Recurrences, Graphs,and MatricesProfessor Lucas BangHarvey Mudd CollegeDepartment of Computer Science

OverviewRecurrencesGraphsMatrices

3 powerful mathematical tools

1 super tool

Matrices Graphs

Recurrences

Sequences

Sequencesordered lists of numbers1, 1, 1, 1, 1, 1, 1, 1, 1, 1, . . .

Sequencesordered lists of numbers1, 1, 1, 1, 1, 1, 1, 1, 1, 1, . . .1, 2, 3, 4, 5, 6, 7, 8, 9, 10, . . .

Sequencesordered lists of numbers1, 1, 1, 1, 1, 1, 1, 1, 1, 1, . . .1, 2, 3, 4, 5, 6, 7, 8, 9, 10, . . .2, 4, 6, 8, 10, 12, 14, 16, 18, 20, . . .

Sequencesordered lists of numbers1, 1, 1, 1, 1, 1, 1, 1, 1, 1, . . .1, 2, 3, 4, 5, 6, 7, 8, 9, 10, . . .2, 4, 6, 8, 10, 12, 14, 16, 18, 20, . . .1, 2, 4, 8, 16, 32, 64, 128, 256, 512, . . .

Sequencesordered lists of numbers1, 1, 1, 1, 1, 1, 1, 1, 1, 1, . . .1, 2, 3, 4, 5, 6, 7, 8, 9, 10, . . .2, 4, 6, 8, 10, 12, 14, 16, 18, 20, . . .1, 2, 4, 8, 16, 32, 64, 128, 256, 512, . . .2, 3, 5, 7, 11, 13, 17, 19, 23 . . .

Sequencesordered lists of numbers1, 1, 1, 1, 1, 1, 1, 1, 1, 1, . . .1, 2, 3, 4, 5, 6, 7, 8, 9, 10, . . .2, 4, 6, 8, 10, 12, 14, 16, 18, 20, . . .1, 2, 4, 8, 16, 32, 64, 128, 256, 512, . . .2, 3, 5, 7, 11, 13, 17, 19, 23 . . .1, 3, 6, 10, 15, 21, 28, 36, . . .

Sequences111211211111221312211131122211113213211

Sequences111211211111221312211131122211113213211

one one

Sequences111211211111221312211131122211113213211

one one

Sequences111211211111221312211131122211113213211

one onetwo ones

Sequences111211211111221312211131122211113213211

one onetwo ones

Sequences111211211111221312211131122211113213211

one onetwo onesone two, one one

Sequences111211211111221312211131122211113213211

one onetwo onesone two, one one

Sequences111211211111221312211131122211113213211

one onetwo onesone two, one oneone one, one two, two ones

Sequences111211211111221312211131122211113213211

one onetwo onesone two, one oneone one, one two, two onesthree ones, two twos, one one

Sequences111211211111221312211131122211113213211

one onetwo onesone two, one oneone one, one two, two onesthree ones, two twos, one one

Conway’s Look-and-Say Sequence

Sequences

Neil Sloane Sloan’s Notebook

Sequences

Sequences

Sequencesordered lists of numbers1, 1, 1, 1, 1, 1, 1, 1, 1, 1, . . .1, 2, 3, 4, 5, 6, 7, 8, 9, 10, . . .2, 4, 6, 8, 10, 12, 14, 16, 18, 20, . . .1, 2, 4, 8, 16, 32, 64, 128, 256, 512, . . .2, 3, 5, 7, 11, 13, 17, 19, 23 . . .1, 3, 6, 10, 15, 21, 28, 35, . . .

Starting from any number, nIf n is even, compute n2If n is odd, compute 3n + 1Repeat

Collatz Sequences

Starting from any number, nIf n is even, compute n2If n is odd, compute 3n + 1Repeat35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, . . .

Collatz Sequences

Starting from any number, nIf n is even, compute n2If n is odd, compute 3n + 1Repeat35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, . . .

