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Strings

CopyWrite D.Bockus

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Strings

• Def: A string is a sequence (possibly empty) of symbols from some alphabet.

• What do we use strings for? 1) Text processing. Word processing.

2) Grammatical Structure of Languages.

3) Searching, String Sequences

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String Example 1• E.g. Java's "for" statement. (simplistic view)

for (initialization ; condition; increment) u v w

– Where a “for” statement breaks down into ‘for (u;v;w)’.

– We can then define each part:• u » identifier = constant

• v » identifier relational_operator value

• w » identifier++

• In this context we can also define a while loop as:– while(v)

• Deterministic Context Free Languages (programming languages) are defined by breaking rules down into sub-rules, etc.

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Strings Example 2

– Genetic Coding:

– aab cd aab d

Searching for, and matching codes, leads into graph theory. a b b b a a a b b a a c c c d d d a b f g a b f g b bd d

s1s2 s3 s4

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String Example 3

• Compression• Converting a large volume of symbols into a

smaller format.– Huffman Coding

– LZW compression.

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Basics

Given a string v: The length of v can be expressed:

1) |v| = magnitude of v

2) length (v).

– Empty strings v = ' ' or v =

• There are 5 common operations that may be performed on strings.– Insertion, Deletion, Substitution, Concatenation,

Comparison.

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Insertion & Deletion• Insertion

k = ac where

a = (a1, a2 .. am)

c = (c1, c2 .. cn)

insert b = (b1, b2 ... bp) between ac

k = abc

= a1, a2 … am, b1, b2 … bp, c1, c2 … cn

|k| = m + p + n

• Deletion

k = abc

delete c

k = ab

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Substitutionk = u where & maybe null, i.e. || = 0 or || = 0

Search for u & replace with v.

k = v Notice this same operation can be accomplished

with a deletion and insertion.k = u

Delete uk =

Insert vk = v

Note: |u| does not have to equal |v|; |k| before does not have to equal |k| after.

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Concatination

This is the joining of 2 strings a & b.c = a + b

So if a = (a1, a2 .. am) & b = (b1, b2 ... bn)

Then c = (a1, a2 .. am, b1, b2 ... bn)

– Note: concatenation may be performed with insertion, i.e. insert b at end of a, or substitution.

• a where is null.

• substitute for b.

|c| = m + n

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Comparison

– Compare a & b to see if one of the following is true.1) a < b

2) a = b

3) a > b

• 1) a is less then b if a lexicographical comparison is performed on each element of a & b.

• Until the first ak < bk is true.

a b

a1 b1 a1 = b1

a2 b2 a2 = b2

a3 b3 a3 < b3

a4 b4 a4 = b4

b5 (a5 = ) < b5

a3 < b3So, a < b

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Comparison Cont...

– Note: a3 < b3 is the first instance where an element in a differs from b.

a < b.

– If a3 = b3 then a is still less then b because |a| < |b|. Can think of having a value of - for comparison purposes.

• 2). For a = b the following must be true.• |a| = |b|

– and • ak = bk k

• 3). a > b, opposite of (1).– Or b < a

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String Representations

• Consider the string "L1 CMPR BANANAS WATERMELLONS 12”

• There are 6 ways to represent strings in storage noting that 3 criteria must be kept in mind

– Storage Efficiency (1:1 packing ratio)

– Ease of Lookup (Searching)

– Ease of Modification• Insertion

• Deletion

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Fixed Length Strings

L 1C M P RB A N A N A SW A T E R M E L O N S1 2

– Adv: Ease of Modification

– Dis: Storage Efficiency due of wasted space at end of short strings.

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Var Strings

– Adv: Easier to look up strings, we already have the length.

– Dis: Still wastes space.

2 L 14 C M P R7 B A N A N A S

11 W A T E R M E L O N S2 1 2

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Count Delimited

– Adv: Very efficient in space usage, Lookup is not bad.

