[PPT] - Compsci 201 201 More Sorti ting, B Backtra ktracking Par art 1 PowerPoint Presentation

SLIDE 1

Compsci 201 201 More Sorti ting, B Backtra ktracking Par art 1 1 of

f 4

4/3/2020 Compsci 201, Spring 2020 1

Susan Rodger April 3, 2020

SLIDE 2

T is for …

Alan Turing, Turing Test, Turing award
From WWII to philosophy to math to computing
Tree
From Search to Game to …

4/3/2020 Compsci 201, Spring 2020 2

SLIDE 3

Announcements

APT

PT-6 due T Tuesd sday, A Apri ril 7

Assig

ignme ment P t P5 Percola latio tion

Tell us solo or partner form due yesterday
Code due Thursday, April 9
Disc

scuss ssion 11 Monday, A Apri ril 6 6

Pre-discussion coming soon
Exam 2 i

is A s Apri ril 10

Online, flexible hours, extra time, open notes
Your own work
APT Q

Quiz iz 2 2 is April 1 il 11-15 15

Your own work

4/1/2020 Compsci 201, Spring 2020 3

SLIDE 4

If you need an extension for work

Fill

ill o

ut

ut t the f he form o

n

n the c he cour urse w web eb page, und under fo forms ms t tab

Then t

n take t the e e extens ensio ion

If you

u need need m more t e tim ime, t tak ake it it again

Try no

not t to get t too far ar b behin ehind!

4/3/2020 Compsci 201, Spring 2020 4

SLIDE 5

PfFFiA

Finis

ish/R h/Revie iew s sorting ing

Loop invariants with efficient and inefficient

sorts

How a priority queue works
Bubblesort focus, Oh No!
Backtr

ktracki king: NQuee eens ns and and m more

Canonical problem-solving and programming

4/3/2020 Compsci 201, Spring 2020 5

SLIDE 6

Simple, O(n2) sorts

Selectio

tion s sort t --- n2 compa mparis isons, n n swaps ps

Find min, swap to front, increment front, repeat
In

Inse sertion sor sort --- n2 com

mpariso

sons, n no

sw

swap, sh shift

stable

ble, fast on sorted data, slide into place

Bubble

bble s sort --- n2 ever erythi hing ng, s slow*

Catchy name, but slow and ugly*

*this isn't everyone's opinion, but it should be

Shell s

ll sort: q : quasi-in insertio tion, f fast i in practi tice

Not quadratic with some tweaks

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SLIDE 7

Case Study: SelectionSort

Cano

nonical O nical O(n2) a algorit ithm hm/co /code e

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SLIDE 8

Case Study: SelectionSort

Invar

ariant iant: o : on jth

th pass, [0,j) is

is in in final inal s sorted o

rder

er

Nested loop re-establishes invariant

4/3/2020 Compsci 201, Spring 2020 8

public void sort(List<T> list) { for(int j=0; j < list.size()-1; j++) { int min = j; for(int k=j+1; k < list.size(); k++) { if (list.get(k).compareTo(list.get(min)) < 0){ min = k; } } swap(list,min,j); } } Final order unexamined j

SLIDE 9

Reminder: Loop Invariant

Statement

ent: t : true e ue each t ch time l loop begins ins t to execut cute

During loop execution it may become false
The loop re-establishes the invariant
Typically stated in terms of loop index
Pictures can help reason about code/solution
Help

elps t to rea eason f formally and and inf informally ab about the he code e you’re w e writing ing

Can I explain the invariant to someone?

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SLIDE 10

Bubblesort isn't much code

Swap

ap ad adjacent elem elements w when hen o

ut

ut o

f order
From beginning to end, then end-1, end-2, …
After

er n n passes es, las ast n n-ele element nts i in place ce

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SLIDE 11

Timing of n2 and other sorts

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SLIDE 12

More efficient O(n log n) sorts

Div

ivid ide and and co conq nquer s sorts:

Quick s

k sort: fast in practice, O(n2) worst case

Merg

rge s sort: stable, fast, extra storage

Timsort: http://en.wikipedia.org/wiki/Timsort
Other s

sor

rts:
Heap sort: priority queue sorting
Radix sort: uses digits/characters (no

compare)

O(n l

log n) is o

ptimal f

al for c comparing ing

But, Radix is O(n) ??

