cse 250 lecture 3

Fall 2021 ©Oliver Kennedy, Andrew HughesThe University at Buffalo, SUNY 1 / 25

CSE 250Lecture 3Function AnalysisTextbook Ch. 7.3-7.4


What is “Fast”?


When is an algorithm “fast”?

● Wall-clock time?

– Is 10s fast? Is 100ms fast? Is 10µs fast?● It depends on the task

– Algorithm vs Implementation.● Compare Grace Hopper’s implementation vs yours

– CPU: ARM iMX6 vs Intel i9 vs AMD 5950● Different speed/capability trade-offs

– Bottlenecks: CPU vs IO vs Memory vs Network vs …

Wall-clock time isn’t great.



● Take into account the “scale” of the problem

– Consider the number of input records, users, pixels, etc..




– Define the number of steps as a function of the input size.


Functions

0 1 2 3 4 5 6 7

-4

1

6

11

16

21

26

log(x)

x

x^2

x!

Number of Users

Ru

ntim

e

Would you use an algorithm that takes |Users|! steps?

Maybe?No


Functions

0 1 2 3 4 5 6 70

10

20

30

40

50

60

3x+5

x^2

Number of Users

Ru

ntim

e

Which is better? 3|Users|+5 steps or |Users|2 steps?




– Define the number of steps as a function of the input size.● Focus on “large” inputs

– Rank functions based on how they behave at large scales.


What hardware are you using?

vs

Intel i9

Images from openclipart.org, used with permission

Motorola 68000


Who implemented the algorithm?

vs

Epic-Level Implementation Error 23: Cat on Keyboard


Goal: Judge the Algorithm Not the Implementation or Deployment

● How fast is a step?

– Only count number of steps● Can this be done in two steps instead of one?

– “3 steps per user” vs “some number of steps per user”



● Take into account the “scale” of the problem.

– Define the number of steps as a function of the input size.● Focus on “large” inputs.

– Rank functions based on how they behave at large scales.● Decouple from infrastructure / implementation.

– Asymptotic Notation


Logarithms


Logarithms (refresher)

If no base specified, assume base-2(in this class)


Logarithms (refresher)

● Let● Exponent rule:● Product rule:● Division rule:● Change of base from b to c:

– Base changes are only a constant factor off● Log/Exponent are inverses:


Growth Functions


Growth Functions

A growth function must be a non-decreasing function of the form

(non-negative integers) (positive real numbers)

f is a function from ... … to …


Behavior at Scale

Two functions of n grow “the same” if they behave the same when n is large.

vs


Behavior at Scale

1 4 16 64 256

1024

4096

1638

4

6553

61

100

10000

1000000

100000000

10000000000

1000000000000

100000000000000

1E+016

N^3

(1/100)N^3 + ...

1 2 4 8 16 32 64 128

256

512

1024

2048

4096

8192

1638

4

3276

8

6553

6

1310

721

100

10000

1000000

100000000

10000000000

1000000000000

100000000000000

1E+016

N^3

(1/100)N^3 + ...

~100x difference


Behavior At Scale

0 0


Asymptotic Analysis @ 5k ft

● Consider: vs

● Case 1:

– grows faster, is better

● Case 2:

– grows faster, is better

● Otherwise:

– Both “behave the same”


Asymptotic Analysis @ 5k ft

Goal: Before you start optimizing the constantsFigure out if you’re in a different function class.


Another Example

Remember: exponential polynomial log constant≫ ≫ ≫


Why focus on dominating terms?

10 20 50 100 1000

0.43 ns 0.52 ns 0.62 ns 0.68 ns 0.82 ns

0.83 ns 1.01 ns 1.41 ns 1.66 ns 2.49 ns

2.5 ns 5 ns 12.5 ns 25 ns 0.25 µs

8.3 ns 22 ns 71 ns 0.17 µs 2.49 µs

25 ns 0.1 µs 0.63 µs 2.5 µs 0.25 ms

25 µs 0.8 ms 78 ms 2.5 s 2.9 days

0.25 µs 0.26 ms 3.26 days 1013 years 10284 years

0.91 ms 19 years 1047 years 10141 years 🤯

Time to execute instructions on a 4GHz Processor

cse 250 lecture 3

Documents