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Grade School Again: A Parallel Perspective Great Theoretical Ideas In Computer Science Steven RudichCS 15-251 Spring 2004 Lecture 17Mar 16, 2004Carnegie Mellon University

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Plus/Minus Binary (Extended Binary) Base 2: Each digit can be -1, 0, 1, Example: 1 -1 -1 = 4 -2 -1 = 1

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One weight for each power of 3. Left = “negative”. Right = “positive”

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How to add 2 n-bit numbers. ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** +

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*** *** **** **** **** **** **** **** **** **** **** **** **** **** **** **** **** **** **** **** * * * * + * ** *

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How to add 2 n-bit numbers. *** *** **** **** **** **** **** **** **** **** **** **** **** **** **** **** **** **** **** **** * * * * + * ** * Let k be the maximum time that it takes you to do Time < kn is proportional to n

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The time grow linearly with input size. # of bits in numbers timetime kn

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If n people agree to help you add two n bit numbers, it is not obvious that they can finish faster than if you had done it yourself.

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Is it possible to add two n bit numbers in less than linear parallel-time? Darn those carries.

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Fast parallel addition is no obvious in usual binary. But it is amazingly direct in Extended Binary!

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Extended binary means base 2 allowing digits to be from {-1, 0, 1}. We can call each digit a “trit”.

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n people can add 2, n-trit, plus/minus binary numbers in constant time!

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An Addition Party to Add 110-1 to -111-1 -

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An Addition Party 1 11 1 0 1 Invite n people to add two n-trit numbers Assign one person to each trit position

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An Addition Party 0 0 12 1 1 -2 Each person should add the two input trits in their possession. Problem: 2 and -2 are not allowed in the final answer.

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Pass Left 0 0 1 0 -2 1 If you have a 1 or a 2 subtract 2 from yourself and pass a 1 to the left. (Nobody keeps more than 0) Add in anything that is given to you from the right. (Nobody has more than a 1) 1

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After passing left 1 0 11 1 -2 There will never again be any 2s as everyone had at most 0 and received at most 1 more

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Passing left again 1 0 11 1 1 0 If you have a -1 or -2 add 2 to yourself and pass a -1 to the left (Nobody keeps less than 0)

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After passing left again 1 10 0 1 0 No -2s anymore either. Everyone kept at least 0 and received At most -1.

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1 10 -1 0 1 0 1 11 1 0 1 =

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Caution: Parties and Algorithms Do not Mix 1 11 0 1 1

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Is there a fast parallel way to convert an Extended Binary number into a standard binary number?

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Not obvious: Sub-linear addition in standard Binary. Sub-linear EB to Binary

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Let’s reexamine grade school addition from the view of a computer circuit.

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Grade School Addition +

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Grade School Addition a4a3a2a1a0b4b3b2b1b0a4a3a2a1a0b4b3b2b1b0 + c5c4c3c2c1 c5c4c3c2c1

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a 4 a 3 a 2 a 1 a 0 b 4 b 3 b 2 b 1 b 0 + c 5 c 4 c 3 c 2 c 1 s1s1

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Ripple-carry adder a i b i cici c i+1 sisi 0 cncn a i b i cici sisi a0b0a0b0 a 1 b 1 s0s0 c1c1 … s1s1 … a n-1 b n-1 s n-1

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Logical representation of binary: 0 = false, 1 = true s 1 = (a 1 XOR b 1 ) XOR c 1 c 2 = (a 1 AND b 1 ) OR (a 1 AND c 1 ) OR (b 1 AND c 1 ) a i b i cici c i+1 sisi

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OR AND XOR cici bibi aiai OR c i+1 sisi a i b i cici c i+1 sisi

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0 How long to add two n bit numbers? Propagation time through the circuit will be (n) Ripple-carry adder

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Circuits compute things in parallel. We can think of the propagation delay as PARALLEL TIME.

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Is it possible to add two n bit numbers in less than linear parallel-time?

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I suppose the EB addition algorithm could be helpful somehow.

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Plus/minus binary means base 2 allowing digits to be from {- 1, 0, 1}. We can call each digit a “trit”.

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n people can add 2, n-trit, plus/minus binary numbers in constant time!

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1 11 1 0 1

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Can we still do addition quickly in the standard representation?

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Yes, but first a neat idea… Instead of adding two numbers to make one number, let’s think about adding 3 numbers to make 2 numbers.

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Carry-Save Addition The sum of three numbers can be converted into the sum of 2 numbers in constant parallel time! + +

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XOR Carries Carry-Save Addition The sum of three numbers can be converted into the sum of 2 numbers in constant parallel time! + + +

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Cool! So if we if represent x as a+b, and y as c+d, then can add x,y using two of these (this is basically the same as that plus/minus binary thing). (a+b+c)+d=(e+f)+d=g+h

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Even in standard representation, this is really useful for multiplication.

