CPS 100, Fall 2011 12.1

PF-AT-EOS

Data compression Seriously important, why?

Priority Queue Seriously important, why?

Assignment overview Huff _______

Bits, Bytes, Atoms What's an int? ASCII? double?

CPS 100, Fall 2011 12.2

PQ Application: Data Compression

Compression is a high-profile application .zip, .mp3, .jpg, .gif, .gz, … What property of MP3 was a significant factor in

what made Napster work (why did Napster ultimately fail?)

Who invented Napster, how old, when?

Why do we care? Secondary storage capacity doubles every year Disk space fills up quickly on every computer

system More data to compress than ever before Will we ever need to stop worrying about storage?

CPS 100, Fall 2011 12.3

More on Compression

Different compression techniques .mp3 files and .zip files? .gif and .jpg? Lossless and lossy

Impossible to compress/lossless everything: Why?

Lossy methods Good for pictures, video, and audio (JPEG,

MPEG, etc.) Lossless methods

Run-length encoding, Huffman, LZW, …

CPS 100, Fall 2011 12.4

Huffman Coding

Understand Huffman Coding Data compression Priority Queue Bits and Bytes Greedy Algorithm

Many compression algorithms Huffman is optimal, per-character compression Still used, e.g., basis of Burrows-Wheeler Other compression 'better', sometimes slower? LZW, GZIP, BW, …

CPS 100, Fall 2011 12.5

Compression and Coding

What gets compressed? Save on storage, why is this a good idea? Save on data transmission, how and why?

What is information, how is it compressible? Exploit redundancy, without that, hard to

compress Represent information: code (Morse cf.

Huffman) Dots and dashes or 0's and 1's How to construct code?

CPS 100, Fall 2011 12.6

Coding/Compression/Concepts

For ASCII we use 8 bits, for Unicode 16 bits Minimum number of bits to represent N

values? Representation of genomic data (a, c ,g, t)? What about noisy genomic data?

We can use a variable-length encoding, e.g., Huffman How do we decide on lengths? How do we

decode? Values for Morse code encodings, why? … - - - …

CPS 100, Fall 2011 12.7

Huffman Coding

D.A Huffman in early 1950’s: story of invention Analyze and process data before compression Not developed to compress data “on-the-fly”

Represent data using variable length codes Each letter/chunk assigned a codeword/bitstring Codeword for letter/chunk is produced by

traversing the Huffman tree Property: No codeword produced is the prefix of

another Frequent letters/chunk have short encoding,

while those that appear rarely have longer ones Huffman coding is optimal per-character coding

method

CPS 100, Fall 2011 12.8

Mary Shaw Software engineering and

software architecture Tools for constructing

large software systems Development is a small

piece of total cost, maintenance is larger, depends on well-designed and developed techniques

Interested in computer science, programming, curricula, and canoeing, health-care costs

ACM Fellow, Alan Perlis Professor of Compsci at CMU

CPS 100, Fall 2011 12.9

Huffman coding: go go gophers

choose two smallest weights combine nodes +

weights Repeat Priority queue?

Encoding uses tree: 0 left/1 right How many bits?

ASCII 3 bitsg 103 1100111 000 ??o 111 1101111 001 ??p 112 1110000 010h 104 1101000 011e 101 1100101 100r 114 1110010 101s 115 1110011 110sp. 32 1000000 111

g o e r s *3 3

h1 211 1

2

p1

h1

2

e1

r1

3

s1

*2

2

p1

h1

2

e1

r1

4

g

3

o

3

6

1

p

CPS 100, Fall 2011 12.10

Huffman coding: go go gophers

Encoding uses tree/trie: 0 left/1 right How many bits? 37!! Savings? Worth it?

ASCII 3 bitsg 103 1100111 000 00o 111 1101111 001 01p 112 1110000 010 1100h 104 1101000 011 1101e 101 1100101 100 1110r 114 1110010 101 1111s 115 1110011 110 100sp. 32 1000000 111 101

3

s1

*2

2

p1

h1

2

e1

r1

4g

3

o

3

6

3

2

p1

h1

2

e1

r1

4

s1

*2

7

g3

o3

6

13

CPS 100, Fall 2011 12.11

Building a Huffman tree

Begin with a forest of single-node trees/tries (leaves) Each node/tree/leaf is weighted with character

count Node stores two values: character and count

Repeat until there is only one node left: root of tree Remove two minimally weighted trees from forest Create new tree/internal node with minimal trees

as children,•Weight is sum of children’s weight (no char)

How does process terminate? Finding minimum? Remove minimal trees, hummm……

CPS 100, Fall 2011 12.12

How do we create Huffman Tree/Trie?