Try this out, starting withn = your age

Collatz Sequences

Collatz SequencesCollatz’s Conjecture: starting from anynumber, the pattern always reaches therepeating sequence . . . , 4, 2, 1, . . .

Collatz SequencesCollatz’s Conjecture: starting from anynumber, the pattern always reaches therepeating sequence . . . , 4, 2, 1, . . .

Q: Is it true?

Collatz SequencesCollatz’s Conjecture: starting from anynumber, the pattern always reaches therepeating sequence . . . , 4, 2, 1, . . .

Q: Is it true?A: Nobody knows!

Recurrences

Recurrencessequence numbers computed from earlier valuesa0 = 0a1 = 2a2 = 4a3 = 6a4 = 8a5 = 10a6 = 12

Recurrencessequence numbers computed from earlier valuesa0 = 0a1 = 2a2 = 4a3 = 6a4 = 8a5 = 10a6 = 12

Recurrencessequence numbers computed from earlier valuesa0 = 0a1 = 2a2 = 4a3 = 6a4 = 8a5 = 10a6 = 12

a0 = 0

Recurrencessequence numbers computed from earlier valuesa0 = 0a1 = 2a2 = 4a3 = 6a4 = 8a5 = 10a6 = 12

a0 = 0base case

Recurrencessequence numbers computed from earlier valuesa0 = 0a1 = 2a2 = 4a3 = 6a4 = 8a5 = 10a6 = 12

a0 = 0base casea1 = a0 + 2 = 0 + 2 = 2

Recurrencessequence numbers computed from earlier valuesa0 = 0a1 = 2a2 = 4a3 = 6a4 = 8a5 = 10a6 = 12

a0 = 0base casea1 = a0 + 2 = 0 + 2 = 2a2 = a1 + 2 = 2 + 2 = 4

Recurrencessequence numbers computed from earlier valuesa0 = 0a1 = 2a2 = 4a3 = 6a4 = 8a5 = 10a6 = 12

a0 = 0base casea1 = a0 + 2 = 0 + 2 = 2a2 = a1 + 2 = 2 + 2 = 4a3 = a2 + 2 = 4 + 2 = 6

Recurrencessequence numbers computed from earlier valuesa0 = 0a1 = 2a2 = 4a3 = 6a4 = 8a5 = 10a6 = 12

a0 = 0base casea1 = a0 + 2 = 0 + 2 = 2a2 = a1 + 2 = 2 + 2 = 4a3 = a2 + 2 = 4 + 2 = 6

In general,an = an−1 + 2

Recurrencessequence numbers computed from earlier valuesa0 = 0a1 = 2a2 = 4a3 = 6a4 = 8a5 = 10a6 = 12

a0 = 0base casea1 = a0 + 2 = 0 + 2 = 2a2 = a1 + 2 = 2 + 2 = 4a3 = a2 + 2 = 4 + 2 = 6

In general,an = an−1 + 2recurrence

Recurrencessequence numbers computed from earlier valuesa0 = 1a1 = 3a2 = 5a3 = 7a4 = 9a5 = 11a6 = 13

a0 = 1base casea1 = a0 + 2a2 = a1 + 2a3 = a2 + 2

an = an−1 + 2recurrence

Recurrencessequence numbers computed from earlier valuesa0 = 1a1 = 3a2 = 5a3 = 7a4 = 9a5 = 11a6 = 13

a0 = 1base casea1 = a0 + 2a2 = a1 + 2a3 = a2 + 2

an = an−1 + 2recurrence

Recurrencessequence numbers computed from earlier valuesa0 = 1a1 = 2a2 = 4a3 = 8a4 = 16a5 = 32a6 = 64

a0 = 1base casea1 = a0 × 2a2 = a1 × 2a3 = a2 × 2

an = an−1 × 2recurrence

Recurrencessequence numbers computed from earlier valuesa0 = 1a1 = 2a2 = 4a3 = 8a4 = 16a5 = 32a6 = 64

