Parallel+Prefix+Algorithms

toc  =Parity Words= [|Parity] is often related to [|error detection and correction]. The modulo 2 sum aka exclusive or (xor) of any number of inputs is sufficient to determine even or odd parity. On a bitboard for each bit 0..63 we like to know, whether the number of ones including all trailing bits is odd or even, xi is the bit in x with bit-index i: code p0 = x0; p1 = x0 ^ x1; p2 = x0 ^ x1 ^ x2; ... p63 = x0 ^ x1 ^ x2 ^ .... ^ x63; code Or recursively code p[i] = p[i-1] ^ x[i]; code
 * Home * Programming * Algorithms * Parallel Prefix Algorithms**
 * [[image:sum.png link="http://www.toves.org/books/distalg/"]] ||~ || **Parallel prefix algorithms** compute all [|prefixes] of a input sequence in [|logarithmic time], and are topic of various SIMD and SWAR techniques applied to bitboards . This page provides some basics on simple parallel prefix problems, like parity words and [|Gray code] with some interesting properties, followed by some theoretical background on more complex parallel prefix problems, like Kogge-Stone by Steffan Westcott, and a comparison of three similar Kogge-Stone routines for different purpose. ||
 * Parallel Prefix Sum ||~ ||^ ||

p63 indicates whether the whole cardinality is odd or even. The parallel prefix solution looks that way: code format="cpp" x ^= x << 1; x ^= x << 2; x ^= x << 4; x ^= x << 8; x ^= x << 16; x ^= x << 32; code and only need log2(64) == 6 steps to perform all the xor instructions.

Surprisingly, thanks to xor, we can restore the initial value by a final: code format="cpp" x ^= x << 1; code  =Gray Code= The reflected binary code, or [|Gray code] has some similar properties. The Gray code is defined by the modulo 2 sum or exclusive or of a word with its value shifted right by one. code format="cpp" gray = x ^ (x >> 1); code This ensures that consecutive Gray codes have only one bit changed, that is a Hamming distance of one. For instance for a 4-bit or nibble Gray code:
 * ~ x decimal ||~ x binary ||~ x>>1 ||~ Gray bin ||~ decimal ||
 * > 0 ||= 0000 ||= 0000 ||= 0000 ||> 0 ||
 * > 1 ||= 0001 ||= 0000 ||= 0001 ||> 1 ||
 * > 2 ||= 0010 ||= 0001 ||= 0011 ||> 3 ||
 * > 3 ||= 0011 ||= 0001 ||= 0010 ||> 2 ||
 * > 4 ||= 0100 ||= 0010 ||= 0110 ||> 6 ||
 * > 5 ||= 0101 ||= 0010 ||= 0111 ||> 7 ||
 * > 6 ||= 0110 ||= 0011 ||= 0101 ||> 5 ||
 * > 7 ||= 0111 ||= 0011 ||= 0100 ||> 4 ||
 * > 8 ||= 1000 ||= 0100 ||= 1100 ||> 12 ||
 * > 9 ||= 1001 ||= 0100 ||= 1101 ||> 13 ||
 * > 10 ||= 1010 ||= 0101 ||= 1111 ||> 15 ||
 * > 11 ||= 1011 ||= 0101 ||= 1110 ||> 14 ||
 * > 12 ||= 1100 ||= 0110 ||= 1010 ||> 10 ||
 * > 13 ||= 1101 ||= 0110 ||= 1011 ||> 11 ||
 * > 14 ||= 1110 ||= 0111 ||= 1001 ||> 9 ||
 * > 15 ||= 1111 ||= 0111 ||= 1000 ||> 8 ||
 * > 0 ||= 0000 ||= 0000 ||= 0000 ||> 0 ||

Decoding Gray code is a parallel prefix problem similar to the ring-wise operations of the Parity words. For a nibble Gray code: code format="cpp" x = gray; x ^= x >> 2; x ^= x >> 1; code Or for 64-bit Gray-Codes: code format="cpp" x = gray; x ^= x >> 32; x ^= x >> 16; x ^= x >> 8; x ^= x >> 4; x ^= x >> 2; x ^= x >> 1; code =Fill Stuff= Parallel prefix front and rear-fills, for instance to determine pawn spans in bitboards, are quite simple to understand. code format="cpp" U64 nortFill(U64 gen) { gen |= (gen << 8); gen |= (gen << 16); gen |= (gen << 32); return gen; }

U64 soutFill(U64 gen) { gen |= (gen >> 8); gen |= (gen >> 16); gen |= (gen >> 32); return gen; } code This is actually a simplified Kogge-Stone algorithm, whose general form even considers a second bitboard to occlude the flood accordantly.

 =Elaborations on Kogge-Stone= Elaborations on Kogge-Stone parallel prefix algorithm by Steffan Westcott :

A **parallel prefix** problem is of the sort: code Given the terms x1, x2, x3, ... , xN and an associative operator # find the values q1 = x1                  q2 = x1 # x2                   q3 = x1 # x2 # x3. .                  .                   qN = x1 # x2 # x3 # ... # xN code Associative expressions can be bracketed any way you like, the result is the same. To see why this is interesting for chess programming, let's define x1, x2, ... and # to be code xI = < gI, pI >    (a two element tuple) where gI = square aI has a rook on it        and   pI = square aI is empty for all I = 1, 2, 3, ... , 8

and xI # xJ = < gI, pI > #  < gJ, pJ > = < ((gI && pJ) || gJ), (pI && pJ) > code All this algebra looks very scary at first, so here's an example: code q2 = x1 # x2 = < rook_on_a1, a1_is_empty > # < rook_on_a2, a2_is_empty > = < ((rook_on_a1 && a2_is_empty) || rook_on_a2), (a1_is_empty && a2_is_empty) > code
 * The first tuple of q2 tells us if a rook is on a2 or can move up to a2 along empty squares.
 * The second tuple of q2 tells us if a rook could move up freely through a1-a2 ie. a1-a2 are empty.

In general, code xI # x(I+1) # ... # xJ = < a_rook_somewhere_on_aI_to_aJ_is_either_on_aJ_or_can_move_up_to_aJ, all_squares_aI_to_aJ_are_empty > code and so code qJ = < a_rook_somewhere_in_file_a_is_either_on_aJ_or_can_move_up_to_aJ, all_squares_a1_to_aJ_are_empty > code The tuples g and p are known as the "generator" and "propagator" terms in the literature of fast carry propagation circuits. But we are not dealing with carry bits in a carry propagate adder! Here, we are generating and propagating upward rook attacks along a file of a chessboard.

Why all this theory? Well, prefix computation is a heavily researched area, researched by many folk smarter than me :) Its of interest because it has many applications, such as VLSI design, pattern matching, and others. There are many different ways of going about it, with different implementation characteristics. code format="cpp" U64 FillUpOccluded(U64 g, U64 p) {          g |= p & (g <<  8);           p &=     (p <<  8);           g |= p & (g << 16);           p &=     (p << 16);    return g |= p & (g << 32); }

