arrays and strings

Arrays and Strings

CSCI 2720University of GeorgiaSpring 2007

The Array ADT

Stores a sequence of consecutively numbered objects

Each object can be accessed (selected) using its index

More formally ….

Given integers l and u with u >= l-1, the interval l ..u is defined to be the set

of integers i such that l <=i<=u An array is a function

from any interval(the index set of the array)

to a set of objects or elements the value set of the array

Formally, continued …

If X is an array and i is a member of its index set, We write X[i] to denote the value of X

at i The members of the range of X are

known as the elements of X

The Array ADT

Access(X,i) Length(X) Assign(X,i,v) Initialize(X,v) Iterate(X,F)

Access(X,i)

Return X[i]

Length(X)

Return u – l + 1, the number of elements in I (the interval on X)

Assign(X,i,v)

Replace array X with a function whose value on i is v (and whose value on all other arguments is unchanged).

We also write this as: X[i] <- v

Initialize(X,v)

Assign v to every element of array X

Iterate(X,F)

Apply F to each element of array X in order, from smallest index to largest index. F is an action on a single array element. for i = l to u do

F(X[i])

String

A special type of array If is any finite set, then a string over

is an array whose value set is and whose index

set is 0..n-1 for some non-negative n The set is called an alphabet Each element of is called a character often consists of the Roman alphabet,

plus digits, the space, and common punctuation marks

Strings

If w is a string, then Length(w) = n

Also written |w|

If w = TREE, then w is a string of length 4 w[0] = T, w[1] = R

The null string is the string whose domain is the empty interval Has no elements Written

String-specific operations

Substring(w,i,m) Concat(w1,w2)

Substring(w,i,m)

w is a string; i,m integers Returns the string of length m containing

the portion of w that starts at i Formally:

returns a string w’ with indices 0 .. m-1 such that w’[k] = w[i+k] for each k satisfying 0 <=k <=m

only applies if 0 <= i <= |w| and 0 <= m <= (|w| -1)

otherwise, returns

Substring …

Example: w = SNICKERING Substring(w,2,3) returns ICK Substring(w,3,0) returns Substring(w,10,3) returns

Prefix each substring(w,0,j) for 0<= j <= |w| is a prefix of w

Suffix each substring(w,j, |w| - j) for 0<= j <= |w| is a

suffix of w

Concat(w1,w2)

returns a string of length |w1| + |w2| whose characters are the characters of

w1 followed by those of w2

Concat(w,) = Concat(,w) = w Example: w1 = BIRD, w2 = DOG,

Concat(w1,w2) = BIRDDOG Concat(w2,w1) = DOGBIRD

Tables vs. Arrays

Table = physical organization of memory into sequential cells

Array = an abstract data type, with specific operations

Arrays frequently implemented using tables, but may be implemented in other ways

Multi-dimensional arrays

a function whose range is any set V and whose domain is the Cartesian product of any number of intervals

the Cartesian product of intervals I1, I2, …Id, written as I1 x I2 x … Id, is the set of all d-tuples <i1, i2, … id> such that ik Ik for each k.

Multi-D arrays

if C is a multidimensional array and if i =<i1, i2, … id> then C[i1, i2, … id] is the value of C at i

The dimension of a multi-D array is the number of intervals whose Cartesian product makes up the index set

The size of the kth dimension of such an array is the number of elements in Ik

Contiguous Representation of Arrays: Why Computer Scientists start counting at 0

Store elements in a table: x x+4 x+8 x+12 x+16 x+20

x[0] x[1] x[2] x[3] x[4] x[5]

Each element begins at x + 4(i-1) x = starting address of the array 4 = sizeof(element) i = index of element of interest

17 43 87 94 101 143

More generally

if X is the address of the first cell in memory of an array with indices l..u, and if each element has size L, then the ith element is stored at address

X + L * (i-1) the element can be retrieved in

constant time

When iterating through the array

can save a few operations by doing “pointer arithmetic” just add L to current address to get

next element don’t have to subtract, multiply, add still linear in number of elements, but

faster linear

Where’s the needed info stored?

Could store L, l, and u at the starting address of X .. but would need to adjust the formula to calculate the location of individual cells.

If language is strongly typed, some or all of L, l, and u may be part of the definition of X and stored elsewhere C/C++ -- L part of typing info, l assumed to be

0, u not stored (programmer needs to keep track)

Where’s the needed info stored?

Can use a sentinel value after the last element of the array C/C++ -- we do this with strings. Store

a ‘\0’ at the end means that you need to iterate through to

find Length, no longer O(1)

What if the elements have different lengths?

allot Max to all elements wasted space can still access in O(1) time

store pointers to elements pointers require memory need 2 accesses (calculate location of pointer,

then follow it), but still O(1) pointer to element is at X + P * (i-1) easy to swap even large or complex elements

… just swap their pointers

2D arrays

can also represent in contiguous memory … but do we keep rows together or do we keep columns together??