Insert weighted values into priority queue What are initial weights? Why?

Remove minimal nodes, weight by sums, re-insert Total number of nodes?

PriorityQueue<TreeNode> pq = new PriorityQueue<TreeNode>(); for(int k=0; k < freq.length; k++){ pq.add(new TreeNode(k,freq[k],null,null)); } while (pq.size() > 1){ TreeNode left = pq.remove(); TreeNode right = pq.remove(); pq.add(new TreeNode(0,left.weight+right.weight, left,right)); } TreeNode root = pq.remove();

CPS 100, Fall 2011 12.13

Creating compressed file

Once we have new encodings, read every character Write encoding, not the character, to compressed

file Why does this save bits? What other information needed in compressed file?

How do we uncompress? How do we know foo.hf represents compressed

file? Is suffix sufficient? Alternatives?

Why is Huffman coding a two-pass method? Alternatives?

CPS 100, Fall 2011 12.14

Uncompression with Huffman

We need the trie to uncompress 000100100010011001101111

As we read a bit, what do we do? Go left on 0, go right on 1 When do we stop? What to do?

How do we get the trie? How did we get it originally? Store 256

int/counts•How do we read counts?

How do we store a trie? 20 Questions relevance?• Reading a trie? Leaf indicator? Node values?

CPS 100, Fall 2011 12.15

Other Huffman Issues

What do we need to decode? How did we encode? How will we decode? What information needed for decoding?

Reading and writing bits: chunks and stopping Can you write 3 bits? Why not? Why? PSEUDO_EOF BitInputStream and BitOutputStream: API

What should happen when the file won’t compress? Silently compress bigger? Warn user?

Alternatives?

CPS 100, Fall 2011 12.16

Huffman Complexities How do we measure? Size of input file, size of

alphabet Which is typically bigger?

Accumulating character counts: ______ How can we do this in O(1) time, though not

really Building the heap/priority queue from counts ____

Initializing heap guaranteed Building Huffman tree ____

Why? Create table of encodings from tree ____

Why? Write tree and compressed file _____

CPS 100, Fall 2011 12.17

Good Compsci 100 Assignment?

Array of character/chunk counts, or is this a map? Map character/chunk to count, why array?

Priority Queue for generating tree/trie Do we need a heap implementation? Why?

Tree traversals for code generation, uncompression One recursive, one not, why and which?

Deal with bits and chunks rather than ints and chars The good, the bad, the ugly

Create a working compression program How would we deploy it? Make it better?

Benchmark for analysis What’s a corpus?

CPS 100, Fall 2011 12.18

Anita Borg 1949-2003 “Dr. Anita Borg tenaciously

envisioned and set about to change the world for women and for technology. … she fought tirelessly for the development technology with positive social and human impact.”

“Anita Borg sought to revolutionize the world and the way we think about technology and its impact on our lives.”

http://www.youtube.com/watch?v=1yPxd5jqz_Q

CPS 100, Fall 2011 12.19

YAQ, YAQ, haha! (Yet Another Queue)

What is the dequeue policy for a Queue? Why do we implement Queue with LinkedList

• Interface and class in java.util Can we remove an element other than first?

How does queue help word-ladder/shortest path? First item enqueued/added is the one we want What if different element is “best”?

PriorityQueue has a different dequeue policy Best item is dequeued, queue manages itself to

ensure operations are efficient

CPS 100, Fall 2011 12.20

PriorityQueue raison d’être

Algorithms Using PQ for efficiency Shortest Path: Google Maps/Garmin to Internet

Routing• How is this like word-ladder? How different?

Event based simulation• Coping with explosion in number of particles or things

Optimal A* search, game-playing, AI,• Can't explore entire search space, can estimate good

move

Data compression facilitated by priority queue Alltime best assignment in a Compsci 100 course?

• Subject to debate, of course From A-Z, soup-to-nuts, bits to abstractions

CPS 100, Fall 2011 12.21

Priority Queue Compression motivates ADT priority queue

Supports two basic operations• add/insert -– an element into the priority queue• remove/delete – the minimal element from the

priority queue Implementations allow getmin/peek as well as

delete• Analogous to top/pop, peek/dequeue in stacks,

queues

Think about implementing the ADT, choices? Add compared to min/remove Balanced search tree is ok, but can we do

better?