– Dis: Modification is hard , Replacing a string must be same length or readjustment of array is needed.

02 L1 04 CMPR 07 BANANAS 11 WATERMELLONS 02 12

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Indexed List

– Adv: Good Storage and Search capabilities

– Dis: Modification is poor

Strategies include: always adding new strings and never reclaiming space except during a repack.

1 2 3 4 5 ...2 4 7 11 21 3 7 14 25

L1 CMPR BANANAS WATERMELLONS 12

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Linked List

– Adv: Modification is simple pointer manipulation.

– Dis: Storage overhead. • one address per character

– Note: Lookup can be improved by adding additional length field to table or by imploring a hash function.

Linked List

2 1

4 2

7 3

11 4

2 5

L 1

C M P R

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Blocked Linked List

– Adv: Better storage then linked list. • More characters per node

– Note: A trade off between dealing with single characters and blocks of characters during modification.

• Note: If modification is not required then methods such as indexed lists are quite useful. Applications include symbol tables in compilers.

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Implementation

– In most cases a variable length string structure is desirable. i.e. the most versatile.

– Consider a string type as:

String {int size;char data[];

}– Java declares string objects with methods to determine

length and other attributes.

– Declaring Variables:

String S1, S2

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Basic Functions

s.length(); -- Returns the length of S1

• Other Usefull functions– String s.concat(String t);– String new String(s);– String s.substring(int i);– int indexOf(String t, int index);

See Java api.

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Variable Length Coding

Old TreeChar Prob. Bits (MRC) Bits (Fixed) (pi)lg(pi) pi(bits)

a 0.15 3 3 -0.41 0.45e 0.25 2 3 -0.50 0.5i 0.13 3 3 -0.38 0.39

m 0.09 4 3 -0.31 0.36s 0.15 2 3 -0.41 0.3z 0.02 4 3 -0.11 0.08t 0.09 4 3 -0.31 0.36r 0.12 4 3 -0.37 0.48

H(U) 2.81Ave. Redundency

Fixed 3 0.06MRC 2.92 0.04

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Huffman Coding Algorithm

1) Collect a history of the frequencies of the characters i.e. determine the probabilities.

2) Arrange the characters in an ordered list (priority queue) based on increasing probabilities (frequency)

3) While (More then 1 node in List) Do i) Remove first 2 Nodes

ii) Combine into a tree and have the tree root represent the sum of the frequencies of the children

iii) Insert into List maintaining proper List order

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Variable Length Coding New TreeBased on new tree

Char Prob. Bits (MRC) Bits (Fixed) (pi)lg(pi) pi(bits)a 0.15 3 3 -0.41 0.45e 0.25 2 3 -0.50 0.5i 0.13 3 3 -0.38 0.39

m 0.09 3 3 -0.31 0.27s 0.15 3 3 -0.41 0.45z 0.02 4 3 -0.11 0.08t 0.09 4 3 -0.31 0.36r 0.12 3 3 -0.37 0.36

H(U) 2.81Ave. Redundency

Fixed 3 0.06MRC 2.86 0.02

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LZW Compression

• Lempel-Ziv Welch (LZW)• Uses a method of finding the largest known prefix

in a character string. • Typical uses.

– LossLess

– Compressed file can be reconstructed without data loss• GIF, TIFF

• zip & unzip

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LZW Compression

• Idea is to build a code table, where codes are added as they are discovered.

• Look at the prefix for a given character.

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Compressor Pseudo Code http://marknelson.us/1989/10/01/lzw-data-compression/

STRING = get input characterWHILE there are still input characters DO    CHARACTER = get input character    IF STRING+CHARACTER is in the string table then        STRING = STRING+character    ELSE        output the code for STRING        add STRING+CHARACTER to the string table        STRING = CHARACTER    END of IF

END of WHILE

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DeCompressor Pseudo Codehttp://marknelson.us/1989/10/01/lzw-data-compression/

Read OLD_CODEoutput OLD_CODECHARACTER = OLD_CODEWHILE there are still input characters DO    Read NEW_CODE    IF NEW_CODE is not in the translation table THEN        STRING = get translation of OLD_CODE        STRING = STRING+CHARACTER    ELSE        STRING = get translation of NEW_CODE    END of IF    output STRING    CHARACTER = first character in STRING    add OLD_CODE + CHARACTER to the translation table    OLD_CODE = NEW_CODEEND of WHILE

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Compressor Example

• Assume we have an alphabet of a and b.

