Download +1 - TU Delft

Arithmetic and Data Representation
http://www.pds.ewi.tudelft.nl/~iosup/Courses/2011_ti1400_3.ppt
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Lecture 1
Making functions
nand gates
A
Y
ADD
B
A,B
Y

time
2
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Lecture 1
Making functions
Circuit Diagram
http://xkcd.com/730/
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Lecture 2
Programmable device
2,1
input stream
Programmable
Device
READ(X)
READ(Y)
ADD(X,Y,Z)
WRITE(Z)
3
output stream
program
• READ(X) means read next input value from input stream and
store it internally as variable X
• WRITE(X) means put value in variable X on output stream
• ADD(X,Y,Z) means assign value of X+Y to Z
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Lecture 2
Von Neumann Architecture
Memory
X: 1
Y: 2
Z: 3
•
•
READ(X)
READ(Y)
ADD(X,Y,Z)
WRITE(Z)
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Central Processing Unit
TEMP_A:
TEMP_B:
RESULT:
IR:
arithmetic
unit
Input
CONTROL
Output
PC:
TU-Delft
Problem: How to Represent and Use Data?
1. Representation
2. Arithmetic
3. Conversion
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The Y2K Problem/The Millenium Bug (2000)
Representation (Gone Wrong)
• Abbreviating the year as two-digits instead of four
-
99 instead of 1999
00 instead of 2000
Cost saving (space is money)
How to represent 2096?
98 < 99, but 99 < 00?!
• Problem identified by Spencer Bolles in 1985
- Y2K acronym used by David Eddy in 1995
• $150B in US (IDC estimate)
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The FDIV Problem (1993)
Arithmetic (Gone Wrong)
• 4195835.0/3145727.0 = 1.333 820 449 136 241 002 5
4195835.0/3145727.0 = 1.333 739 068 902 037 589 4
• Pentium used the radix-4 SRT algorithm (which is fast)
• “The [problem] can be traced to five missing entries in a
lookup table”
• Solution: recall defective processors ($350M) or
scale operands and use 80-bit extended-precision internal
Pentium format (formats are discussed in this lecture)
• … oh, and hardware errors still happen (latest, Intel’s Cougar/
Sandy Chipsets in February 1, 2011—est. cost $700M)
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Ariane 5 (1996)
Conversion (Gone Wrong)
June 4, 1996: Ariane 501 Satellite Launch
explodes 40s into the flight sequence
Cost: $370M
“The internal SRI* software exception was
caused during execution of a data
conversion from 64-bit floating point to 16bit signed integer value. The floating point
number which was converted had a value
greater than what could be represented by a
16-bit signed integer.” Inquiry Board Report
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Data Type Representation
1.
2.
3.
4.
5.
6.
Integer numbers
Real numbers
Booleans
Characters
Composite types (e.g. arrays)
Objects
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Number systems
0
1
10110
66
277
-15
0100
1110
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Question
• Which decimal number is associated with the
bitstring <110> ?
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Definition and Representation
Integers (Z) = Set of natural numbers including 0
and their non-zero negatives
Integer numbers are generally represented by:
• An alfabet 
• A string X of n elements from : Xn  n
• Applying Xn to a valuation function F
- signature: F : n  Z
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Examples
 R = {I, V, M, ....}
•
-
VI = 6
IV = 4

•
-
•
10
= {0, 1, 2, 3, .....,9}
27710 = 2*102 + 7*101 + 7*100
radix
Many sets  and valuation functions F possible
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Question
What value has 267 ?
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Binary representation
• In computer, we have 2 ={0, 1}
• There are, however, many valuation functions F
1110
F1
66
277
-15
F2
F3
0011
1010
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Criteria for representations
•
•
•
•
Let Xn = < xn-1, xn-2 , ....., x0 >
Range of natural numbers of X
Representation of 010
X
Efficiency of implementation
- Sign inversion : A 
-A

- Addition: A+B
- Extension: n bit  m bit representation, m>n
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Data Type Representation
1.