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SLIDE 13

Stable, Stability

Stab

able le: r : respec ect o

rder

er o

f equal k

l keys when s sorting ing

First sort by shape, then by color: Stable!
Triangle < Square < Circle; Yellow < Green < Red

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SLIDE 14

Merge Sort

Idea: Divide and Conquer
Divide list into two halves
Sort both halves (smaller problem)
Merge the two sorted halves

9 5 1 4 3 6 2 7

Compsci 201, Spring 2020 14 4/3/2020

SLIDE 15

What does recursively sort mean? Merge Sort

Use the same Merge Sort algorithm

– Divide list into two halves – Sort both halves (smaller problem) – Merge the two sorted halves 9 5 1 4 9 5 1 4 divide list into 2 halves 5 9 1 4 recursively sort each half 1 4 5 9 merge the two sorted list

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SLIDE 16

Merge two sorted lists

Both lists are sorted.

1 4 5 9 2 3 6 7 Find the smallest from front of two lists

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SLIDE 17

MergeSort idea for code

mergesort(data) n = length of data if n is 1: return data else: d1 = mergesort(first half of data) d2 = mergesort(second half of data) return merge(d1, d2)

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SLIDE 18

Time for MergeSort n items: T(n)

mergesort(data) n = length of data if n is 1: return data else: d1 = mergesort(first half of data) d2 = mergesort(second half of data) return merge(d1, d2)

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SLIDE 19

Quicksort - Idea

Pivot – select and adjust < pivot, pivot, > pivot

– Select one of the elements – Put it where it belongs in sorted order – Put elements less than it, to its left – Put elements greater than it, to its right

Recursively sort the elements to its left
Recursively sort the elements to its right

Compsci 201, Spring 2020 30 4/3/2020

SLIDE 20

Quicksort - Idea

Pivot – select and adjust < pivot, pivot, > pivot
Recursively sort the elements to its left
Recursively sort the elements to its right

5 9 1 4 3 6 2 7

Compsci 201, Spring 2020 31 4/3/2020

SLIDE 21

Quicksort: fast in practice

Inv

nvented in in 1962 b 1962 by Tony ny H Hoar are, d did idn' n't und under erstand recur cursio ion: n:

Canonical T(n) = 2T(n/2)+O(n), but
Worst case is O(n2), bad pivot. Shuffle first?

<= X > X X

pivot index

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void doQuick(List<T> list, int first, int last) { if (first >= last) return; int piv = pivot(list,first,last); doQuick(list,first,piv-1); doQuick(list,piv+1,last); }

SLIDE 22

Pivot is O(n)

Invar

ariant iant: : [first,p] <= list.get(first)

Invar

ariant iant: : (p,k) > list.get(first)

<= X > X X

pivot index

4/3/2020 Compsci 201, Spring 2020 34

private int pivot(List<T> list, int first, int last){ T piv = list.get(first); int p = first; for(int k=first+1; k <= last; k++){ if (list.get(k).compareTo(piv) <= 0){ p++; swap(list,k,p); } } swap(list,p,first); return p; }

<= [first]

??? k first

> [first]

p

SLIDE 23

https://en.wikipedia.org/wiki/Timsort

Stable, O

O(n n lo log n) n) in in average and and w worst, O O(n) n) b bes est!

In practice lots of data is "close" to sorted
Invent

ented by Tim P Peter ers for P Pytho hon, n, now i in J Java

Replaced merge sort which is also stable
Engineer

ineered ed t to be correct ect, f , fas ast, u useful i ul in pract ctic ice

Theory and explanation not so simple

https://www.youtube.com/watch?v=NVIjHj-lrT4

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SLIDE 24

Summary of O(n log n) sorts

Ti

Timsort: hy hybrid o

f mer

erge and and ins insertion?

Fast in real world: Python, Java 7+, Android
Wha

hat’s the b he bes est O O(n lo n log n) n) sort t to call all?

The one in the library you have access to
Arrays.sort or Collections.sort
Chang

anging ing how y you s sort: :

.compareTo() or .compare()

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SLIDE 25

sortingwoto or ginooorsttw

http://b

//bit it.l .ly/2 /201sprin ing20-04 0403 03-1

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SLIDE 26

Brian Fox

GNU Bash Shell (developer) First employee at Free Software Foundation First online banking system at Wells Fargo

There’s nothing that I am better at than everyone else, except being me. There’s no secret to being me. Follow your interests and work hard at them. Then you will play bass better, program better, cook better, ride motorcycles better, or anything else that you really want to do.

4/3/2020 Compsci 201, Spring 2020 38

https://lifehacker.com/im-brian-fox-author-of-the-bash-shell-and-this-is-how-1820510600

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Compsci 201 201 More Sorti ting, B Backtra ktracking Par art 2 2 of

f 4

4/3/2020 Compsci 201, Spring 2020 39

Susan Rodger April 3, 2020

SLIDE 28

How does a Priority Queue work?

Imple

lemented w wit ith a h a Heap eap

Tree that is stored in an array.
It is NOT a tree
But easier to think of it as a tree
It REALLY is an array

4/3/2020 Compsci 201, Spring 2020 40

SLIDE 29

How does a priority queue work?

Imple

lementation w wit ith a h a heap heap

Tree that is stored in an array
Min heap, remove always returns the min
Could make it a max heap
Use comparator to change
Root is in index 1
Left child: index 2*k
Right child: index 2*k + 1

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0 1 2 3 4 5 6 7 8 9 10 11

25 19 17 7 10 6 9 21 13

SLIDE 30

Heap is an array, visualize as Tree

Ro Root a at 1 Left child: 2k Right child: 2k + 1

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6

0 1 2 3 4 5 6 7 8 9 10 11

10 7 25 25 19 19 17 17 7 10 6 13 9 9 21 21 13

SLIDE 31

Heap is an array, visualize as Tree

No Node 10 10: i index 2 Left child: Right child:

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6

0 1 2 3 4 5 6 7 8 9 10 11

10 7 25 25 19 19 17 17 7 10 6 13 9 9 21 21 13

Node at e at i index ex k k Left: index 2k Right: index 2k+1

SLIDE 32

Heap is an array, visualize as Tree

No Node 17 17: i index 4 Left child: Right child:

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6

0 1 2 3 4 5 6 7 8 9 10 11

10 7 25 25 19 19 17 17 7 10 6 13 9 9 21 21 13

Node at e at i index ex k k Left: index 2k Right: index 2k+1

SLIDE 33

Properties of a Min-Heap

Each

h node i is smaller aller t than i n its c child ildren en

Wher

here is is t the m he mini inimal no node? e?

At the root of the tree
At the front of the array
A hea

heap is is alw always bala alanced!

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SLIDE 34

Min-Heap – Where is smallest?

Note each node smaller than children

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6

0 1 2 3 4 5 6 7 8 9 10 11

10 7 25 25 19 19 17 17 7 10 6 13 9 9 21 21 13

SLIDE 35

Remove the minimum element

Can

an’t rem emove t the r he root

Swap

ap root wit ith la h last elem element in in ar array.

Rem

emove m min ( in (no now la last elem element in in ar array)

Fix

ix the heap he heap

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SLIDE 36

Remove the minimum

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6

0 1 2 3 4 5 6 7 8 9 10 11

10 7 25 25 19 19 17 17 7 10 6 13 9 9 21 21 13

SLIDE 37

N elements – Time to remove min?

Swa

wap

Remo

move mi min

Adj

Adjust

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0 1 2 3 4 5 6 7 8 9 10 11

10 7 19 19 17 17 7 10 25 13 9 25 21 21 13 9

SLIDE 38

Add element to a min-heap

Put

ut t the he elem element a at t the end he end o

f the ar

he array

Not

a a hea heap any anymore!

Bubble e

le element nt u up path t h to fix heap ap!

Compare element to parent, swap if needed

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SLIDE 39

Add Element to min-Heap

Add 8

dd 8

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6

0 1 2 3 4 5 6 7 8 9 10 11

10 7 25 25 19 19 17 17 7 10 6 13 9 9 21 21 13

SLIDE 40

N elements – Time to add element?

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6

0 1 2 3 4 5 6 7 8 9 10 11

8 7 25 25 19 19 17 17 7 8 6 10 9 9 21 21 10 13 13

Add t

dd to a array

Adj

Adjust

SLIDE 41

Review ew – Sort with PriorityQueues

How can w

n we sort N N elem ement ents u using ng P Priorit ity Q Queue eue?

Add all elements to pq, then remove them
Every operation is O(log N), so this sort?
O(N log N) – basis for hea

heap s sort

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SLIDE 42

WOTO – Priority Queues

http://b

://bit.l .ly/20 /201s 1spring2 g20-040 403-2

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SLIDE 43

Compsci 201 201 More Sorti ting, B Backtra ktracking Par art 3 3 of

f 4

4/3/2020 Compsci 201, Spring 2020 73

Susan Rodger April 3, 2020

SLIDE 44

A Story about Bubble Sort

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SLIDE 45

Prof. Owen

Astrachan

Ph

PhD at at Duke, , Stayed at ed at D Duke

Ha

Hates B Bubble so sort, t thinks it is s the worst rst so sort ever

Obse

sessed w with Bubbl bles esort!

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SLIDE 46

Steve Wolfman and Rachel Pottinger, Duke 1997, now Profs at UBC

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SLIDE 47

Wrote a paper

“W

“Why bu bubbl bble i is not m my favorite sort”

Submit

itted i it t to a conferenc nce

It was

as r reject cted!

4/3/2020 Compsci 201, Spring 2020 77

SLIDE 48

Another paper accepted!

4/3/2020 Compsci 201, Spring 2020 78

SLIDE 49

11/08/77

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SLIDE 50

17 Nov 75 Not needed Can be tightened considerably

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SLIDE 51

Donald Knuth (Turing 1974)

“In s

short, t , the b e bubble le s sort seems t to have n e nothing hing to recommend end i it, e , exce cept a a catchy nam name, and and t the he fac act t tha hat it it lea leads t to some inter eresting ing t theo eoretica ical l problem lems.” .”

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SLIDE 52

Jim Gray (Turing 1998)

Bubble sort is a good

argument for analyzing algorithm performance. It is a perfectly correct

algorithm. But it's

performance is among the worst imaginable. So, it crisply shows the difference between correct algorithms and good algorithms.

(italics ola’s)

SLIDE 53

Brian Reid (Hopper Award 1982)

Feah. I love bubble sort,

and I grow weary of people who have nothing better to do than to preach about it. Universities are good places to keep such people, so that they don't scare the general public.

(continued)

SLIDE 54

Brian Reid (Hopper 1982)

I am quite capable of squaring N with or without a calculator, and I know how long my sorts will bubble. I can type every form of bubble sort into a text editor from memory. If I am writing some quick code and I need a sort quick, as opposed to a quick sort, I just type in the bubble sort as if it were a statement. I'm done with it before I could look up the data type of the third argument to the quicksort library. I have a dual-processor 1.2 GHz Powermac and it sneers at your N squared for most interesting values of N. And my source code is smaller than yours.

Brian Reid who keeps all of his bubbles sorted anyhow.