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Grade School Multiplication X 101 1 0111 * * * * * * * * * * * * * * * * * * * * * * * * * * * *

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We need to add n 2n-bit numbers: a 1, a 2, a 3,…, a n

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A tree of carry-save adders a1a1 anan + Add the last two + + + + + + + + + + + +

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A tree of carry-save adders T(n) log 3/2 (n) + [last step] + Add the last two + + + + + + + + + + + +

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So let’s go back to the problem of adding two numbers. In particular, if we can add two numbers in O(log n) parallel time, then we can multiply in O(log n) parallel time too!

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If we knew the carries it would be very easy to do fast parallel addition 0

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What do we know about the carry- out before we know the carry-in? a b c in c out s abC out 000 01C in 10 111

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What do we know about the carry- out before we know the carry-in? a b c in c out s abC out 000 01 10 111

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Hey, this is just a function of a and b. We can do this in parallel. abC out 000 01 10 111

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Idea #1: do this calculation first. + This takes just one step!

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Idea #1: do this calculation first. + Also, once we have the carries, it only takes one step more: s i = (a i XOR b i ) XOR c i

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So, everything boils down to: can we find a fast parallel way to convert each position to its final 0/1 value? Called the “parallel prefix problem”

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So, we need to do this quickly….

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01 0 000 1 111 01 ( ( ( ( ( Can think of as all partial results in: for the operator : Idea #2: x = x 1 x = 1 0 x = 0

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( ( ( ( ) ( ) And, the operator is associative. Idea #2 (cont):

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Just using the fact that we have an Associative, Binary Operator Binary Operator: an operation that takes two objects and returns a third. A B = C Associative: ( A B ) C = A ( B C )

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Examples Addition on the integers Min(a,b) Max(a,b) Left(a,b) = a Right(a,b) = b Boolean AND Boolean OR

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In what we are about to do “+” will mean an arbitrary binary associative operator.

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Prefix Sum Problem Input: X n-1, X n-2,…,X 1, X 0 Output:Y n-1, Y n-2,…,Y 1, Y 0 where Y 0 = X 0 Y 1 = X 0 + X 1 Y 2 = X 0 + X 1 + X 2 Y 3 = X 0 + X 1 + X 2 + X 3 Y n-1 = X 0 + X 1 + X 2 + X 3 + … + X n-1......

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Prefix Sum example Input: 6, 9, 2, 3, 4, 7 Output:31, 25, 16, 14, 11, 7 where Y 0 = X 0 Y 1 = X 0 + X 1 Y 2 = X 0 + X 1 + X 2 Y 3 = X 0 + X 1 + X 2 + X 3 Y n-1 = X 0 + X 1 + X 2 + X 3 + … + X n-1......

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Example circuitry (n = 4) + X1X1 X0X0 y1y1 X0X0 X2X2 + X1X1 + y2y2 X0X0 y0y0 X3X3 + ++ X2X2 X1X1 X0X0 y3y3

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y n-1 + Divide, conquer, and glue for computing y n-1 X n/2 -1 … X1X1 X0X0 sum on n/2 items X n-1 …X n/2 X n-2 sum on n/2 items T(1)=0 T(n) = T( n/2 ) + 1 T(n) = log 2 n

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Modern computers do something slightly different. This algorithm is fast, but how many components does it use?

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y n-1 + Size of Circuit (number of gates) X n/2 -1 … X1X1 X0X0 Sum on n/2 items X n-1 …X n/2 X n-2 Sum on n/2 items S(1)=0 S(n) = S( n/2 ) + S( n/2 ) +1 S(n) = n-1

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Sum of Sizes X0X0 y0y0 + X1X1 X0X0 y1y1 X0X0 X2X2 + X1X1 + y2y2 X3X3 + ++ X2X2 X1X1 X0X0 y3y3 S(n) = 0 + 1 + 2 + 3 + + (n-1) = n(n-1)/2

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Recursive Algorithm n items (n = power of 2) If n = 1, Y 0 = X 0 ; + X3X3 X2X2 + X1X1 X0X0 + X5X5 X4X4 + X n-3 X n-4 + X n-1 X n-2 …

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Recursive Algorithm n items (n = power of 2) If n = 1, Y 0 = X 0 ; + X3X3 X2X2 + X1X1 X0X0 + X5X5 X4X4 + X n-3 X n-4 + X n-1 X n-2 … … Prefix sum on n/2 items

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+ Recursive Algorithm n items (n = power of 2) If n = 1, Y 0 = X 0 ; + X3X3 X2X2 + X1X1 X0X0 + X5X5 X4X4 + X n-3 X n-4 + X n-1 X n-2 … +++ Prefix sum on n/2 items Y n-1 Y1Y1 Y0Y0 Y2Y2 Y n-2 Y4Y4 Y3Y3 … …