2.
3.
4.
5.
6.
Integer numbers 1.1. Sign-Magnitude
1.2. Two’s Complement (2C)
Real numbers
1.3. One’s Complement (1C)
Booleans
1.4. Excess-X
Characters
Composite types (e.g. arrays)
Objects
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1. Integer Representation
1.1. Sign Magnitude
• F: sign(xn-1){xn-22n-2 + ....+ x020}
sign(1) = - and sign(0)= +
• Range: [-(2n-1 -1), 2n-1 -1]
• Zero: 100... and 000...

• Inversion: change only
sign bit
X
sign bit
1 1 0
V = - 21 = -2
0 1 0
V = +21 = 2
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1. Integer Representation
1.2. Two’s complement (2C)
• F: -xn-12n-1 + xn-22n-2 + ....+ x020
• Range: [-2n-1, 2n-1-1]
• Zero: 000... (only a single representation)
X
sign
1 1 0
V = - 22 +21 = -2
0 1 0
V = +21 = 2
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1. Integer Representation 1.2. 2C
2C negative
100 101 110 111 000 001 010 011
-4 -3
-2 -1
0
1
2
3
Inversion rule:
- invert all bits
- add 1
- General: A + Å = 2n
110
001
1
+
010
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1. Integer Representation
1.3. One’s Complement (1C)
• F: -xn-1(2n-1-1) + xn-22n-2 + ....+ x020
• Range: [-(2n-1-1), 2n-1-1]
• Zero: 111... and 000...
X
sign
1 1 0
V = - 22 +1+21 = -1
0 1 0
V = +21 = 2
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1. Integer Representation 1.3. 1C
1C negative
100 101 110 111 000 001 010 011
-3 -2
-1 0
0
1
2
3
110
Inversion rule:
- invert all bits
- General: A + Å = 2n-1
001
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1. Integer Representation
1.4. Excess-x
• F: xn-12n-1 + xn-22n-2 + ....+ x020 - 2n-1
• Range: [-2n-1, 2n-1-1]
• Zero: 10000.....
X
1 1 0
V = 22 + 2 1 - 2 2 = 2
0 1 0
V = +21 - 22= -2
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Data Type Representation
1.
2.
3.
4.
5.
6.
Integer numbers 1.5. Addition
1.6. Sign Extension for 2C
Real numbers
1.7. Other Representations
Booleans
1.8. Conversion
Characters
Composite types (e.g. arrays)
Objects
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1. Integer Representation
1.5. Addition
• SM
- Sign and Magnitude are processed separately
- Opposite sign operands: subtraction
• 2C
- All bits can be added, regardless of position
• 1C
- All bits can be added, regardless of position
- When carry at position 2n , add 1 to result
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1. Integer Representation 1.5. Addition
Example
X
000
001
010
011
100
101
110
111
2C
0
1
2
3
-4
-3
-2
-1
1C
0
1
2
3
-3
-2
-1
0
SM Ex-4
0
-4
1
-3
2
-2
3
-1
0
0
-1
1
-2
2
-3
3
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1. Integer Representation
1.6. Sign Extension (for 2C)
• Represent n-bit in k-bit number
• If k>n, extend sign bit to the left
• If k<n,
- n-k+1 bits equal, remove lefmost n-k bits
- if not, overflow
• Example: n=6, k=4
4 = 000100 -> 0100
9 = 001001 -> overflow
-6 = 111010 -> 1010
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1. Integer Representation 1.7. Other Representations
Binary Coded Decimal
decimal X
0
0000
1
0001
2
0010
3
0011
4
0100
5
0101
6
0110
7
0111
8
1000
9
1001
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1. Integer Representation 1.7. Other Representations
Radix 2, 8, 10, 16
binair octal dec. hex
0
0
0
0
1
1
1
1
10
2
2
2
11
3
3
3
100
4
4
4
101
5
5
5
110
6
6
6
111
7
7
7
1000
10
8
8
1001
11
9
9
1010
12 10 A
1011
13 11
B
1100
14 12 C
1101
15 13 D
1110
16 14
E
1111
17 15
F
L33T-Speak
h4x0r
hacker
1007
money (loot)
1337
elite (leet)
http://www.jayssite.com/stuff/l33t/l33t_translator.html
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1. Integer Representation
1.8. Conversion [1/2]
• Simple, if one radix is power of other radix
- binary to octal
10100101001 = 010 100 101 001 = 2451oct
- binary to hexadecimal
10100101001 = 0101 1010 1001 = 5A9hex = 0x05A9
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1. Integer Representation
1.8. Conversion [2/2]
• From binary to decimal
10110101001 = 210 + 28 + 27 + 25+ 23+ 20 = 1449
• From decimal to binary (first method)
1449 = 210 +425 = 210 +28 +169 =..= 10110101001
• From decimal to binary (second method)
1449 = 2*724 +1 LSB
724 = 2*362 +0
...... =......