Example: array with logical orderingA B C D E F G HI J K L

Row major v. column-major

ABCDEFGHIJKL

AEIBFJCGKDHL

Where are 2D elements stored?

Row-major: R[i,j] stored at: R + L * (NPR(i-1) + (j-1)), where

R is starting address of the array L is the size of each element NPR is the number of elements per row i is the row number j is the column number

Where are 2D elements stored?

Column-major: C[i,j] stored at: C + L * (NPC(j-1) + (i-1)), where

C is starting address of the array L is the size of each element NPC is the number of elements per

column i is the row number j is the column number

Multi-dimensional arrays

Constant-time initialization

procedure Initialize(ptr M, value v)//Initialize each element of M to v

Count(M) <- 0Default(m) <- v

function Valid(int I, ptr M): boolean//return true if M[i] has been modified //since last Initialize return (0 <= When(M)[i] < Count(M)) and

(Which(m)[When(M)[i]] == i)

Constant time initialization

function Access(int i, ptr M):value// return M[i]if Valid(I,M) then

return Data(M)[i]else

return Default(M)

procedure Assign(ptr M, int I, value v)// Set M[i] <- v

if not Valid(i, M) thenWhen(M)[i] <- Count(M)Which(M)[Count(M)] <- iCount(M) <- Count(M) + 1

Data(M)[i] <- v

But requires 3x memory …

Which(M)

When(M)

Data(M)

Sparse Arrays

Definitions List Representations Hierarchical Tables Arrays with Special Shapes

Sparse Arrays

some arrays contain only a few elements … wouldn’t it be more efficient to store only the non-null values? same idea when only a few values differ from the majority

some arrays have a special shape … upper diagonal matrix, symmetric matrix

sparse array : an array in which only a small fraction of the elements are significant in some way

null element: doesn’t need to be stored; is either actually null, or well-known, or easily calculated

List representations

Hierarchical tables

Upper-triangular matrix

Representation of Strings

Background Huffman Encoding Lempel-Ziv Encoding

Representing Strings

How much space do we need? Assume we represent every

character. How many bits to represent each

character? Depends on ||

Bits to encode a character

Two character alphabet{A,B} one bit per character:

0 = A, 1 = B Four character alphabet{A,B,C,D}

two bits per character: 00 = A, 01 = B, 10 = C, 11 = D

Six character alphabet {A,B,C,D,E, F} three bits per character: 000 = A, 001 = B, 010 = C, 011 = D, 100=E,

101 =F, 110 =unused, 111=unused

More generally

The bit sequence representing a character is called the encoding of the character.

There are 2n different bit sequences of length n,

ceil(lg||) bits required to represent each character in

if we use the same number of bits for each character then length of encoding of a word is |w| * ceil(lg||)

Can we do better??

If is very small, might use run-length encoding

What if …

the string we encode doesn’t use all the letters in the alphabet?

log2(ceil(|set_of_characters_used|) But then also need to store / transmit

the mapping from encodings to characters

… and is typically close to size of alphabet

Huffman Encoding:

Still assumes encoding on a per-character basis

Observation: assigning shorter codes to frequently used characters can result in overall shorter encodings of strings

requires assigning longer codes to rarely used characters

Problem: when decoding, need to know how many bits to

read off for each character. Solution:

Choose an encoding that ensures that no character encoding is the prefix of any other character encoding. An encoding tree has this property.

A Huffman Encoding Tree

12

21

9

7

43

5

23

A T R N

E

0 1

0 1

0 1 0 1

12

21

9

7

43

5

23

A T R N

E

0 1

0 1

0 1 0 1

A 000

T 001

R 010

N 011

E 1

Weighted path length

A 000

T 001

R 010

N 011

E 1

Weighted path = Len(code(A)) * f(A) +

Len(code(T)) * f(T) + Len(code(R) ) * f(R) +

Len(code(N)) * f(N) + Len(code(E)) * f(E)

= (3 * 3) + ( 2 * 3) + (3 * 3) + (4 *3) + (9*1)

= 9 + 6 + 9 + 12 + 9 = 45

Claim (proof in text) : no other encoding can result in a shorter weighted path length

