Algorithm Programming 1 89-210 Some Topics in Compression Bar-Ilan University 2006-2007 תשס"ז by Moshe Fresko

Huffman Coding Variable-length encoding Works on probabilities of symbols (characters, words, etc.) Build a tree Get two least frequent symbols/nodes Join them into a parent node Parent node’s frequency is sum of child nodes’ Continue until the tree contains all nodes and symbols The path of a leaf indicates its code Frequent symbols are near the root giving them short codes

LZ77 Introduced in 1977 by Abraham Lempel and Jacob Ziv Dictionary based Works in a window size n Decoding is easy and fast (but not Encoding) Produces a list of tuples (Pos,Len,C) Pos : Position backwards from the current position Len : Number of symbols to be taken C : Next character

LZ77 Based on strings that repeat themselves An outcry in Spain is an outcry in vain An outcry in Spa(6,3)is a(22,12)v(21,3) aaaaaaaaaa a(1,9)

LZ77 - Example Window size : 5 ABBABCABBBBC NextSeq Code A (0,0,A) BA (1,1,A) BC (3,1,C) ABB (3,2,B) BBC (2,2,C)

LZ77 - Some Variations LZSS - A flag bit for distinguishing pointers from the other items. LZR - No limit on the pointer size. LZH - Compress the pointers in Huffman coding.

LZ78 Instead of a window to previously seen text, a dictionary of phrases will be build Both encoding and decoding are simple From the current position in the text, find the longest phrase that is found in the dictionary Output the pair (Index,NextChar) Index : The dictionary phrase of that index NextChar : The next character after that phrase Add to the dictionary the new phrase by appending the next character

LZ78 - Example ABBABCABBBBC Dictionary size Input Output Add to dictionary A (0,A) 1 = “A” B (0,B) 2 = “B” BA (2,A) 3 = “BA” BC (2,C) 4 = “BC” AB (1,B) 5 = “AB” BB (2,B) 6 = “BB” BC (4,EOLN) Dictionary size

LZW Produces only a list of dictionary entry indexes Encoding Starts with initial dictionary For example, possible ascii characters (0..255) From the input, find the longest string that exists in the dictionary Output this string’s index in the dictionary Append the next character in the input to that string and add it into the dictionary Continue from that character on from (2)

LZW - Example ABBABCABBBBC Dictionary size : ? Initial dictionary 0=“A”, 1=“B”, 2=“C” Input NextChar Output Add to dictionary A B 0 3 = “AB” B B 1 4 = “BB” B A 1 5 = “BA” AB C 3 6 = “ABC” C A 2 7 = “CA” AB B 3 8 = “ABB” BB B 4 9 = “BBB” B C 1 10 = “BC” C - 2 - Dictionary size : ?

LZW – Encoding Example T=ababcbababaaaaaaa Initial Dictionary Entries :1=a 2=b 3=c Input Output NextSymbol Add To Dictionary a 1 b 4 = ab b 2 a 5 = ba ab 4 c 6 = abc c 3 b 7 = cb ba 5 b 8 = bab bab 8 a 9 = baba a 1 a 10 = aa aa 10 a 11 = aaa aaa 11 a 12= aaaa a 1 - -

LZW – Encoding Algorithm w = Empty while ( read next symbol k ) { if wk exists in the dictionary w = wk else add wk to the dictionary; output the code for w; w = k; }

LZW – Decoding Algorithm read a code k output dictionary entry for k w = k while ( read a code k ) { entry = dictionary entry for k output entry add w + entry[0] to dictionary w = entry }

LZW – Decoding There is a special case problem with the previous algorithm It can be confronted on every decoding process of a big file It is the case where the index number read is not in the dictionary yet Example : ABABABA Initially : A=1,B=2 Output=1 2 3 5 In decoding above algorithm will not find the dictionary entry ABA=5 An additional small check will solve the problem Be careful to do it in the Exercise 3

LZW – Dictionary Length Typically : 14 bits = 16384 entries (first 256 of them are single bytes) What if we are out of dictionary length Don’t add to the dictionary any more Delete the whole dictionary (This will be used in the exercise) LRU : Throw those that are not used recently Monitor performance, and flush dictionary when the performance is poor. Double the dictionary size

Exercise 3 – Compression Algorithms Define an interface “Compression” in “Compression.java” as: interface Compression { void compress(InputStream is, OutputStream os) throws IOException ; void decompress(InputStream is, OutputStream os) throws IOException ; void stop() ; } // Note that these definitions are as general as possible to let flexibility in the input and output media chosen. Define the LZW compression algorithm in file LZW.java This class must implement the above Compress interface.

Exercise 3 – Compression Algorithms Write an application program “main()” in “LZW.java” that will run in the following form java LZW compress/decompress InputFile OutputFile It will run the given Algorithm for compressing/decompressing the given input file into the given output file. Example: java LZW compress MyTextFile.txt MySmallFile.lzw will compress MyTextFile.txt into MySmallFile.lzw java LZW decompress MySmallFile.lzw MyNewTextFile.txt will uncompress the given file into a new file

Exercise 3 – Compression Algorithms You can use the following two classes to keep a dynamic list and an associative array ArrayList (with interface List) HashMap (with interface Map) Example: // For keeping a dynamic list List myList = new ArrayList() ; // For keeping an associative array Map myMap = new HashMap() ; List boolean add(Object) int size() Object get(int) Map Object put(Object key,Object value) boolean containsKey(Object key) Object get(Object key) For more information, look at the Java API specification page.

Exercise 3 – Compression Algorithms Algorithm Parameters: LZW: The maximal initial Dictionary size is 512 entries which requires 9 bits for writing. First 256 entries are the ascii characters. After reaching the end of the dictionary it will double the dictionary size. Which means from that point on the representation of an entry will be 10 bits and will be a dictionary of max size 1024. This will continue until 14 bits. After reaching 214 entries, it will clean the dictionary by starting from the beginning with 9-bit represenations. Please take the two numbers 9 and 14 (start Bit Count, last Bit Count) as a parameter in the constructor. The default constructor will start them with 9 and 16. LZW() { this(9,14); } LZW(int startNumOfBits, int endNumOfBits) { /*Initialization*/ } You have to use BitInputStream/BitOutputStream from the 1st Exercise to write the bits into an output or to read from input.

Exercise # 2 Compression Algorithms Important: You submit via submitex. Course number: 89-210 Exercise: ex2 Two files will be delivered. Compression.java and LZW.java It must compile/work under Java 1.4/1.5. You can try it on the Unix environment (on Sunshine). Do not write debugging information to the console or any other place. Write enough comments. Write wherever you think there is a need for the understanding of the code. Write your code according to the OOP principles. Everybody must do it alone. Deadline: 14 Dec 2006