CPS 100, Fall 2011 12.22

Priority Queue sorting See PQDemo.java,

code below sorts, complexity? String[] array = {...}; // array filled with dataPriorityQueue<String> pq = new PriorityQueue<String>();for(String s : array) pq.add(s);for(int k=0; k < array.length; k++){ array[k] = pq.remove();}

Bottlenecks, operations in code above Add words one-at-a-time to PQ v. all-at-

once What if PQ is an array, add or remove

fast/slow? We’d like PQ to have tree characteristics,

why?

CPS 100, Fall 2011 12.23

Priority Queue top-M sorting

What if we have lots and lots and lots of data code below sorts top-M elements, complexity?

Scanner s = … // initialize; PriorityQueue<String> pq = new PriorityQueue<String>(); while (s.hasNext()) { pq.add(s.next()); if (pq.size() > M) pq.remove(); }

What’s advantageous about this code? Store everything and sort everything? Store everything, sort first M? What is complexity of sort: O(n log n)

CPS 100, Fall 2011 12.24

Priority Queue implementations

Priority queues: average and worst caseInsert average

Getmin(delete)

Insert worst

Getmin(delete)

Unsorted list

Sorted list

Search tree

Balanced tree

Heap

log n log n O(n) O(n)

O(1) log n log nlog n

log nlog n log n log n

O(n)O(1) O(n)O(1)

O(n) O(1) O(n) O(1)

Heap has O(n) build heap from n elements

CPS 100, Fall 2011 12.25

PriorityQueue.java [java.util]

What about objects inserted into pq? Comparable, e.g., essentially sortable How can we change what minimal means? Implementation uses heap, tree stored in an

array

Use a Comparator for comparing entries we can make a min-heap act like a max-heap, see PQDemo Where is class Comparator declaration? How

used? What if we didn't know about

Collections.reverseOrder?• How do we make this ourselves?

CPS 100, Fall 2011 12.26

Big-Oh and a tighter look at inserts log(1) + log(2) + log(3) + … + log(n)

Property of logs, log(a) + log(b) = log(a*b) log(1*2*3*…*n) = log(n!)

We can show using Sterling’s formula:

log(n!) = c1*log(n) + nlog(n) – c2*n We can get O(n log n) easily, this is a

tight bound, lower, n log n) as well

CPS 100, Fall 2011 12.27

Priority Queue implementation

Heap data structure is fast and reasonably simple Why not use inheritance hierarchy as was used with

Map? Trade-offs when using HashMap and TreeMap:

• Time, space, ordering properties, TreeMap support?

Changing comparison when calculating priority? Create object to replace, or in lieu of compareTo

• Comparable interface compares this to passed object • Comparator interface compares two passed objects

Both comparison methods: compareTo() and compare()• Compare two objects (parameters or self and parameter)• Returns –1, 0, +1 depending on <, ==, >

CPS 100, Fall 2011 12.28

Creating Heaps

Heap: array-based implementation of binary tree used for implementing priority queues: add/insert, peek/getmin, remove/deletemin, O(???)

Array minimizes storage (no explicit pointers), faster too, contiguous (cache) and indexing

Heap has shape property and heap/value property shape: tree filled at all levels (except perhaps last)

and filled left-to-right (complete binary tree) each node has value smaller than both children

CPS 100, Fall 2011 12.29

Array-based heap store “node values” in array

beginning at index 1 for node with index k

left child: index 2*k right child: index 2*k+1

why is this conducive for maintaining heap shape?

what about heap property? is the heap a search tree? where is minimal node? where are nodes added?

deleted?

0 1 2 3 4 5 6 7 8 9 10

6 10 7 1713 259 21 19

6

10 7

17 13 9 21

19 25

CPS 100, Fall 2011 12.30

Thinking about heaps Where is minimal

element? Root, why?

Where is maximal element? Leaves, why?

How many leaves are there in an N-node heap (big-Oh)? O(n), but exact?

What is complexity of find max in a minheap? Why? O(n), but ½ N?

Where is second smallest element? Why? Near root?