Building the Huffman Tree

A3

T4

R4

E5

Building the Huffman Tree

A3

T4

R4

E5

7

Building the Huffman Tree

R4

E5

A3

T4

7

Building the Huffman Tree

R4

E5

A3

T4

79

Building the Huffman Tree

A3

T4

7

R4

E5

9

Building the Huffman Tree

A3

T4

7

R4

E5

9

16

Building the Huffman Tree

A3

T4

7

R4

E5

9

160

0 1

1

0 1

00 01 10 11

Taking a step back …

Why do we need compression? rate of creation of image and video data image data from digital camera

today 1k by 1.5 k is common = 1.5 mbytes need 2k by 3k to equal 35mm slide = 6

mbytes video at even low resolution of

512 by 512 and 3 bytes per pixel, 30 frames/second

Compression basics video data rate

23.6 mbytes/second 2 hours of video = 169 gigabytes

mpeg-1 compresses 23.6 mbytesdown to 187 kbytes per second 169 gigabytes down to 1.3 gigabytes

compression is essential for both storage and transmission of data

Compression basics

compression is very widely used jpeg, gif for single images mpeg1, 2, 3, 4 for video sequence zip for computer data mp3 for sound

based on two fundamental principles spatial coherence and temporal

coherence similarity with spatial neighbor similarity with temporal neighbor

Basics of compression

character = basic data unit in the input stream

represents byte, bit, etc. strings = sequences of characters encoding = compression decoding = decompression codeword = data elements used to

represent input characters or character strings

codetable = list of codewords

Codeword

encoding/compression takes characters/strings as input and use

codetable to decide on which codewords to produce

decoder/decompressor takes codewords as input and uses same

codetable to decide on which characters/strings to produce

Codetable

clearly both encoder and decoder must pass the encoded data as a series of codewords

also must pass the codetable the codetable can be passed explicitly

or implicitly that is we either

pass it across agree on it beforehand (hard wired) recreate it from the codewords (clever!)

Basic definitions

compression ratio = size of original data / compressed data basically higher compression ratio the better

lossless compression output data is exactly same as input data essential for encoding computer processed data

lossy compression output data not same as input data acceptable for data that is only viewed or heard

Lossless versus lossy

human visual system less sensitive to high frequency losses and to losses in color

lossy compression acceptable for visual data

degree of loss is usually a parameter of the compression algorithm

tradeoff - loss versus compression higher compression => more loss lower compression => less loss

Symmetric versus asymmetric

symmetric encoding time == decoding time essential for real-time applications (ie.

video or audio on demand) asymmetric

encoding time >> decoding ok for write-once, read-many situations

Entropy encoding

compression that does not take into account what is being compressed

normally is also lossless encoding most common types of entropy

encoding run length encoding Huffman encoding modified Huffman (fax…) Lempel Ziv

Source encoding

takes into account type of data (ie. visual)

normally is lossy but can also be lossless most common types in use:

JPEG, GIF = single images MPEG = sequence of images (video) MP3 = sound sequence

often uses entropy encoding as a sub-routine

Run length encoding

one of simplest and earliest types of compression

take account of repeating data (called runs) runs are represented by a count along with

the original data eg. AAAABB => 4A2B

do you run length encode a single character?

no, use a special prefix character to represent start of runs

Run length encoding

runs are represented as <prefix char><repeat count><run char>

prefix char itself becomes<prefix char>1<prefix char>

want a prefix char that is not too common an example early use is MacPaint file

format run length encoding is lossless and has

fixed length codewords

MacPaint File Format

Run length encoding

works best for images with solid background

good example of such an image is a cartoon

does not work as well for natural images

does not work well for English text however, is almost always a part of a

larger compression system

Huffman encoding

assume we know the frequency of each character in the input stream

then encode each character as a variable length bit string, with the length inversely proportional to the character frequency

variable length codewords are used; early example is Morse code

Huffman produced an algorithm for assigning codewords optimally

Huffman encoding

input = probabilities of occurrence of each input character (frequencies of occurrence)

output is a binary tree each leaf node is an input character each branch is a zero or one bit codeword for a leaf is the concatenation of bits

for the path from the root to the leaf codeword is a variable length bit string

a very good compression ratio (optimal)?

Huffman encoding

Basic algorithmMark all characters as free tree nodesWhile there is more than one free node

Take two nodes with lowest freq. of occurrenceCreate a new tree node with these nodes as

children and with freq. equal to the sum of their freqs.

Remove the two children from the free node list.Add the new parent to the free node list

Last remaining free node is the root of the binary tree used for encoding/decoding

Huffman example

a series of colors in an 8 by 8 screen colors are red, green, cyan, blue,

magenta, yellow, and black sequence is

rkkkkkkk gggmcbrr kkkrrkkk bbbmybbr kkrrrrgg gggggggr kkbcccrr grrrrgrr

Huffman example

Huffman example

Huffman example

Huffman example

Fixed versus variable length codewords

run length codewords are fixed length Huffman codewords are variable length length inversely proportional to frequency all variable length compression schemes

have the prefix property one code can not be the prefix of another binary tree structure guarantees that this

is the case (a leaf node is a leaf node!)

