[Algorithms II] Week 5-2 Data Compression

[TOC]

1. Introduction to Data Compression

pb: reduce the size of a file, to save space/time for storing/transmitting.
applications: generic file compression(gzip), multimedia (mp3), communication(skype).

From binary data B, ⇒ generate a compressed representation C(B).

lossless compression: get exactly B from C(B)
compression ratio: |C(B)|/|B|(||means number of bits)


Tools

extension to stdio libraries: read/write bits

public class BinaryStdIn{  
    boolean readBoolean(); // read 1 bit and return as a boolean  
    char readChar(); // read 8 bits and return a char  
    char readChar(int r); // read r (<=8) bits and return a char  
    // similar method for int/long/double  
    boolean isEmpty(); // is bitstream empty?  
    void close();  
}


public class BinaryStdOut{  
    void write(boolean b); // write 1 bit  
    void write(char c); // write 8 bits  
    void write(char c, int r); // write r (least-significant) bits of c  
    // similar method for int/long/double  
    void close(); // might add some byte alignment bits   
}

usage example: store a date 12/31/1999

universal date compression?
prop. NO algorithm can compress every bitstring.
pf. by contradiction: repeatedly compress the bitstring ⇒ bit length goes to 0.

2. Run-Length Coding

one simple type of redundancy in bitstream: long runs of repeated bits.
⇒ use 4-bit counts to represent alternating 1s and 0s.


  • question1: how many bits to represent counts ? ⇒ pick 8 bits (just tradeoff)
  • question2: what if run length is bigger than max length(2^8)? ⇒ just add length=0 of the alternating bit...

application: JPEG...

code:

public class RunLength{  
    private final static int R = 256; // max run length  
    private final static int lgR = 8;  
    public static void compress(){  
        boolean bit, oldbit = false;  
        int l=0;  
        while(!BinaryStdIn.isEmpty()){  
            bit = BinaryStdIn.readBoolean();  
            if(bit!=old){  
                oldbit = bit;  
                BinaryStdOut.write(l,lgR);  
                l = 0;    
            }  
            else if(l==R-1){  
                BinaryStdOut.write(l,lgR);  
                BinaryStdOut.write(0,lgR);  
                l = 0;  
            }  
            l ++;  
        }  
        BinaryStdOut.write(l,lgR);  
        BinaryStdOut.close();

    }  
    public static void expand(){  
        boolean bit = false;  
        while(!BinaryStdIn.isEmpty()){  
            int l = BinaryStdIn.readInt(lgR);  
            for(int i=0;i<l;i++)  
                BinaryStdOut.write(bit);  
            bit != bit;  
        }  
        BinaryStdOut.close();  
    }  
}

3. Huffman Compression

classique...

idea: variable length code.
ex. Morse code, more freq chars use less chars.
→ pb: ambiguity? one code is the prefix of another...
→ need to use prefix-free code


⇒ use a trie to represent the prefix-free code, in bitstream, use a binary trie:

Huffman Trie Node

define trie nodes having frequences:

public static class Node implements Comparable<Node>{  
    private char ch; // only used for leaves, null for internal nodes  
    private int freq;  
    private final Node left, right; // left-->0, right-->1  
    public Node(char ch, int freq, Node left, Node right){//...}  
    public boolean isLeaf(){  
        return left==null && right==null;}  
    public int compareTo(Node that){  
        return this.freq-that.freq;}  
}

implementation

expansion
in the bitstream, we first put the (serialized) trie in the head.