1 = 0*1 +1
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Data Type Representation
1.
2.
3.
4.
5.
6.
Integer numbers
1.9. Binary Addition
Real numbers
1.10. Operation Overflow
Booleans
Characters
Composite types (e.g. arrays)
Objects
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1. Integer Representation 1.9. Binary Addition
Addition of binary digits
• Addition of two bits
0
0 +
0
1
0
+
1
0
1
+
1
1
x
1
y
+
+
10
cs
• Sum s and Carry c
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1. Integer Representation 1.9. Binary Addition
More positions
xi
0
0
1
yi
0
1
1
ci
0
1
1
si
0
0
1
ci+1
0
1
1
.
.
ci+1
.
xi
yi
ci
si
.
.
.
.
truth table (partial)
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1. Integer Representation 1.9. Binary Addition
Basic Adder
xi
ci+1
yi
Full Adder
ci
si
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1. Integer Representation 1.9. Binary Addition
Ripple Carry Adder
FA
FA
FA
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1. Integer Representation 1.10. Overflow
Addition/subtraction
• 2C
- Simple, all bits can be treated the same
- Carry last position can be ignored
- Subtraction by first complementing the subtrahend
(minuend – subtrahend = difference)
• Example:
0111
1101
------10100
(7)
(-3)
+
(4)
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1. Integer Representation 1.10. Overflow
Addition/subtraction
• 1C
- all bits can be treated the same
- when carry in last position add +1 to
Least Significant Bit (LSB)
0111
1100
------10011
1
-------0100
(7)
(-3)
+
+
(4)
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1. Integer Representation 1.10. Overflow
Overflow detection
• Suppose we have 2C notation
- Carry-out MSB is not sufficient to indicate overflow
0111 (7)
0100 (4)
-------- +
01011 (-5?)
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1. Integer Representation 1.10. Overflow
Overflow
• Overflow occurs in 2C iff
- Both operand have the same sign
- The sum bit in the MSB position has the opposite sign
OVFL  xn1yn1sn1  xn1yn1sn1
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Data Type Representation
1.
2.
3.
4.
5.
6.
Integer numbers
2.1. Fixed Point
Real numbers
2.2. Floating Point
Booleans
2.3. IEEE 754
Characters
Composite types (e.g. arrays)
Objects
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2.1. Fixed Point (Fractions)
• Real numbers (R) = value in a continuous space
(vs. Integers—discrete space)
• F: -xn-120 + xn-22-1 + ....+ x 02-n+1
• Range: [-1, 1-2-n+1]
• But: very small numbers cannot be represented, …
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Question
Can conversion between decimal and
binary numbers always be done exacly in
a. integers ?
b. fractions ?
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Answer
a) Can conversion between decimal and binary
numbers always be done exacly in integers?
-> Yes
b) Can conversion between decimal and binary
numbers always be done exacly in fractions?