• We start by building a code book initialized to all characters in the alphabet, in this case a and b.

• We can now compress the string:a a a b b b b b b a a b a a b a

Code String

2

3

4

5

6

7

8

9

0

1

a

b

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Compressor Example …

a a a b b b b b b a a b a a b a Code String

2

3

4

5

6

7

8

9

0

1

a

b

Output Code

0

Find largest prefix in code book

a a

Add code + next char to code bookFind largest prefix in code book

2

Add code + next char to code booka a b

Find largest prefix in code book

1

Add code + next char to code book b bFind largest prefix in code book

4

Add code + next char to code bookb b b

Find largest prefix in code book

5

Add code + next char to code book

b b b a

Find largest prefix in code book

3

a a b a

Add code + next char to code bookFind largest prefix in code book

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No more input to compress so stop

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De-compressor Example

• We have an encoded string. 0 2 1 4 5 3 7

• To decode we need two things,– knowledge of the alphabet.

– An initialized code book based on the alphabet.

• Headers on say GIF files contain the alphabet information.

• The code book is re-build during de-compression

Code String

2

3

4

5

6

7

8

9

0

1

a

b

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De-compressor Example..

• During De-compression a code is read and an attempt is made to find this code in the code book.

• There are two cases:– The code is found in the code book.– The code is not found in the code book.

• Code found:– output the string from found code.– make an entry based on:

previous string + firstChar of current string.

• Not found:– make an entry into the code book based on:

previous string + firstChar of previous string.– output the string of new entry.

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De-compressor Example...

• Notice that a code which is not found is a special case:

• E.g. during compression of a a a b b b ….– a is coded to 0, but the compressor now enters aa into

the code book.

– aa is the next code to be used.

– During de-compression, we can guess at this code.

– Text(previous) + FC(previous).

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De-compressor Example….

• More formally:– We encounter a string P[…]P[…]PQ.– If P[…] is in the code book and P[…]P is not, then the

compressor outputs P[…] and adds P[…]P to the code book.

– When the de-compressor sees P[…]P it will not of added this code yet.

– We know from the pattern that P[…] is already in the code book and it was the last code encountered, and that P[…]P would normally be added next (during compression).

– So…. We can accurately guess and enter P[…]P into the code book.

• Taken from: http://www.danbbs.dk/~dino/whirlgif/lzw.html

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De-compressor Example….

Code String

2

3

4

5

6

7

8

9

0

1

a

b

0 2 1 4 5 3 7

Output Text

a a

Found - No code book entry is made for first code

a

Not Found - Enter Text(previous) + FC(Previous).Output last code entered into code book

a a

Found - Enter Text(previous) + FC(current).

b

a a bNot Found - Enter Text(previous) + FC(Previous).b b

Output last code entered into code book

b b

Not Found - Enter Text(previous) + FC(Previous).

b b b

Output last code entered into code book

b b b

Found - Enter Text(previous) + FC(current).

a a b

b b b a

Not Found - Enter Text(previous) + FC(Previous).

a a b a

Output last code entered into code book

a a b a

No more code to de-compress - STOP

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Links

http://www.cs.sfu.ca/cs/CC/365/li/squeeze/

Squeeze Page - Applets dealing with compression Algorithms

http://www.geocities.com/yccheok/lzw/lzw.html

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Finite State Machine for KMP Pattern 1010110

011

0 1 2 3 4 5 6 7

0

0

0

0

1

1

1 0111 00