SLIDE 55

Niklaus Wirth (Turing award 1984)

I have read your article and share your view that Bubble Sort has hardly any

merits. I think that it is so often

mentioned, because it illustrates quite well the principle of sorting by exchanging. I think BS is popular, because it fits well into a systematic development of sorting

algorithms. But it plays no role in

actual applications. Quite in contrast to C, also without merit (and its derivative Java), among programming codes.

SLIDE 56

Obama on Sorting

When he was a senator running for President

What is the most efficient way to sort a million 32-bit integers?

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SLIDE 57

Wikipedia page on BubbleSort

a

4/3/2020 Compsci 201, Spring 2020 88

SLIDE 58

Musical Bubblesort

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SLIDE 59

Compsci 201 201 More Sorti ting, B Backtra ktracking Par art 4 4 of

f 4

4/3/2020 Compsci 201, Spring 2020 91

Susan Rodger April 3, 2020

SLIDE 60

Backtracking and Blob Fill

Explo

lore a a move ( (blo lob f fill ill) if if it it works? F Fab abulous!

If it does not work? Undo the move, try again
Similar

ilar t to Sudoku s u solving ng? C Crossword p puz uzzles les?

Tentatively try number of word, follow through
May need to undo and try alternative

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SLIDE 61

Exhaustive Search

Can e

n explore e ever ery p possib ible le m move i e in tic-ta tac-to toe

Cannot explore every possible move in chess
Brute

te-force d doesn sn't w 't work

Be smart, try move? Then undo, try another
Bac

acktrackin ing in in sea earch t tree ee

Smart pruning

4/3/2020 Compsci 201, Spring 2020 93

SLIDE 62

N-Queens: Know History

Can

an w we e plac lace N N quee ueens o

n

n NxN xN boa

ard s

so

no
queen at

en attack cks a another her

4x4 or 8x8 or …

4/3/2020 Compsci 201, Spring 2020 94

SLIDE 63

8 Queens – What fun!

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SLIDE 64

Nqueen Concepts

For eac

each c column c c in in [0.. 0..N)

Try to place queen in each row of column c
grid[r][c] ok? Place queen, try c+1
If not ok? Or Doesn't work? Try next row, r+1
When

hen ha have all q all quee ueens b been een p plac laced?

If c == N and success? Done!
Can't do column c? return false, c-1 continues

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SLIDE 65

Code for Nqueens Backtracking

https://coursework.cs.duke.edu/201spring20/backtracking-sp20/

Done w

ne when hen c c == N

Place queens, recurse, unplace and try again
Retur

urn t n true

All placed
Recursive
Us

Use myBoard

Track moves

4/3/2020 Compsci 201, Spring 2020 97

SLIDE 66

Backtracking Summary

Enumer

erate a all p possible m le moves/cho choices ices

Nqueen? Each column and each row in column
Blob-fill? Each neighbor: fill, and unfill
Boar

ard often en t two-dimens nsio iona nal a l array/grid id

Record move, recurse, undo if not done

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SLIDE 67

Backtracking APTs

Often us

use e grid[][] to store s e state/ e/moves

In Java this is actually an array of arrays
int[][] a = new int[4][4] fo

for e examp mple

What is a[0]? What is a[0][0]?
Often m

n move m must be explicit licitly u undone ne

Sometimes just try everything

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SLIDE 68

Collaborative APT Solving

https://w

://www2.cs cs.duke.edu .edu/c /csed/n ed/newap apt/g /gri ridg dgam ame.h .html

Sha

hall ll w we e pla lay a a gam ame?

Each player plays perfectly
If I go her

here, w will y ill you u win? in?

4/3/2020 Compsci 201, Spring 2020 101

SLIDE 69

All-seeing and All-knowing

Giv

iven a a boar ard, ho how many any w winni inning m moves?

Can't tell, can determine all possible moves
Make a move, ask if it's a winner
Can't tell? Make move and repeat

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SLIDE 70

What can we do with a board?

Can y

you d u det eter ermine e if [r][c] i is legal al?

