Compsc sci 201 201 Bi Big-Oh, I Interfac aces es, M , Maps - - PowerPoint PPT Presentation

Compsc sci 201 201 Bi Big-Oh, I Interfac aces es, M , Maps

2/7/2020 CompSci 201, Spring 2020 1

Susan Rodger February 7, 2020

I is for …

• Interface
• LinkedList implements List
• Inheritance
• LinkedList extends AbstractSequentialList

2/7/2020 CompSci 201, Spring 2020 2

Announcements

• Assignment P

P2 2 out ut t today, due Thur ue Thur. F Feb eb 13 13

• Get it done early, great practice for exam
• APT

APT-3 d 3 due y e yesterday

• Last chance to turn in today til 11:59pm
• Dis

iscussion 5 5 on n Feb eb 10 10

• Prepare for exam
• Exam

am nex next w week eek, F Feb eb 14 14

2/7/2020 CompSci 201, Spring 2020 3

PFOWBE

• Big

ig-Oh and and O O-Not

• tation
• Building a mathematical formalism with intuition
• Interfaces

aces: L List, S , Set, a , and M Map

• When it makes sense to use general type
• Empirical and Analytical measures of efficiency
• Maps: API and P

Problem lem S Solving ving

• Keys and Values

2/7/2020 CompSci 201, Spring 2020 4

Review ListSplicer.java, removeFirst

• https://c

//course sework.cs.d s.duke.e .edu/2 /201sp spring20/classc sscode/

• Decla

claratio ions ns – using ing L List<> i inter erfac ace

• Method removeFirst, parameter list
• Method removeFirst pass either list

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list.remove(0)

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• lis

list is is Link inkedList o

• r Arra

rrayList, c , call L ll List<> m metho hods

• If lis

list is is Ar ArrayList – cal all r l rem emove f for Ar ArrayList

• If lis

list is is L Link inkedList, call all r rem emove f for L Link inkedList

list.remove(0)

2/7/2020 CompSci 201, Spring 2020 7

• Wha

hat is is “faster”? LinkedLis ist or

• r Arra

rrayList

RemoveFirst

• Poly. (array)

Access all elements randomly

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• Wha

hat is is “faster”? LinkedLis ist or

• r Arra

rrayList

Random Access

• Poly. (linked)

Measuring Efficiency

• Whic

hich is is f faster, L Link inkedList o

• r Ar

ArrayList?

• What does it depend on?
• Empirically: depend on computer used?
• ArrayLis

ist.r .rem emove(0) 0): :

• y = 0.0064x2 – 0.0156x + 0.0238
• R2 = 0.9984

2/7/2020 CompSci 201, Spring 2020 9

Random Access Efficiency

• Random A

Access f s for L Lists ts

• alist.get(N/2) versus llist.get(N/2)
• Does this depend on N?
• Link

inkedList rand andom acc access o

• f n

n elem elements n n tim imes

• y = 0.0129x2 – 0.7137x + 1.3337
• R2 = 0.9889
• Arra

rrayList rand andom ac access of n elem n elements n t n tim imes

• y – 0.0002x + 5E-05, R2 = 0.8169

2/7/2020 CompSci 201, Spring 2020 10

Big-Oh aka O-Notation

• Intuit

uitio ion: b n: beha havio vior in the l limit it m mat atters

• What happens as N gets large, where we

measure performance in terms of N

• For polynomials: leading term, no coefficients

y = 3x y = 6x - 2 y = 15x + 44 y = x2 y = x2 - 6x+ 9 y = 3x2 + 4x

• The first family is O(n), the second is O(n2)

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More on O-Notation

• Provid

vides es t theo eoretic ical a l analysis

• is. Independent of, and

can obscure some, empirical details

• Compare: 20N hours v N2 microseconds
• Which is better? Does it depend?
• If an alg

an algorithm is is O O(N) it it’s als also O(N2) f fro rom a m a technic chnical, m , mat athe hematical p ical perspect ectiv ive

• O is an upper bound, in the limit
• We try to provide tight, or best bounds/analysis

2/7/2020 CompSci 201, Spring 2020 12

Big-Oh for Algorithms

• Binary s

search: h: g gues ess number 1 1-1024, 024, hi, i,lo lo,correct

• # of guesses? O(log N) note 210 = 1024
• If 12 seconds for 210 then 24 seconds for 220
• Seque

uent ntial/line ial/linear ar s sear arch: ch: e every e elem ement nt o

• f list
• # elements examined? O(N)
• If 12 seconds for 210 then 24 seconds for 211
• Double input, double time

2/7/2020 CompSci 201, Spring 2020 13

Big-Oh for More Algorithms

• Efficie

icient nt s sorting ing: : merge, q , quick ck, Tim

• # elements examined? O(N log N)
• More time than linear, but not terrible
• Looking

ng at at e ever ery p pair, o

• r slow s

sorting ing, e e.g., ., bubble

• # elements examined? O(N2)
• 12 seconds for 210 then 144 seconds for 211
• Double the input, square the time

2/7/2020 CompSci 201, Spring 2020 14

Running times in seconds machine: 109 instructions/sec

N O(log N) O(N) O(N log N) O(N2) 10 3E-9 1E-8 3.3E-8 0.0000001 100 7E-9 1E-7 6.64E-7 0.0001 1,000 1E-8 1E-6 0.00001 0.001 10,000 1.3E-8 0.00001 0.0001329 0.102 100,000 1.7E-8 0.0001 0.001661 10.008 1,000,000 0.00000002 0.001 0.0199 16.7 min 1,000,000,000 0.00000003 1.002 65.8 31.8 years

2/7/2020 CompSci 201, Spring 2020 15

WOTO

http:// //bi bit.ly/2 /201spr pring20-02 0207 07-1

2/7/2020 CompSci 201, Spring 2020 19

What do they all have in common?