public void expand{  
    Node root = getTrie(); //   
    int N = BinaryStdIn.readInt(); // number of chars in string  
    for(int i=0; i<N; i++){  
        Node x = root;  
        while (!x.isLeaf()){  
            if(BinaryStdIn.readBoolean()) x = x.right;  
            else x = x.left;  
        }  
        BinaryStdOut.write(x.ch);  
    }  
    BinaryStdOut.close();  
}

output trie
ie. to serialize a trie.
⇒ use preorder traversal:
0 for internal nodes, and 1 for leaf (followed by the corresponding char)

recursive method:




private static void writeTrie(Node x){  
    if(x==null) return;  
    if(x.isLeaf()){  
        BinaryStdOut.write(true); // leaf node  
        BinaryStdOut.write(x.ch); // followed by the char  
    } else {  
        BinaryStdOut.write(false); // internal node  
        writeTrie(x.left);  
        writeTrie(x.right);  
    }  
}

read trie
Reconstruct the trie from the serialized bitstring.
preorder(x) = x+preorder(x.left)+preorder(x.right) ⇒ recursive method...

这个递归函数还是蛮有代表性的, 值得看.

private static Node readTrie(){  
    if(BinaryStdOut.readBoolean()) {  
        char ch = BinaryStdOut.readChar();  
        return new Node(ch, 0, null, null);  
    } else {  
        Node left = readTrie();  
        Node right = readTrie();  
        return new Node('\0', 0, left, right); // subtle recursion  
    }  
}

Huffman algorithm

pb: how to find the best prefixless code?

Shannon-Fano algo:

  • partition symbols S into 2 subsets: S1, S0, with roughly equal freq
  • code in S1 start with 1 and in S0 start with 0
  • recur on S1 and S0

[Huffman algo] (1950)

  • count letter freq in input
  • build a node for each char
  • select the 2 tries with minimum weight(freq), merge them, and put it back
  • repeat until we get only 1 trie

application: jpeg, pdf, mp3, ...

implementation

private static Node buildTrie(int[] freq){  
    MinPQ<Node> pq = new MinPQ<Node>();  
    for(char i=0; i<R; i++)  
        pq.insert( new Node(i, freq[i], null, null) );  
    while(pq.size()>1){  
        Node x = pq.pop(), y = pq.pop();  
        pq.insert( new Node('\0',x.freq+y.freq, x, y);  
    }  
    return pq.top();  
}

prop. Huffman algorithm produces the optimal prefixless code.

running time: N + RlgR

4. LZW Compression

idea:
adaptive model: learn and update the model as you read the text.
decoder will do the same thing.

build a table of not only mapping chars to codes, but also mapping >=2-char combinations to codes(of fixed width).
when encoding strings, look for the longest prefix that is in our table.


LZW compression

LZW compression algo

  • create a symbol table mapping string keys to W-bit code, initialize as mapping only chars to codes
  • find longest prefix s of the unscanned input
  • write out the corresponding code
  • add s+c into the symbol table, where c is the next char in input

Use a trie for representing the code table → because support longest prefix match.

implementation

public void compress(String input){  
    TST<Integer> st = new TST<Integer>(); // a trie symbol table  
    for(char i=0; i<R; i++)  
        st.put(""+(char)i, i);  
    int code = R+1; // R is used as "stop"  
    while(input.length()>0){  
        String s = st.longestPrefixOf(input);  
        BinaryStdOut.write(st.get(s), W);  
        int l = s.length();  
        if(l<input.length())  
            st.put(s+input.charAt(l), code++);  
        input = input.substring(l);  
    }  
    BinaryStdOut.write(R, W); // write "stop"  
    BinaryStdOut.close();

}

LZW expansion

no need to get the codeword table, the input is just the compressed bitstring.

  • maintain a (reverse)table mapping int(code) to strings.
  • as we decode the string, we add new entries to our table.

to represent the table: just an array.

note when add new entries: add key = s+c, where s is the last decoded string, c is the first char of the currently decoded string.

example: when decoded ABR, s = BR, currently decoded string = ABR, so we add BRA.

tricky case: ABABABA
compression

problem in expansion when reading the "83":




→ need to be able to update the table when encountering a code not yet defined...

Summary

  • Huffman: fixed-length symbols, with variable-length codes.
  • LZW: variable-length symbols, with fixed-length codes.

theoretical limit:

还剩下一周的内容, 可能要等到二月底考完以后有时间再看了, 现在先复习备考...

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