[1][0] is legal, why?
[3][1] is NOT legal, why?
Suppose

e ther ere a e are n e no legal al m moves es? Answer er: Z Zero/0

Sup

uppose I I place a an n 'X' a and nd the hen ask

How many ways to win does opponent have?
If answer is Zero/0, what does placing 'X' do?
This l

leads ds to to backtr ktrackin ing, b belie lieve the c code de!!!

".X.X" "..X." ".X.." "...."

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SLIDE 71

APT GridGame Ideas

Have bool
olean metho

hod winWithBoard(board)

For each legal move
Place on board – board now changed
Call winWithBoard(board) -- if true? I lose
Undo move --- take off board and continue
If you

u could uld s see ee ahe ahead, y you' u'd k kno now if if the he m move wa was g good.

d. winWithBoard is

is an an orac acle

4/3/2020 Compsci 201, Spring 2020 104

SLIDE 72

What Do We Believe?

Predicti

tions b s by oracles … s …

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SLIDE 73

All-seeing and All-knowing

Giv

iven a a boar ard, ho how many any w winni inning m moves?

Can't tell, can determine all possible moves
Make a move, ask if it's a winner
Can't tell? Make move and repeat

4/3/2020 Compsci 201, Spring 2020 106

SLIDE 74

All-seeing and All-knowing

Given a

n a board, a are t e there a e any m moves possib ible? le?

If no moves possible … last move was winner!
Last person to place an X wins
So, t

, try e every X, w , wit ith eac h each o

ne …

ne …

Ask recursively if winner
Make move and ask …

– Make move and ask …

4/3/2020 Compsci 201, Spring 2020 107

SLIDE 75

GridGame backtracking, count wins

private int winCount(char[][] board) { int wins = 0; for(int r=0; r < 4; r++){ for(int c=0; c < 4; c++){ if (canMove(board,r,c)){ board[r][c] = 'X'; int opponentWins = winCount(board); if (opponentWins == 0){ wins += 1; } board[r][c] = '.'; } } } return wins; }

".X.X" "X.X." ".X.." "...." ".X.X" "..X." ".X.." "...." ".X.X" "X.X." ".X.X" "...."

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SLIDE 76

Red to try each open space and …

".X.X" "X.X." ".X.." "...." ".X.X" "..X." ".X.." "...." ".X.X" "X.X." ".X.X" "...." ".X.X" "X.X." ".X.X" "X..." ".X.X" "..X." ".X.." "...." ".X.X" "..X." ".X.." "...X" ".X.X" "X.X." ".X.." "...X" Red loses Red wins!

Make a move, ask oracle (recursion) how many winners after move? Oracle tries all moves and says none … meaning made move winner

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SLIDE 77

GridGame backtracking, count wins

private int winCount(char[][] board) { int wins = 0; for(int r=0; r < 4; r++){ for(int c=0; c < 4; c++){ if (canMove(board,r,c)){ board[r][c] = 'X'; int opponentWins = winCount(board); if (opponentWins == 0){ wins += 1; } board[r][c] = '.'; } } } return wins; }

".X.X" "X.X." ".X.." "...." ".X.X" "..X." ".X.." "...." ".X.X" "X.X." ".X.X" "...."

4/3/2020 Compsci 201, Spring 2020 111

SLIDE 78

Don't Know Much about History

Usene

net, C , Ches hess, C Chec ecker ers, … …

Alan Biermann, Tom Truscott: Internet Pioneer

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SLIDE 79

WOTO on Backtracking

http://b

://bit.l .ly/20 /201s 1spring2 g20-040 403-3

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SLIDE 80

Donald Knuth (Turing 1974)

Writin

ting multip ltiple le v volume mes “The he Art o

f Comput

uter er Program amming ng”

Title i

is “Prof

fessor

sor o

f The

Ar Art of

f Comp
mputer

Program amming ng” at ” at Stanf nford

Pipe O

e Organ i an in h his h home e wi with 812 812 pipes

Played

ed t the o

rgan

an i in the e Duke C e Chap apel el!

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