• The

They all t all took a a CompSci course a at Duke! e!

2/7/2020 CompSci 201, Spring 2020 20

ArrayList Methods

Arra rrayList<String> words ds = = new w Arraylis list<>(); words ds.add dd(“ (“cat”); ); words ds.add dd(“ (“fish”); ); words ds.add dd(“do dog”) ”); Stri ring b g b = word rds.ge get(1 (1); ); wo words.set(2, “ 2, “frog”); int int c c = words.ind index exOf(“ (“cat”); ); wo words.set(1, 1, word rds.ge get(c)); ));

2/7/2020 CompSci 201, Spring 2020 22

Problems and Solutions

• Str

tring th that o

• ccu

ccurs m most st i in a list o

• f str

trings?

• CountingStringsBenchmark.java, two ideas
• See also CountingStringsFile for same ideas
• https://coursework.cs.duke.edu/201spring20/classcode
• Parallel arrays: word[k] occurs count[k] times
• Use ArrayLists: 2 “the”, 3 “fat”, 4 “fox”

2/7/2020 CompSci 201, Spring 2020 31

the fox cried fat tears 2 4 1 3 5

0 1 2 3 4

How does the code work?

• Process

ss each s string s s

• First time words.add(s),counter.add(1)
• Otherwise, increment count corresponding to s
• c[x] += 1 ?

2/7/2020 CompSci 201, Spring 2020 33

What is complexity of this code?

2/7/2020 CompSci 201, Spring 2020 34

• Sea

earch f for eac each w word and and … … if if occurs a at k k

• +1 to counter.get(k), else add at end
• Sear

arch c h complex lexit ity? O O(M) w when M n M differ erent nt words

• One search is O(M) – what about all searches?
• Tracking all words. First time zero, then one, …
• Avoid analyzing duplicates for the moment
• Will take longer if we have multiple occurrences of

some of M words

Tracking N strings

2/7/2020 CompSci 201, Spring 2020 35

• Complex

lexit ity o

• f search?

h? O O(M) f for M differ erent nt w words

• Complexity of words.indexOf(..) is O(M)
• what about all calls? 1 + 2 + … N is N(N+1)/2

O(N2)

Should we be more precise?

• Addin

ing M M dif ifferent w words w will b ill be e O(M2)

• 1 + 2 + … + M = M(M+1)/2
• Addin

ing d dup upli licates: w we e need need t to be e prec ecise ab about adding ng N N total w al words.

• Sometimes word will be found, still O(M) for M

different words

• We have both M and N here, but treat M == N

for easier analysis.

2/7/2020 CompSci 201, Spring 2020 36

CountingStringsFile.java

• Gene

enerate an an Ar ArrayList of S Stri rings gs

• Find the word that occurs the most often
• See three different methods

2/7/2020 CompSci 201, Spring 2020 37

Understanding O-notation

• This

This is is an an up upper er b bound und and and in in the lim he limit it

• Coefficients don’t matter, ord

rder o r of growth

• N + N + N + N = 4N is O(N) --- why?
• N*N is O(N2) – why?
• O(1) means independent of N, constant time
• In

n anal analyzing c code and and c code f e fragments

• Account for each statement
• How many times is each statement executed?

2/7/2020 CompSci 201, Spring 2020 38

Why coeffficients don’t matter

2/7/2020 CompSci 201, Spring 2020 39

Just Say No.. When you can

O(n2)

2/7/2020 CompSci 201, Spring 2020 41

Example: Analyze using big-Oh

2/7/2020 CompSci 201, Spring 2020 42

• Wha

hat is is runt untime o

• f stuff(N)
• How to reason about this
• What is retur

urn v n value ue o

• f stuff(N)
• What if code changes to sum += k

Counting for O-notation

• Why

hy is is O O(1) 1) c complexity o

• f sum

um + += n

• Is this O(1) for any x += y ?
• Loop executes N times, doing O(1) per iteration
• Total runtime for method? O(n)

2/7/2020 CompSci 201, Spring 2020 43

Example 2: Analyzing O-Notation

• Wha

hat is is big ig-Oh o

• f run

untim ime o

• f cal

all calc(N) ?

• Num. of statements executed, O(1) for line 146?
• Use calc(16) and generalize
• Wha

hat is is big ig-Oh o h of va value ue r retur urned ned b by calc(N) ?

• Table? k = 1, 2, 4, 8, 16, 32, 64

2/7/2020 CompSci 201, Spring 2020 44

WOTO

http:// //bi bit.ly/2 /201spr pring20-02 0207 07-2

2/7/2020 CompSci 201, Spring 2020 52