-> No
- Counter-example: 0.210 =0.00110011001100.....2
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2.2. Floating Point (1)
• Need a represention for:
- very large numbers
- real numbers
• With a limited number of bits
• Floating point notation:
G = m.re
r = radix
m = mantissa, significant, or fraction
e = exponent
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2.2. Floating Point (2)
• r is normally 10 or 2
• mantissa m is normalized: (1/r) ≤ m < 1
- e.g., if r = 2 then 0.5 ≤ m < 1
- e.g., if r = 10 then
6.543 x 1010 is normalized, 654,300 x 105 is not
• Normalization is used to save as much as possible
significant digits
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2.2. Floating Point
Fixed or Floating Point?
• Metrics
- Range = ratio largest:smallest number
- Precision = how many significant bits for representation
• Floating point idea: separate the exponent from the
mantissa. Vs Fixed point:
- Wider range (through exponent)
- Lower precision (mantissa has fewer bits)
• There’s no free lunch in computer science!
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2.3. FP Representation/IEEE 754
Birth of a Standard (IEEE 754)
• Need
- Many competing hardware makers
IBM, Cray, CDC, Intel, …
- Various range and accuracy
• Work Group initiated by Intel 1976, led by W. Kahan
- Correct FP
- Developers can write portable code AND prove correctness
• Result: IEEE 754
Read: IEEE 754: An Interview with William Kahan, IEEE Computer 1998
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2.3. FP Representation/IEEE 754
IEEE Floating point standard (IEEE 754)
value = ±1s x 2e’-127 x m
s
e’
8 bit
m
32 bit
23 bit
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2.3. FP Representation/IEEE 754
Mantissa
value = ±1s x 2e’-127 x m’
m’=1.m, m=[01][01]…
• Implicit 1 bit just left of the point, so 24-bit total
• Mantissa m is normalized: (1/r) ≤ m < 1
- So, including implicit 1., 1 ≤ m’ < 2
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2.3. FP Representation/IEEE 754
Exponent
value = ±1s x 2e’-127 x m
• e: [1,254] in excess-127 representation->[-126,127]
• e=0 and e=255 are reserved values
• Floating point (radix point is “floating”)
- e=35 => m+1 means +235 (+34 359 738 368)
- e=0 => m+1 means +20 (+1)
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2.3. FP Representation/IEEE 754
Sign
value = ±1s x 2e’-127 x m
• Sign is 0 for positive, 1 for negative
• Floating-point numbers that in value differ only in
sign differ in IEEE 754 representation in the s bit
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Recap
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Lecture 2
Von Neumann Architecture
Memory
X: 1
Y: 2
Z: 3
•
•
READ(X)
READ(Y)
ADD(X,Y,Z)
WRITE(Z)
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Central Processing Unit
TEMP_A:
TEMP_B:
RESULT:
IR:
arithmetic
unit
Input
CONTROL
Output
PC:
TU-Delft
Definition and Representation
Integers (Z) = Set of natural numbers including 0
and their non-zero negatives
Integer numbers are generally represented by:
• An alfabet 
• A string X of n elements from : Xn  n
• Applying Xn to a valuation function F
- signature: F : n  Z
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Criteria for representations
•
•
•
•
Let Xn = < xn-1, xn-2 , ....., x0 >
Range of natural numbers of X
Representation of 010
X
Efficiency of implementation
- Sign inversion : A 
-A

- Addition: A+B
- Extension: n bit  m bit representation, m>n
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Data Type Representation
1.
2.
3.
4.
5.
6.
Integer numbers 1.1. Sign-Magnitude
1.2. Two’s Complement (2C)
Real numbers
1.3. One’s Complement (1C)
Booleans
1.4. Excess-X
Characters
Composite types (e.g. arrays)
Objects
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1. Integer Representation 1.9. Binary Addition
Basic Adder
xi
ci+1
yi
Full Adder
ci
si
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2.1. Fixed Point (Fractions)
• Real numbers (R) = value in a continuous space
(vs. Integers—discrete space)
• F: -xn-120 + xn-22-1 + ....+ x 02-n+1
• Range: [-1, 1-2-n+1]
• But: very small numbers cannot be represented, …
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2.3. FP Representation/IEEE 754
IEEE Floating point standard (IEEE 754)
value = ±1s x 2e’-127 x m
s
e’
8 bit
m
32 bit
23 bit
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Recap over
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Memory Jog 1: Select the correct answer
1 0011 0010 1101 1011 1101 11012
is …
1. 321BE0010
2. 123EBDD16
3. 231DB9916
4. 132DBDD16
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Memory Jog 2: Select the correct answer
10510 is to 101012 as 07FD16 is to …
1. 40910
2. 204510
3. 90410
4. 204511
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2.3. FP Representation/IEEE 754
An Example: Number to IEEE 754
•
•
•
•
Input: 8.75
Step 1 (binary fixed-point): +1000.11 (.75=1x2-1+1x2-2)
Step 2 (normalize): 1.0001100… x 23
Step 3 (bias the exponent): 3 + 127 = 130 (10000010)
Source: C. Hecker, Let’s Go to the (Floating) Point, Game Developer, 1996
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2.3. FP Representation/IEEE 754
An Example: IEEE 754 to Number
In: 110000000 01010000 00000000 0000000
• Step 1: Split into IEEE 754 components
1 10000000 01010000 00000000 0000000
• Step 2 (Sign): 0 is positive, 1 is negative. Sign is …?
• Step 3 (exponent): 128 (10000000)-127 (bias)=1
• Step 4 (mantissa): 1.01010000… = 1.3125
(1x2-1=.5 1x2-2 =.25 1x2-3 =.125 1x2-4 =.0625)
• Step 5 (result): (-1)1 x 1.3125 x 21 = -2.625
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Excercise 1: Select the correct answer
6.510 in IEEE 754 Single Precision is …
1. 0 10000100
10101000000000000000000
2. 0 10000001
10100000000000000000000
3. 1 01000010
00101010100000000000000
4. 0 01000010
00101010100000000000002
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Excercise 2: Select the correct answer
0
1000 0100
0101 0000 0000 0000 0000 000
is the IEEE 754 Single Precision for …
1. 3210
2. 4216
3. 4210
4. 3216
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Hint: http://www.h-schmidt.net/FloatApplet/IEEE754.html
TU-Delft
2.3. FP Representation/IEEE 754
FP exceptions
[see wiki ieee 754]
= ±1s x 2e’-126 x 0.fraction
= ±1s x 2e’-127 x 1.fraction
0 00000000
1 00000000
0 11111111
1 11111111
0 11111111
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00000000000000000000000
00000000000000000000000
00000000000000000000000
00000000000000000000000
00000000000001000001000
= +0
=-0
= +Infinity
= - Infinity
= NaN
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2.3. FP Representation/IEEE 754
Smallest Denormalized Positive Number
In: 000000000 00000000 00000000 0000001
• Step 1: Split into IEEE 754 components
0 00000000 00000000 00000000 0000001
•
•
•
•
Step
Step
Step
Step
2
3
4
5
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(Sign): 0 is positive, 1 is negative. Sign is +
(exponent): 0 (00000000)-126 (bias)=-126
(mantissa): 0.00…1 = 2-23(23 bits)