a0 = 1base casea1 = a0 × 2a2 = a1 × 2a3 = a2 × 2

an = an−1 × 2recurrence

RecurrencesFibonacci NumbersF0 = 1F1 = 1F2 = 2F3 = 3F4 = 5F5 = 8F6 = 13

F0 = 1base casesF1 = 1F2 = F1 + F0F3 = F2 + F1

Fn = Fn−1 + Fn−2recurrence

F4 = F3 + F2

RecurrencesFibonacci NumbersF0 = 1F1 = 1F2 = 2F3 = 3F4 = 5F5 = 8F6 = 13

F0 = 1base casesF1 = 1F2 = F1 + F0F3 = F2 + F1

Fn = Fn−1 + Fn−2recurrence

F4 = F3 + F2

RecurrencesLucas numbersL0 = 2L1 = 1L2 = 3L3 = 4L4 = 7L5 = 11L6 = 18

L0 = 2base casesL1 = 1

Fn = Fn−1 + Fn−2recurrence

Recurrencesmake up your own base casesQ0 =?base casesQ1 =?Qn = Qn−1 + Qn−2recurrence

Compute the first 10 values

RecurrencesLucas numbers ratiosL0 = 2L1 = 1

RecurrencesLucas numbers ratiosL0 = 2L1 = 1 1 ÷ 2 = 0.5

RecurrencesLucas numbers ratiosL0 = 2L1 = 1L2 = 3

1 ÷ 2 = 0.53 ÷ 1 = 3

RecurrencesLucas numbers ratiosL0 = 2L1 = 1L2 = 3L3 = 4

1 ÷ 2 = 0.53 ÷ 1 = 34 ÷ 3 = 1.3333

RecurrencesLucas numbers ratiosL0 = 2L1 = 1L2 = 3L3 = 4L4 = 7

1 ÷ 2 = 0.53 ÷ 1 = 34 ÷ 3 = 1.33337 ÷ 4 = 1.75

RecurrencesLucas numbers ratiosL0 = 2L1 = 1L2 = 3L3 = 4L4 = 7L5 = 11

1 ÷ 2 = 0.53 ÷ 1 = 34 ÷ 3 = 1.33337 ÷ 4 = 1.7511 ÷ 7 = 1.5714

RecurrencesLucas numbers ratiosL0 = 2L1 = 1L2 = 3L3 = 4L4 = 7L5 = 11L6 = 18

1 ÷ 2 = 0.53 ÷ 1 = 34 ÷ 3 = 1.33337 ÷ 4 = 1.7511 ÷ 7 = 1.571418 ÷ 11 = 1.6364

RecurrencesLucas numbers ratiosL0 = 2L1 = 1L2 = 3L3 = 4L4 = 7L5 = 11L6 = 18

1 ÷ 2 = 0.53 ÷ 1 = 34 ÷ 3 = 1.33337 ÷ 4 = 1.7511 ÷ 7 = 1.571418 ÷ 11 = 1.6364Try this for your sequence

RecurrencesLucas numbers ratiosL0 = 2L1 = 1L2 = 3L3 = 4L4 = 7L5 = 11L6 = 18

1 ÷ 2 = 0.53 ÷ 1 = 34 ÷ 3 = 1.33337 ÷ 4 = 1.7511 ÷ 7 = 1.571418 ÷ 11 = 1.6364Try this for your sequence

Matrices

[6 8 44 1 3]

Matricesrectangular arrangements of numbers

[6 8 44 1 3]

Matricesrectangular arrangements of numbers3 columns2 rows

[3 12 7]

Matricesrectangular arrangements of numbers2 columns2 rows

Matricesrectangular arrangements of numbers

[3 12 7] + [8 23 2

] =adding two matrices[? ?? ?

