Data storage

Charles McAnany

What are the ones and zeroes?

Hard driveComputer

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1011010

Definitions

A bit is a single one or zero. A byte is eight bits. Numbers stored as ones and zeroes are stored in binary.

Binary numbers

• A list of ones and zeroes is a number. • It’s just like a number in base ten.

1576 10110Ones p

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Thousands place

Hundreds place

Tens place

Ones p

lace

Eight

s plac

eFo

urs p

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Twos

plac

e

Sixte

ens p

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Converting from binary to decimal

10110One

s plac

e

Eight

s plac

eFo

urs p

lace

Twos

plac

e

Sixte

ens p

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Place Power Value Present?1 2^0 1 No2 2^1 2 Yes3 2^2 4 Yes4 2^3 8 No5 2^4 16 Yes

Then, add up all the present values. 16 + 4 + 2 = 22

From decimal to binary• Harder!• Find the largest power of two that fits in the decimal number. • Subtract that number, and mark it as present in the binary. • Repeat until the decimal number is zero.

1577

1577

Place Power Value Present?1 2^0 12 2^1 23 2^2 44 2^3 85 2^4 166 2^5 327 2^6 648 2^7 1289 2^8 256

10 2^9 51211 2^10 1,02412 2^11 2,04813 2^12 4,09614 2^13 8,19215 2^14 16,384

Largest power of two that fits:1024 (not 2048, because 2048 > 1577.)

1577-1024= 553

Mark the 1024 spot, and continue with 553.

553

Place Power Value Present?1 2^0 12 2^1 23 2^2 44 2^3 85 2^4 166 2^5 327 2^6 648 2^7 1289 2^8 256

10 2^9 51211 2^10 1,024 yes12 2^11 2,04813 2^12 4,09614 2^13 8,19215 2^14 16,384

Largest power of two that fits:512

553- 512= 41

Mark the 512 spot, and continue with 41.

41

Place Power Value Present?1 2^0 12 2^1 23 2^2 44 2^3 85 2^4 166 2^5 327 2^6 648 2^7 1289 2^8 256

10 2^9 512 yes11 2^10 1,024 yes12 2^11 2,04813 2^12 4,09614 2^13 8,19215 2^14 16,384

Largest power of two that fits:32

41- 32= 9

Mark the 32 spot, and continue with 9.

9

Place Power Value Present?1 2^0 12 2^1 23 2^2 44 2^3 85 2^4 166 2^5 32 yes7 2^6 648 2^7 1289 2^8 256

10 2^9 512 yes11 2^10 1,024 yes12 2^11 2,04813 2^12 4,09614 2^13 8,19215 2^14 16,384

Largest power of two that fits:8

9- 8= 1

Mark the 8 spot, and continue with 1.

1

Place Power Value Present?1 2^0 12 2^1 23 2^2 44 2^3 85 2^4 166 2^5 32 yes7 2^6 648 2^7 1289 2^8 256

10 2^9 512 yes11 2^10 1,024 yes12 2^11 2,04813 2^12 4,09614 2^13 8,19215 2^14 16,384

Largest power of two that fits:1

1- 1= 0

Mark the 1 spot, the number is zero, so you’re done.

Place Power Value Present?1 2^0 1 yes2 2^1 23 2^2 44 2^3 85 2^4 166 2^5 32 yes7 2^6 648 2^7 1289 2^8 256

10 2^9 512 yes11 2^10 1,024 yes12 2^11 2,04813 2^12 4,09614 2^13 8,19215 2^14 16,384

Wherever you marked present, that’s a 1. If there’s no mark, that number’s a 0. Starting at the bottom, fill in the binary number.The number is 000011000100001.

Playing for money.

• The first person to convert the following number to binary will receive a cash prize.

• The number is: 200,000,000.

Text

• Each byte is a character. So, each character has a number. The capital letters are in the handout.

• The following string (bytes separated by commas) is:

• 01001000, 01000101, 01001100, 01001100, 01001111

• H E L L O• Please take a moment to write your name in

binary.

Images:Images are broken into pixels.

Each color is given a number.Blue = 0, Light blue = 1, red = 2, white = 3

00 01 00 01 10 10 10

00 00 00 00 11 11 11

00 01 00 01 10 10 10

11 11 11 11 11 11 11

10 10 10 10 10 10 10

Then, store the numbers, and any other info needed to make the image.The image format might specify, for instance, the first eight bits is the row length. So, our file would be

Glue the rows together. (remembering howlong a row is elsewhere.)

00 01 00 01 10 10 10 00 00 00 00 11 11 11 00 01 00 01 10 10 10 11 11 11 11 11 11 11 10 10 10 10 10 10 10

000001110001000110101000000000111111000100011010101111111111111110101010101010

Recreating an image. • The first eight bits are the row length. Use the coloring

scheme in the handout. • Here’s the file as it appears on the disk. Recreate the

image. (I’ve broken it into bytes for ease of reading.)• 00000101• 00010001• 00000010• 00001100• 00001100• 11111100

Compression

• If a particular pattern occurs often in a file, it may be possible to compress the contents. We’ll use Huffman coding to deflate a text file.

• The original text is “this is an example of a huffman tree”.

• We start by analyzing letter frequency. The most common letters should have the shortest codes.

Huffman codingChar Freq Code

space 7 111

a 4 010

e 4 000

f 3 1101

h 2 1010

i 2 1000

m 2 0111

n 2 0010

s 2 1011

t 2 0110

l 1 11001

o 1 00110

p 1 10011

r 1 11000

u 1 00111

x 1 10010

To encode, we replace each character with itsHuffman code. The word “tree” is originally 01010100 01010010 01000101 01000101Replacing it with the codes, we get:011011000000000

Image compression using Huffman coding

• Our image example used two bits for every pixel. That’s great for images with four colors. But most images are stored using 32 or even 64 bits per pixel.

• If most of the image is one color, we can give that color a code of very few bits, converting all of those 64-bit pixels into 1 bit pixels.

Other compression methods

• Huffman coding is very widely used in lossless compression. When you view the data that was encoded, you get the EXACT same data back.

• But some things (music and images in particular) may not need to be stored with perfect accuracy.

• For these, we use lossy compression.

Lossy compression

Credits

• Maru the cat http://catsnco.wordpress.com/2011/10/19/i-am-maru/

• Huffman tree http://en.wikipedia.org/wiki/Huffman_coding

• Lossy compression NMR http://nmr.cemhti.cnrs-orleans.fr/Dmfit/Howto/Top/Default.aspx