(result): (-1)0 x 2-23 x 2-126 = 2-149 ~ 10-45
TU-Delft
2.3. FP Representation/IEEE 754
Smallest Normalized Positive Number
In: 000000001 00000000 00000000 0000000
• Step 1: Split into IEEE 754 components
0 00000001 00000000 00000000 0000000
•
•
•
•
Step
Step
Step
Step
2 (Sign): 0 is positive, 1 is negative. Sign is +
3 (exponent): 1 (00000001)-127 (bias)=-126
4 (mantissa): 1.00…0 = 1
5 (result): (-1)0 x 1 x 2-126 = 2-126 ~ 10-38
(smallest denormalized was 2-149 ~ 10-45)
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2.3. FP Representation/IEEE 754
FP Operations
• Multiplication f1 x f2:
m1  r  m2  r  m1  m2  r
e1
e2
e1 e 2
• Addition f1 + f2 (e2>e1, shift f1 by e1-e2):
m1  r  m2  r  m1  r  r  m2  r
 (m1  r shift  m2 )  r e 2
e1
e2
shift
e2
e2
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2.3. FP Representation/IEEE 754
FP accuracy
• Rounding: danger of false accuracy
• e.g., 1.5*106 + 0.1 *10-3
1.500000
0.000000 0001
------------------- +
1.500000
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2.3. FP Representation/IEEE 754
FP rounding
• Rounding = convert to lower precision (w/ error)
• Operations use guard bits -> used for rounding
0.101
0.110
--------- *
0.011 110
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2.3. FP Representation/IEEE 754
FP-Rounding [1/2]
• Chopping: cut at guards bits (Round to zero)
- error: 0 to almost -1 in LSB + Bias towards zero
Truncation error
• 0.011110  0.011
grows with the #ops
• Von Neumann Rounding/Round to Nearest
- all guard bits 0 cut; else LSB:=1
- error (-1,1) in LSB
• 0.01000  0.010
error: 0
• 0.01110
 0.011
error: 1/16
• 0.0101110  0.011
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2.3. FP Representation/IEEE 754
FP-Rounding [2/2]
• Round to nearest EVEN number (default IE3754)
- Add 1 to LSB if truncated MSB is 1
Decimal:
- error [-1/2,1/2] in LSB
• 10.5  10
• 0.001000  0.001
• 11.5  12
• 0.001011  0.001
• -10.5  -10
• 0.000111  0.001
• -11.5  -12
• 0.001111  0.010
• 0.001100  0.010 (to nearest EVEN number)
• Round downward/upward/to nearest ODD number
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2.2. Floating Point
Fixed or Floating Point? (Revisited)
staff.ee.sun.ac.za/whsteyn/Papers2/CONPRA2299_final.pdf
• Problem
- What range is needed?
- What precision is needed?
- Need to replicate results later? (standard format needed)
• Platform AND Algorithm AND Input
- How to optimize instructions/memory requirements?
- What can the Platform do for this Algorithm AND Input?
• There’s no free lunch in computer science!
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Data Type Representation
1.
2.
3.
4.
5.
6.
Integer numbers
Real numbers
Booleans
Characters
Composite types (e.g. arrays)
Objects
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3. Booleans
• Booleans have two values: true/false
• Used in programming languages
- if COND then ACTION1 else ACTION2
• Storage takes usually a complete byte
- false is often represented as 00000000
- true by any other pattern
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Data Type Representation
1.
2.
3.
4.
5.
6.
Integer numbers
Real numbers
4.1. ASCII
Booleans
4.2. Unicode
Characters
4.3. Processing Characters
Composite types (e.g. arrays)
Objects
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4. Characters
4.1. ASCII
• American Standard Code for
Information Interchange
• ASCII set 128 Characters
(7 bit)
• Alphabet {a-zA-Z}
• Numbers {0,...,9}
• Special symbols, eg. <, =, >, $
• Punctuation marks, ! ?, .