6

10 7

17 13 9 21

19 25

0 1 2 3 4 5 6 7 8 9 10

6 10 7 1713 259 21 19

CPS 100, Fall 2011 12.31

Adding values to heap to maintain heap shape,

must add new value in left-to-right order of last level could violate heap

property move value “up” if too

small

change places with parent if heap property violated stop when parent is

smaller stop when root is reached

pull parent down, swapping isn’t necessary (optimization)

13

610 7

17 9 21

19 25 8

13

610 7

17 9 21

19 25

610 7

17 9 21

19 25 13

8

insert 8

bubble 8 up

67

17 9 21

19 25

8

13

10

CPS 100, Fall 2011 12.32

Adding values, details (pseudocode)void add(Object elt){ // add elt to heap in myList myList.add(elt); int loc = myList.size()-1;

while (1 < loc && elt < myList.get(loc/2)){ myList.set(loc,myList.get(loc/2)); loc = loc/2; // go to parent } // what’s true here?

myList.set(loc,elt);}

13

610 7

17 9 21

19 25

8

13

610 7

17 9 21

19 25

0 1 2 3 4 5 6 7 8 9 10

6 10 7 1713 259 21 19

array myList

CPS 100, Fall 2011 12.33

Removing minimal element Where is minimal element?

If we remove it, what changes, shape/property?

How can we maintain shape? “last” element moves to

root What property is violated?

After moving last element, subtrees of root are heaps, why? Move root down (pull child

up) does it matter where? When can we stop “re-heaping

”? Less than both children Reach a leaf

13

610 7

17 9 21

19 25

13

2510 7

17 9 21

19

13

710 25

17 9 21

19

13

710 9

17 25 21

19

CPS 100, Fall 2011 12.34

Views of programming

Writing code from the method/function view is pretty similar across languages Organizing methods is different, organizing code

is different, not all languages have classes, Loops, arrays, arithmetic, …

Program using abstractions and high level concepts Do we need to understand 32-bit twos-

complement storage to understand x =x+1? Do we need to understand how arrays map to

contiguous memory to use ArrayLists?

CPS 100, Fall 2011 12.35

Bottom up meets Top down

We teach programming by teaching abstractions What's an array? What’s an ArrayList? What's an array in C? What's a map, Hashmap? Treemap?

How are programs stored and executed in the JVM? Differences on different architectures? Similarities between Java and C++ and Python and

PHP? What about how the CPU works? How memory

works? Shouldn't we study this too?

CPS 100, Fall 2011 12.36

From bit to byte to char to int to long

Ultimately everything is stored as either a 0 or 1 Bit is binary digit a byte is a binary term (8

bits) We should be grateful we can deal with Strings

rather than sequences of 0's and 1's. We should be grateful we can deal with an int

rather than the 32 bits that comprise an int

If we have 255 values for R, G, B, how can we pack this into an int? Why should we care, can’t we use one int per

color? How do we do the packing and unpacking?

CPS 100, Fall 2011 12.37

More information on bit, int, long

int values are stored as two's complement numbers with 32 bits, for 64 bits use the type long, a char is 16 bits Standard in Java, different in C/C++ Facilitates addition/subtraction for int values We don't need to worry about this, except to

note:• Infinity + 1 = - Infinity (see Integer.MAX_VALUE)• Math.abs(-Infinity) > Infinity

Java byte, int, long are signed values, char unsigned What are values for 16-bit char? 8-bit byte? Why will this matter in Burrows Wheeler?

CPS 100, Fall 2011 12.38

More details about bits

How is 13 represented? … _0_ _0_ _1_ _1_ _0_ _1_ 24 23 22 21 20

Total is 8+4+1 = 13 What is bit representation of 32? Of 15? Of

1023? What is bit-representation of 2n - 1? What is bit-representation of 0? Of -1?

• Study later, but -1 is all 1’s, left-most bit determines < 0

Java signed byte: -128..127, # bits? What if we only want 0-255? (Huff, pixels, …) Convert negative values or use char, trade-offs?

Java char unsigned: 0..65,536 # bits? Why is char unsigned? Why not as in C++/C?

CPS 100, Fall 2011 12.39

How are data stored?

To facilitate Huffman coding we need to read/write one bit Why do we need to read one bit? Why do we need to write one bit? When do we read 8 bits at a time? 32 bits?

We can't actually write one bit-at-a-time. We can't really write one char at a time either. Output and input are buffered,minimize memory

accesses and disk accesses Why do we care about this when we talk about

data structures and algorithms?• Where does data come from?

CPS 100, Fall 2011 12.40

How do we buffer char output?

Done for us as part of InputStream and Reader classes InputStreams are for reading bytes Readers are for reading char values Why do we have both and how do they interact?Reader r = new InputStreamReader(System.in); Do we need to flush our buffers?

In the past Java IO has been notoriously slow Do we care about I? About O? This is changing, and the java.nio classes help

• Map a file to a region in memory in one operation