CS 177 Week 9 Recitation Slides

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Matrix EncodingMatrix Multiplication

Trees as ListsColumnar Transposition

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Announcements

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Matrix Encoding: Row by Row

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A

1 1 0

3 2 8

5 -3 5

0,0 0,1 0,2

1,0 1,1 1,2

2,0 2,1 2,1

A = [1, 1, 0, 3, 2, 8, 5, -3, 5]rowLength = 3A[rowLength*y +x]

Matrix Encoding: Column by Column

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A

1 1 0

3 2 8

5 -3 5

0,0 0,1 0,2

1,0 1,1 1,2

2,0 2,1 2,1

A = [1, 3, 5, 1, 2, -3, 0, 8, 5]columnLength = 3A[columnLength*y +x]

Matrix Multiplication

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0,0 0,1 0,2

1,0 1,1 1,2

2,0 2,1 2,1

0,0 0,1 0,2

1,0 1,1 1,2

2,0 2,1 2,1

0,0 0,1 0,2

1,0 1,1 1,2

2,0 2,1 2,1

= *

C A B

C A B

Matrix Multiplication

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1 1 1

2 2 2

3 3 3

1 2 3

4 5 6

7 8 9

12

C A B

C0,0 = A

0,0 * B0,0 + A

0,1 * B1,0 + A

0,2 * B2,0

12 1 1 1 4 1 7

Matrix Multiplication

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1 1 1

2 2 2

3 3 3

1 2 3

4 5 6

7 8 9

12 15

C A B

C0,1 = A

0,0 * B0,1 + A

0,1 * B1,1 + A

0,2 * B2,1

15 1 2 1 5 1 8

Matrix Multiplication

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1 1 1

2 2 2

3 3 3

1 2 3

4 5 6

7 8 9

12 15 18

C A B

C0,2 = A

0,0 * B0,2 + A

0,1 * B1,2 + A

0,2 * B2,2

18 1 3 1 6 1 9

Matrix Multiplication

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1 1 1

2 2 2

3 3 3

1 2 3

4 5 6

7 8 9

12 15 18

24

C A B

C1,0 = A

1,0 * B0,0 + A

1,1 * B1,0 + A

1,2 * B2,0

24 2 1 2 4 2 7

Matrix Multiplication

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1 1 1

2 2 2

3 3 3

1 2 3

4 5 6

7 8 9

12 15 18

24 30

C A B

C1,1 = A

1,0 * B0,1 + A

1,1 * B1,1 + A

1,2 * B2,1

30 2 2 2 5 2 8

Matrix Multiplication

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1 1 1

2 2 2

3 3 3

1 2 3

4 5 6

7 8 9

12 15 18

24 30 36

C A B

C1,2 = A

1,0 * B0,2 + A

1,1 * B1,2 + A

1,2 * B2,2

36 2 3 2 6 2 9

Matrix Multiplication

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1 1 1

2 2 2

3 3 3

1 2 3

4 5 6

7 8 9

12 15 18

24 30 36

36

C A B

C2,0 = A

2,0 * B0,0 + A

2,1 * B1,0 + A

2,2 * B2,0

36 3 1 3 4 3 7

Matrix Multiplication

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1 1 1

2 2 2

3 3 3

1 2 3

4 5 6

7 8 9

12 15 18

24 30 36

36 45

C A B

C2,1 = A

2,0 * B0,1 + A

2,1 * B1,1 + A

2,2 * B2,1

45 3 2 3 5 3 8

Matrix Multiplication

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1 1 1

2 2 2

3 3 3

1 2 3

4 5 6

7 8 9

12 15 18

24 30 36

36 45 54

C A B

C2,2 = A

2,0 * B0,2 + A

2,1 * B1,2 + A

2,2 * B2,2

54 3 3 3 6 3 9

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1,0 = 1,0 * 0,0 + 1,1 * 1,0 + 1,2 * 2,0

2,2 = 2,0 * 0,2 + 2,1 * 1,2 + 2,2 * 2,22,1 = 2,0 * 0,1 + 2,1 * 1,1 + 2,2 * 2,12,0 = 2,0 * 0,0 + 2,1 * 1,0 + 2,2 * 2,0

1,1 = 1,0 * 0,1 + 1,1 * 1,1 + 1,2 * 2,11,2 = 1,0 * 0,2 + 1,1 * 1,2 + 1,2 * 2,2

0,1 = 0,0 * 0,1 + 0,1 * 1,1 + 0,2 * 2,10,0 = 0,0 * 0,0 + 0,1 * 1,0 + 0,2 * 2,0

0,2 = 0,0 * 0,2 + 0,1 * 1,2 + 0,2 * 2,2

Indexes in Matrix Multiplication

Matrix Multiplication

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A = [[1, 1, 1] ,[2,2,2] ,[3,3,3]] B = [[1, 2, 3], [4,5,6] ,[7,8, 9]] C = [3*[0] for i in range(3)] for i in range(3): for j in range(3): for k in range(3): C[i][j] = C[i][j] + ( A[i][k] * B[k][j])

Trees as Lists Lists are powerful structures Lists can be used to build trees

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Root

Leaf1 Leaf2 Leaf3

Root

L1

L2

L3 L4 L5L0

L1

L2

L3 L4

L6

L0

L5

Root

Simple trees>>> Tree1 = ['L0', 'L1']

>>>Tree2 = ['L0', 'L1', 'L2']

>>>Tree3 = ['L0', 'L1', 'L2','L3', 'L4', 'L5']

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L1

Tree1

L0 L1 L2L0

Tree2

L0

Tree3

L1 L2 L3 L4 L5

>>> Tree = [['L0', 'L1'], ['L2','L3']]

What is tree[1][0]?

What is tree[0][1]?

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1

Tree

L1L0

0 1

0

L2 L30 1

QuestionBuild the list representing the following tree, (and call it

tree)

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Tree

BA E FC D

Answer>>>Tree = [[‘A’,’B’],[’C’,’D’],[’E’,’F’]]

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Tree

BA E FC D

QuestionBuild the following tree:

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Tree

B E FD

X Y Z

AnswerBuild the following tree

>>> Tree = [[['X','Y'],'B'],[['Z'],'D'] ,['E','F'] ]

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Tree

B E FD

X Y Z

0

1 2

0

0 1

10 0

0

11

QuestionHow to access: X, Z, and E?

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Tree

B E FD

X Y Z

0

1 2

0

0 1

10 0

0

11

Answer>>>Tree[0][0][0]

‘X’

>>>Tree[1][0][0]

‘Z’

>>>Tree[2][0]

‘E’

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Tree

B E FD

X Y Z

0

1 2

0

0 1

10 0

0

11

Columnar Transposition

S = [[‘P’, ‘R’, ‘O’], [‘G’, ‘R’, ‘A’], [‘M’, ‘~’, ‘~’]]

How to get “PROGRAM”? How to get “MGP~RP~AO”?

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P R O

G R A

M ~ ~

D = ""for i in range(0,3):       for j in range(0,3):               D = D + S[i][j]print(D)

D = ""for i in range(0,3):       for j in range(0,3):               D = D + S[3-1-j][i]print(D)

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ANY QUESTIONS?

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