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CIS 115

Compression & Error Checking

What is Compression?
Why is it useful?


Compression

Run Length Encoding

RLE Example

WWWWWWWWWWWWBWWWW
WWWWWWWWBBBWWWWWW
WWWWWWWWWWWWWWWWW
WBWWWWWWWWWWWWWWW

Image Source: Wikipedia

RLE Example

WWWWWWWWWWWWBWWWW
WWWWWWWWBBBWWWWWW
WWWWWWWWWWWWWWWWW
WBWWWWWWWWWWWWWWW
12W1B12W3B24W1B15W

Image Source: Wikipedia

Wait, isn't text just numbers (ASCII)?

How can we tell which numbers are text and which are not?


RLE Example

WWWWWWWWWWWWBWWWW
WWWWWWWWBBBWWWWWW
WWWWWWWWWWWWWWWWW
WBWWWWWWWWWWWWWWW

Image Source: Wikipedia

Escape Coding

WWWWWWWWWWWWBWWWW
WWWWWWWWBBBWWWWWW
WWWWWWWWWWWWWWWWW
WBWWWWWWWWWWWWWWW
WW12BWW12BB3WW24BWW15

Image Source: Wikipedia

Huffman Coding

Huffman Example

THIS IS AN EXAMPLE OF
A HUFFMAN TREE 288 bits (8 * 36)

Image Source: Wikipedia

Huffman Example

THIS IS AN EXAMPLE OF
A HUFFMAN TREE

Image Source: Wikipedia

Huffman Example

Image Source: Wikipedia

Huffman Example

THIS IS AN EXAMPLE OF
A HUFFMAN TREE 0110 1010 1000 1011 111 1000 1011 111 010 0010 111 000 10010 ... (135 bits)

Image Source: Wikipedia

What is Error Checking?
Why is it useful?


What is Error Checking?
Why is it useful?


Intuition - Squeezing

Hello there
Hello there

Hello ghere
Hello ghere

Simple - Addition

Hello there

72 101 108 108 111 32
116 104 101 114 101 33
= 1101
      

Simple - Addition

Better - Pinpoint

4837543622563997
4 8 3 7 ?
5 4 3 6 ?
2 2 5 6 ?
3 9 9 7 ?
? ? ? ?

Better - Pinpoint

4 8 3 7 2
5 4 3 6 8
2 2 5 6 5
3 9 9 7 8
4 3 0 6
483725436822565399784306

Better - Pinpoint

483725436827565399784306
4 8 3 7 2 2
5 4 3 6 8 8
2 7 5 6 5 0
3 9 9 7 8 8
4 3 0 6
4 8 0 6

Better - Fletcher's Checksum

INPUT:  a data word (e.g., a sequence of ASCII-numbers)
OUTPUT: two checksums for the word, each sized to fit
        in one byte

ALGORITHM:
1. divide the  Word  into a sequence of equally-sized
   blocks, b1 b2 ... bn
2. define two checksums, starting at  C1 = 0  and  C2 = 0
3. for each block, bi,
        add  bi to  C1
        add the new value of  C1  to  C2
4.  compute  Checksum1 = C1 mod 255 and
    Checksum2 = C2 mod 255
5.  return  Checksum1  and  Checksum2

Better - Fletcher's Checksum

72 101 108 108 111 32 116 104 101 114 101 33
Block C1 C2
72 72 72
101 173 245
108 281 526
108 389 915
Total: 1101 (81) 7336 (96)

Testing Fletcher's

Can we do better?

Cyclic Redundancy Check (CRC)

Input: 010100001001
Check: 1011

    010100001001
XOR  1011
    ------------
        10001001
    XOR 1011
       ---------
        00111001
      XOR 1011
         -------
           10101
       XOR 1011
          ------
  Checksum: 011

Hash Codes

Choose a "hash base", b (e.g., b= 2 or b= 10 or b= 37)

For a word of integers of length n+1:
    w =  x0 x1 x2 ... xn-1 xn,

Compute this hash number:
    hash(w) =  (x0 * bn) + (x1 * bn-1) + (x2 * bn-2) + ...
               (xn-1 * b1) + xn
             

Hash Codes

For  word = 4 5 6,
when  b = 10,
     hash(word) = (4 * 102) + (5 * 101) + 6  =
     400 + 50 + 6  =  456
when b = 100,
     hash(word) = (4 * 1002) + (5 * 1001) + 6  =
     40000 + 500 + 6  =  40506
when b = 5,
     hash(word) = (4 * 52) + (5 * 51) + 6  =
     100 + 20 + 6  =  126
when b = 2,
     hash(word) = 16 + 10 + 6 =  32
when b = 1,
     hash(word) = 4 + 5 + 6 = 15
             

Hash Codes

For word = 12 33 08,
when b = 10,
     hash(word) = (12 * 102) + (33 * 101) + 8  =
     1200 + 330 + 8  =  1538
when b = 100,
     hash(word) = (12 * 1002) + (33 * 1001) + 8  =
     120000 + 3300 + 8  =  123308
when b = 5,
     hash(word) = (12 * 52) + (33 * 51) + 8  =
     300 + 165 + 8  =  473
when b = 2,
     hash(word) = 48 + 66 + 8 =  122
       

Hamming Codes

BIT#:    1       2        3  4       5  6  7
        -------------------------------------------
PURPOSE: P3,5,7  P3,6,7   D  P5,6,7  D  D  D

where  D  is a data bit, and
Pa,b,c,...  is the parity bit for data bits at  a,b,c,...

Examples:
DATA 3 5 6 7  P3,5,7  P3,6,7  P5,6,7   HAMMING CODE
------------------------------------------------------
     0 1 0 0     1       0       1     1 0 0 1 1 0 0

     1 0 1 1     0       1       0     0 1 1 0 0 1 1
      

Assignments

Blog 7: 9 Algorithms - Making Meaning

Now that we’ve finished reading the last textbook, it is time to step back and think about what we read. Write about your reactions to it and what you learned from it. I’d recommend almost treating this like an in-depth book review for others who are interested in reading the book, but don’t mind some spoilers. Some questions I’d like you to answer: