leaf1’], ‘Leaf2’, [‘Leaf3’, ‘Leaf4’, [‘Leaf5’, ‘Leaf6’]]]

Leaf0

Leaf5

What is the intuitionEach sub list encodes the ‘branches’ of the treeWe can think of each sub list as a ‘sub tree’We can use indexes (the bracket notation []) to

select out elements or ‘sub trees’

How can we select out the leaves?

Root

Leaf1

Tree[0]

Leaf2

Tree[1]

Leaf3

Tree[2]

Tree = [‘Leaf1’, ‘Leaf2’, ‘Leaf3’]

Indexes provide us a way to “traverse” the tree

Root

Leaf1

Leaf2

Leaf3 Leaf4

Leaf6Tree = [[‘Leaf0’,‘Leaf1’], ‘Leaf2’, [‘Leaf3’, ‘Leaf4’, [‘Leaf5’, ‘Leaf6’]]]

Leaf0

Leaf5

0

0

0

0 1

1

1

1

2

2

Indexes provide us a way to “traverse” the tree

Root

Leaf1

Leaf2

Leaf3 Leaf4

Leaf6Tree = [[‘Leaf0’,‘Leaf1’], ‘Leaf2’, [‘Leaf3’, ‘Leaf4’, [‘Leaf5’, ‘Leaf6’]]]Tree[2]

Leaf0

Leaf5

0

0

0

0 1

1

1

1

2

2

Indexes provide us a way to “traverse” the tree

Root

Leaf1

Leaf2

Leaf3 Leaf4

Leaf6Tree = [[‘Leaf0’,‘Leaf1’], ‘Leaf2’, [‘Leaf3’, ‘Leaf4’, [‘Leaf5’, ‘Leaf6’]]]Tree[2][2]

Leaf0

Leaf5

0

0

0

0 1

1

1

1

2

2

Indexes provide us a way to “traverse” the tree

Root

Leaf1

Leaf2

Leaf3 Leaf4

Leaf6Tree = [[‘Leaf0’,‘Leaf1’], ‘Leaf2’, [‘Leaf3’, ‘Leaf4’, [‘Leaf5’, ‘Leaf6’]]]Tree[2][2][1]

Leaf0

Leaf5

0

0

0

0 1

1

1

1

2

2

CQ: How do we select ‘Leaf4’ from the Tree?

Tree = [[‘Leaf0’,‘Leaf1’], ‘Leaf2’, [‘Leaf3’, ‘Leaf4’, [‘Leaf5’, ‘Leaf6’]]]

A: Tree[2][1]B: Tree[3][2]C: Tree[1][2]

Operations on TreesTrees, since they are encoded via lists support

the same operations lists supportWe can “+” two treesWe can embedded two trees within a list

This creates a larger tree with each of the smaller trees as sub trees

Example: Tree1 = [‘Leaf1’, ‘Leaf2’] Tree2 = [‘Leaf3’, ‘Leaf4’] Tree = [Tree1, Tree2]

“+” two trees

Leaf1

Leaf2

Leaf3 Leaf4Leaf0

Leaf6Leaf5Tree1 = [[‘Leaf0’, ‘Leaf1’]]Tree2 = [‘Leaf2’]Tree3 = [[‘Leaf3’, ‘Leaf4’, [‘Leaf5’, ‘Leaf6’]]]

“+” two trees

Leaf1

Leaf2

Leaf3 Leaf4Leaf0

Leaf6Leaf5Tree1 = [[‘Leaf0’, ‘Leaf1’]]Tree2 = [‘Leaf2’]Tree4 = Tree1+Tree2 [[‘Leaf0’, ‘Leaf1’], ‘Leaf2’]

“+” two trees

Leaf1

Leaf2

Leaf3 Leaf4Leaf0

Leaf6Leaf5Tree4 = [[‘Leaf0’, ‘Leaf1’], ‘Leaf2’]Tree3 = [[‘Leaf3’, ‘Leaf4’, [‘Leaf5’, ‘Leaf6’]]]Tree = Tree4+Tree3

Why are trees important?They are a fundamental structure in computer

scienceThey enable us to search very quickly

We will revisit trees later in the courseWhat have we covered so far:

Given a tree diagram we can write a list of lists Given a complex list we can select elements

HomeworkWork on Project 3Read Sections 11.1, 11.2, 11.3 (15 pages)