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Parallel time complexity T(1)=0; T(2) = 1; T(n) = T(n/2) + 2 T(n) = 2 log 2 (n) - 1 + If n = 1, Y 0 = X 0 ; + X3X3 X2X2 + X1X1 X0X0 + X5X5 X4X4 + X n-3 X n-4 + X n-1 X n-2 … + ++ Prefix sum on n/2 items Y n-1 Y1Y1 Y0Y0 Y2Y2 Y n-2 Y4Y4 Y3Y3 … … 1 T(n/2) 1

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Size S(1)=0; S(n) = S(n/2) + n - 1 S(n) = 2n – log 2 n -2 + If n = 1, Y 0 = X 0 ; + X3X3 X2X2 + X1X1 X0X0 + X5X5 X4X4 + X n-3 X n-4 + X n-1 X n-2 … + ++ Prefix sum on n/2 items Y n-1 Y1Y1 Y0Y0 Y2Y2 Y n-2 Y4Y4 Y3Y3 … … n/2 S(n/2) (n/2)-1

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Putting it all together: Carry Look- Ahead Addition To add two n-bit numbers: a and b 1 step to compute x values ( ) 2 log 2 n - 1 steps to compute carries c 1 step to compute c XOR (a XOR b) 2 log 2 n + 1 steps total

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T(n) log 3/2 (n) + 2log 2 2n + 1 + carry look ahead + + + + + + + + + + + + Putting it all together: multiplication

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For a 64-bit word that works out to a parallel time of 22 for multiplication, and 13 for addition.

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Addition requires linear time sequentially, but has a practical 2log2n + 1 parallel algorithm.

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Multiplication (which is a lot harder sequentially) also has an O(log n) time parallel algorithm.

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And this is how addition works on commercial chips….. Processorn2log 2 n +1 80186169 Pentium3211 Alpha6413

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In order to handle integer addition/subtraction we use 2’s compliment representation, e.g., -44= -64-64 3232 1616 8421 1010100

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Addition of two numbers works the same way (assuming no overflow). -64-64 3232 1616 8421 1010100

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To negate a number, flip each of its bits and add 1. -64-64 3232 1616 8421 1010100 -64-64 3232 1616 8421 01 0 1011 -64-64 3232 1616 8421 01 0 1100

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x + flip(x) = -1. So, -x = flip(x)+1. -64-64 3232 1616 8421 1111111

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Most computers use two’s compliment representation to perform integer addition and subtraction.

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Grade School Division * * * * * * * * * * * * * * * * * * * * * * * * * * * * * n bits of precision: n subtractions costing 2log 2 n + 1 each (n logn)

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Let’s see if we can reduce to O(n) by being clever about it.

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Idea: internally, allow ourselves to “go negative” using trits so we can do constant-time subtraction. Then convert back at the end. (technically, called “extended binary”)

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SRT division algorithm 1 0 1 11 1 1 0 1 1 0 1 23711 21 r 6 Rule: Each bit of quotient is determined by comparing first bit of divisor with first bit of dividend. Easy! -1 0 –1 -1 1 0 –1 1 1 = -1 0 0 1 -1 0 –1 -1 -2 0 1 1 0 1 1 1 2 = 1 0 0 0 -1 0 –1 -1 0 –1 –1 1 1 0 r –1 –1 1 23711 22 r -5 Time for n bits of precision in result: 3n + 2log 2 (n) + 1 Convert to standard representation by subtracting negative bits from positive. 1 addition per bit

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Intel Pentium division error The Pentium uses essentially the same algorithm, but computes more than one bit of the result in each step. Several leading bits of the divisor and quotient are examined at each step, and the difference is looked up in a table. The table had several bad entries. Ultimately Intel offered to replace any defective chip, estimating their loss at $475 million.

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If millions of processors, how much of a speed-up might I get over a single processor?

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Brent’s Law At best, p processors will give you a factor of p speedup over the time it takes on a single processor.

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The traditional GCD algorithm will take linear time to operate on two n bit numbers. Can it be done faster in parallel?

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If n 2 people agree to help you compute the GCD of two n bit numbers, it is not obvious that they can finish faster than if you had done it yourself.

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Is GCD inherently sequential?

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No one knows.

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Deep Blue Many processors help DB look many chess moves ahead. It was not obvious that more processors could really help with game tree search. I remember a heated debate in C.B.’s Ph.D. thesis defense.

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1 Modified from Modified from 1998 Morgan Kaufmann Publishers Chapter Three: Arithmetic for Computers citation and following credit line is included:

1 Modified from Modified from 1998 Morgan Kaufmann Publishers Chapter Three: Arithmetic for Computers citation and following credit line is included:

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