]

Matricesrectangular arrangements of numbers

[3 12 7] + [8 23 2

] =adding two matrices[11 35 9

]add entries in the same positions

Matricesrectangular arrangements of numbers

[3 12 7] + [8 23 2

] =adding two matrices[11 35 9

]add entries in the same positions

Matricesrectangular arrangements of numbers

[3 12 7]

×[2 10 4

] =multiplying two matrices[? ?? ?

]

Matricesrectangular arrangements of numbers

[3 12 7]

×[2 10 4

] =multiplying two matrices[6 74 30

]?

Matricesrectangular arrangements of numbers

[3 12 7]

×[2 10 4

] =multiplying two matrices[6 74 30

]3 × 2 + 1 × 0 = 6

?

Matricesrectangular arrangements of numbers

[3 12 7]

×[2 10 4

] =multiplying two matrices[6 74 30

]

Matricesrectangular arrangements of numbers

[3 12 7]

×[2 10 4

] =multiplying two matrices[6 74 30

]3 × 1 + 1 × 4 = 7

Matricesrectangular arrangements of numbers

[3 12 7]

×[2 10 4

] =multiplying two matrices[6 74 30

]3 × 1 + 1 × 4 = 7

Matricesrectangular arrangements of numbers

[3 12 7]

×[2 10 4

] =multiplying two matrices[6 74 30

]

Matricesrectangular arrangements of numbers

[3 12 7]

×[2 10 4

] =multiplying two matrices[6 74 30

]

Matricesrectangular arrangements of numbers

[3 12 7]

×[2 10 4

] =multiplying two matrices[6 74 30

]

Matricesrectangular arrangements of numbers

[3 12 7]

×[2 10 4

] =multiplying two matrices[6 74 30

]

Matricesrectangular arrangements of numbers

[3 12 7]

×[2 10 4

] =multiplying two matrices[6 74 30

]

Matricesrectangular arrangements of numbers

[3 12 7]

×[37

] =multiplying a matrix by a single column matrix[1455

]

Matricesrectangular arrangements of numbers

[1 11 0]

×[10

] =multiplying a matrix by a single column matrix[11

]

Matricesrectangular arrangements of numbers

[1 11 0]

×[11

] =multiplying a matrix by a single column matrix[21

]

Graphs

123

4

Bridges of KönigsbergGraphs

123

4

Bridges of KönigsbergGraphs

123

4

Bridges of KönigsbergGraphs$1 $1 $1

$1$1$1$1

123

4

Bridges of KönigsbergGraphs$1 $1 $1

$1$1$1$1

Q: Starting from the north bank, how manydifferent bridge tours can I take for $4? (and I amhappy to see the same bridge more than once!)

Directed Graphsnodes connected by directed edges123

4

Directed Graphsnodes connected by directed edges123

4

$1 trips1 21 4

$2 trips1 21 2 131 2 41 4 3Q: how many $3 trips are there?

Directed Graphsnodes connected by directed edges123

4$2 trips1 21 2 131 2 41 4 3

1 2 11 2 1 24$3 trips

1 2 31 2 4 231 4 3 2

$2 trips1 21 2 131 2 41 4 3

1 2 11 2 1 24$3 trips

1 2 31 2 4 231 4 3 2

$1 trips1 21 4

$# 1 2 3 4 52 4 5 ? ?Homework: Fill in 4 and 5. What insights doyou have? Is there any pattern to discover?

Directed Graphsnodes connected by directed edges

1 2A slight variation: how many tripsthat cost $n start and end at 1?

Matrices and Graphs[1 11 0

]1 2

123

4

Matrices and Graphs

What does the matrix look like for the Königsberg graph?

Review

Matrices Graphs

Recurrences

[1 11 0] 1 2

Fn = Fn−1 + Fn−2

Review

Matrices Graphs

Recurrences

[1 11 0] 1 2

Fn = Fn−1 + Fn−2

Want to explore more?

These slides at www.cs.hmc.edu/∼bang