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 Susan Rodger February 7, 2020 2/7/2020 CompSci 201, Spring 2020 1

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 otation • 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 2/7/2020 CompSci 201, Spring 2020 5

list.remove(0) • lis list is is Link inkedList o or 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 2/7/2020 CompSci 201, Spring 2020 6

list.remove(0) • Wha hat is is “faster”? LinkedLis ist or or Arra rrayList RemoveFirst 1.4 y = 0.0064x 2 - 0.0156x + 0.0238 1.2 R² = 0.9984 1 0.8 0.6 0.4 0.2 y = -4E-05x + 0.0009 0 linked array Linear (linked) Poly. (array) 2/7/2020 CompSci 201, Spring 2020 7

Access all elements randomly • Wha hat is is “faster”? LinkedLis ist or or Arra rrayList Random Access 25 y = 0.1292x 2 - 0.7137x + 1.3337 20 R² = 0.9889 15 10 5 y = 0.0002x + 5E-05 R² = 0.8169 0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000110000120000130000140000150000 linked array Poly. (linked) Linear (array) 2/7/2020 CompSci 201, Spring 2020 8

Measuring Efficiency • Whic hich is is f faster, L Link inkedList o or Ar ArrayList? • What does it depend on? • Empirically: depend on computer used? • ArrayLis ist.r .rem emove(0) 0): : • y = 0.0064x 2 – 0.0156x + 0.0238 • R 2 = 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 of n n elem elements n n tim imes • y = 0.0129x 2 – 0.7137x + 1.3337 • R 2 = 0.9889 • Arra rrayList rand andom ac access of n elem n elements n t n tim imes • y – 0.0002x + 5E-05, R 2 = 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 = x 2 y = x 2 - 6x+ 9 y = 3x 2 + 4x • The first family is O( n ) , the second is O( n 2 ) 2/7/2020 CompSci 201, Spring 2020 11

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 N 2 microseconds • Which is better? Does it depend? • If an alg an algorithm is is O O(N) it it’s als also O(N 2 ) 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 2 10 = 1024 • If 12 seconds for 2 10 then 24 seconds for 2 20 • Seque uent ntial/line ial/linear ar s sear arch: ch: e every e elem ement nt o of list • # elements examined? O(N) • If 12 seconds for 2 10 then 24 seconds for 2 11 • 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 or slow s sorting ing, e e.g., ., bubble • # elements examined? O(N 2 ) • 12 seconds for 2 10 then 144 seconds for 2 11 • Double the input, square the time 2/7/2020 CompSci 201, Spring 2020 14

Running times in seconds machine: 10 9 instructions/sec N O(log N) O(N) O(N log N) O(N 2 ) 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 occu ccurs m most st i in a list o of 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” the fox cried fat tears 0 1 2 3 4 2 4 1 3 5 2/7/2020 CompSci 201, Spring 2020 31

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? • 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 2/7/2020 CompSci 201, Spring 2020 34

Tracking N strings • Complex lexit ity o of 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(N 2 ) 2/7/2020 CompSci 201, Spring 2020 35

Should we be more precise? • Addin ing M M dif ifferent w words w will b ill be e O(M 2 ) • 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(N 2 ) – 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(n 2 ) 2/7/2020 CompSci 201, Spring 2020 41

Example: Analyze using big-Oh • Wha hat is is runt untime o of stuff(N) • How to reason about this • What is retur urn v n value ue o of stuff(N) • What if code changes to sum += k 2/7/2020 CompSci 201, Spring 2020 42

Counting for O-notation • Why hy is is O O(1) 1) c complexity o of 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