• Data transmission codes (non printable characters)
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4. Characters
4.1. ASCII
• Designed 1960—1963
• Underlies the WWW (actually, as common part of UTF-8)
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4. Characters 4.1. ASCII
An Example
The Demo, http://sloan.stanford.edu/MouseSite/1968Demo.html
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4. Characters 4.1. ASCII
Raster Images as ASCII (Example)
PPM Format Specification, http://netpbm.sourceforge.net/doc/ppm.html
P3
# RGB.ppm
33
255 0 0 255 255 255 0 0 0
255 255 255 0 255 0 255 255 255
0 0 0 255 255 255 0 0 255
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4. Characters
4.2. Unicode
• Large set of international alphabets represented
• Subset ISO 10646
• 16 bits representation (see Joseph Becker,
Unicode 1988,
http://www.unicode.org/history/unicode88.pdf)
“a simple unambiguous fixed-length character encoding”
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4. Characters
4.2. Unicode/UTF-8
• Variable-width encoding for Unicode (1-4 bytes)
• 1 byte: First 128 characters are ASCII (US-ASCII)
• 2 bytes: Next 1,920 characters encode Latin+diacritics, Greek,
Cyrillic, Coptic, …
• 3 bytes: Most others (Chinese, Hindi, tagalog/PH, …)
• 4 bytes: Historical/Exotic variants (Klingon? Actually, Klingon
has an unofficial 2-byte Unicode encryption, see
www.evertype.com/standards/csur/klingon.html )
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4. Characters 4.2. Unicode/UTF-8
An Example: The Romanian Alphabet
• HTML:
&#258; or
&#x102;
http://bucovina.chem.tue.nl/romanian.utf8.htm
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4. Characters
4.3. Processing of a number [1/3]
• User types a number, e.g. 123 (decimal)
• Computer receives ASCII codes
- ASCII-code of 1: 00110001
- ASCII-code of 2: 00110010
- ASCII-code of 3: 00110011
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4. Characters
4.3. Processing of a number [2/3]
• Program converts codes to binary values
1*100
2*10
3*1
123
01100100
00010100
00000011
01111011
• Perform operation (e.g. subtract 50)
123
50
73
01111011
00110010
01001001
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4. Characters
4.3. Processing of a number [3/3]
• The number 73 must be fed back to the screen
0 (*100)
7 (*10)
3 (*1)
00000000
00000111
00000011
• Computer sends following ASCII codes to screen
- ASCII code of 0:
- ASCII code of 7:
00110000
00110111
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Data Type Representation
1.
2.
3.
4.
5.
6.
Integer numbers
Real numbers
Booleans
Characters
Composite types (e.g. arrays)
Objects
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5. Arrays [1/3]
• Most programming languages support arrays
A: array[0..N,0..M] of integer;
• Must be mapped onto linear memory
• Row major order
• Column major order
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5. Arrays [2/3]
array
row major
column major
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5. Arrays [3/3]
• Storage method is stored as part of array
descriptor: so called dope vector.
begin address
no of dimensions
dim #1: low bound
dim #1: high bound
dim #1: multiplier
dim #2: low bound
etc
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address fixed element
no of dimensions
stride (skipped memory
elements)
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Data Type Representation
1.
2.
3.
4.
5.
6.
Integer numbers
Real numbers
Booleans
Characters
Composite types (e.g. arrays)
Objects
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6. Objects
Records
Record
field 1: type 1;
field 2: type 2;
end record
• Records can be stored with fixed off-set
• Records can contain pointers to other data
structures. Results in linked lists
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6. Objects
Raster Images as Records (Example)
• Simple, fixed representation:
- Height, Width
- RGB(Alpha) data
03 03
FF 00 00 FF FF FF 00 00 00
FF FF FF 00 FF 00 FF FF FF
00 00 00 FF FF FF 00 00 FF
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6. Objects
Raster Images as Records (Example)
• Fixed-format representation with look-up table:
- Height, Width
- Color map/look-up table (here, RGB)
- Image = index in color map per pixel
03 03
FF FF FF 00 00 00 FF 00 00
00 FF 00 00 00 FF
02 00 01
Map[02] =
00 03 00
FF 00 00
01 00 04
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6. Objects
Raster Images as Records (Example)
• Variable-format representation with look-up table:
- Image:
Height,
Width,
number of bits per color, …
- Color map structure:
RGB/ARGB,
number of entries, …
- Color map/look-up table
- Image = index in color map per pixel
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Fun With Hexadecimal Numbers
Color Codes [1/2]
http://en.wikibooks.org/wiki/LaTeX/Colors
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Fun With Hexadecimal Numbers
Color Codes [2/2]
http://johncfish.com/bggallery/otherchart/index.htm
http://www.visibone.com/
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