Huffman encoding

advantages maximum compression ratio assuming correct

probabilities of occurrence easy to implement and fast

disadvantages need two passes for both encoder and decoder

one to create the frequency distribution one to encode/decode the data

can avoid this by sending tree (takes time) or by having unchanging frequencies

Modified Huffman encoding

if we know frequency of occurrences, then Huffman works very well

consider case of a fax; mostly long white spaces with short bursts of black

do the following run length encode each string of bits on a line Huffman encode these run length codewords use a predefined frequency distribution

combination run length, then Huffman

Beyond Huffman Coding …

1977 – Lempel & Ziv, Israeli information theorists, develop a dictionary-based compression method (LZ77)

1978 – they develop another dictionary-based compression method (LZ78)

The LZ family

LZ77 LZR LZSS LZB LZH – used by zip and unzip

LZ78 LZW – Unix compress LZC – Unix compress LZT LZMW LZJLZFG

Overview of LZ family

To demonstrate: simple alphabet containing only two

letters, a and b, and create a sample stream of text

LZ family overview

Rule: Separate this stream of characters into pieces of text so that the shortest piece of data is the string of characters that we have not seen so far.

Sender : The Compressor

Before compression, the pieces of text from the breaking-down process are indexed from 1 to n:

indices are used to number the pieces of data. The empty string (start of text) has index 0. The piece indexed by 1 is a. Thus a, together with

the initial string, must be numbered Oa. String 2, aa, will be numbered 1a, because it

contains a, whose index is 1, and the new character a.

the process of renaming pieces of text starts to pay off. Small integers replace what were once

long strings of characters. can now throw away our old stream of

text and send the encoded information to the receiver

Bit Representation of Coded Information

Now, want to calculate num bits needed each chunk is an int and a letter num bits depends on size of table

permitted in the dictionary every character will occupy 8 bits because

it will be represented in US ASCII format

Compression good?

in a long string of text, the number of bits needed to transmit the coded information is small compared to the actual length of the text.

example: 12 bits to transmit the code 2b instead of 24 bits (8 + 8 + 8) needed for the actual text aab.

Receiver: The Decompressor (Implementation

receiver knows exactly where boundaries are, so no problem in reconstructing the stream of text.

Preferable to decompress the file in one pass; otherwise, we will encounter a problem with temporary storage..

Lempel-Ziv applet

Seehttp://www.cs.mcgill.ca/~cs251/Ol

dCourses/1997/topic23/#JavaApplet

Lempel Ziv Welsch (LZW)

previous methods worked only on characters LZW works by encoding strings some strings are replaced by a single

codeword for now assume codeword is fixed (12 bits) for 8 bit characters, first 256 (or less) entries

in table are reserved for the characters rest of table (257-4096) represent strings

LZW compression

trick is that string-to-codeword mapping is created dynamically by the encoder

also recreated dynamically by the decoder need not pass the code table between the

two is a lossless compression algorithm degree of compression hard to predict depends on data, but gets better as

codeword table contains more strings

LZW encoder

Initialize table with single character stringsSTRING = first input characterWHILE not end of input stream

CHARACTER = next input characterIF STRING + CHARACTER is in the string table

STRING = STRING + CHARACTERELSE

Output the code for STRINGAdd STRING + CHARACTER to the string

tableSTRING = CHARACTER

END WHILEOutput code for string

Demonstrations

Another animated LZ algorithm … http://www.data-compression.com/lempelziv.html

LZW encoder example

compress the string BABAABAAA

LZW decoder

Lempel-Ziv compression

a lossless compression algorithm All encodings have the same length

But may represent more than one character

Uses a “dictionary” approach – keeps track of characters and character strings already encountered

LZW decoder example

decompress the string <66><65><256><257><65><260>

LZW Issues

compression better as the code table grows

what happens when all 4096 locations in string table are used?

A number of options, but encoder and decoder must agree to do the same thing do not add any more entries to table (as is) clear codeword table and start again clear codeword table and start again with

larger table/longer codewords (GIF format)

LZW advantages/disadvantages

advantages simple, fast and good compression can do compression in one pass dynamic codeword table built for each file decompression recreates the codeword

table so it does not need to be passed disadvantages

not the optimum compression ratio actual compression hard to predict

Entropy methods

all previous methods are lossless and entropy based

lossless methods are essential for computer data (zip, gnuzip, etc.)

combination of run length encoding/huffman is a standard tool

are often used as a subroutine by other lossy methods (Jpeg, Mpeg)

Lempel-Ziv compression

a lossless compression algorithm All encodings have the same length

But may represent more than one character

Uses a “dictionary” approach – keeps track of characters and character strings already encountered

String Searching

Background Knuth-Morris-Pratt algorithm Boyer-Moore algorithm Fingerprinting and the Karp-Rabin

algorithm