U64 RookAttacksUp(U64 rooks, U64 empty_squares) { return ShiftUp(FillUpOccluded(rooks, empty_squares)); } code The method chosen in FillUpOccluded is based on a Kogge-Stone parallel prefix network, because it can be implemented very easily in software. The diagram below (trust me, it really _is_ supposed to look like that) is an illustration of how it works. The corresponding lines of program code are shown on the right. The inputs are fed into the network at the top, pass along the connecting lines, are combined by the # operator at various points, and the outputs appear at the bottom. code x1 x2 x3 x4 x5 x6 x7 x8        Input : g, p V  V  V  V  V  V  V  V V  V  V  V  V  V  V  V q1 q2 q3 q4 q5 q6 q7 q8         Output : g code To convince yourself this works, select any output qI and trace upwards from the bottom of the diagram. You'll see it leads back to x1, x2, ..., xI, so qI is formed by some computation of x1 # x2 # ... # xI.
 * # #  #  #  #  #  #          g |= p & (g <<  8);
 * | |  |  |  |  |  |          p &=     (p <<  8);
 * | #  #  #  #  #  #          g |= p & (g << 16);
 * | |  |  |  |  |  |          p &=     (p << 16);
 * | |  |  #  #  #  #          g |= p & (g << 32);
 * | |  |  |  |  |  |          p &=     (p <<  8);
 * | #  #  #  #  #  #          g |= p & (g << 16);
 * | |  |  |  |  |  |          p &=     (p << 16);
 * | |  |  #  #  #  #          g |= p & (g << 32);
 * | #  #  #  #  #  #          g |= p & (g << 16);
 * | |  |  |  |  |  |          p &=     (p << 16);
 * | |  |  #  #  #  #          g |= p & (g << 32);
 * | |  |  |  |  |  |          p &=     (p << 16);
 * | |  |  #  #  #  #          g |= p & (g << 32);
 * | |  |  #  #  #  #          g |= p & (g << 32);
 * | |  |  #  #  #  #          g |= p & (g << 32);
 * | |  |  #  #  #  #          g |= p & (g << 32);
 * | |  |  #  #  #  #          g |= p & (g << 32);
 * | |  |  #  #  #  #          g |= p & (g << 32);
 * | |  |  #  #  #  #          g |= p & (g << 32);
 * | |  |  #  #  #  #          g |= p & (g << 32);
 * | |  |  #  #  #  #          g |= p & (g << 32);
 * | |  |  #  #  #  #          g |= p & (g << 32);
 * | |  |  #  #  #  #          g |= p & (g << 32);
 * | |  |  #  #  #  #          g |= p & (g << 32);

[Implementor's note : The 2 program lines that assign to variable p are not strictly correct. By the end of the routine, p1 and p2 have been trashed (reset). However, they are trashed after they can affect the correctness of the routine result.]

 =Add/Sub versus Attacks= As a comparison three Kogge-Stone routines. A "classical" byte-wise SWAR adder, byte-wise subtraction, and byte-wise rook-attacks in east direction. Note that the inner parallel prefix instructions are the same in all three routines: code format="cpp" gen |= pro & (gen << 1); pro &=      (pro << 1); gen |= pro & (gen << 2); pro &=      (pro << 2); gen |= pro & (gen << 4); code The difference are the assignments of generator and propagator, and the final result.

Add
The software approach of a [|Kogge-Stone] hardware adder, requires the initial carry bits as generator, thus the intersection of both summands, while the propagator is the modulo 2 sum, the exclusive or of both summands. To avoid inter byte overflows, the propagator has the least significant bits of each byte cleared. However the final carries need one further masked shift and the modulo two sum with the initial propagator. code format="cpp" // SIMD bytewise add a + b U64 koggeStoneByteAdd(U64 a, U64 b) { U64 gen, pro; gen = a & b;   pro  = a ^ b;   pro &= ~aFile; gen |= pro  & (gen << 1); pro &=        (pro << 1); gen |= pro  & (gen << 2); pro &=        (pro << 2); gen |= pro  & (gen << 4); gen =~aFile & (gen << 1); return gen ^ a ^ b; } code

Sub
Subtraction is based on the two's complement which is ones' complement plus one aka the A-File assuming little-endian rank-file mapping. Thus, it is actually the sum of {a, ~b, one}, which is considered in calculating propagator and generator, which is the majority of all three inputs. Note that code format="cpp" gen =~aFile & (gen << 1); return gen ^ a ^ ~b ^ aFile; code can be re-written like in following routine: code format="cpp" // SIMD bytewise sub a - b U64 koggeStoneByteSub(U64 a, U64 b) { U64 gen, pro; gen = a & ~b;      // based on -b = ~b + 1 pro = a ^ ~b; gen |= aFile & pro; // majority gen |= pro  & (gen << 1); pro &=        (pro << 1); gen |= pro  & (gen << 2); pro &=        (pro << 2); gen |= pro  & (gen << 4); gen = aFile | (gen << 1); return gen ^ a ^ ~b; } code

East Attacks
Here the Kogge-Stone algorithm generates the sliding piece attacks along the empty squares. Generator are the rooks or queens, propagator the empty squares without the A-file to avoid wraps. code format="cpp" U64 eastAttacks(U64 occ, U64 rooks) { U64 gen, pro; pro = ~occ  & ~aFile; gen = rooks; gen |= pro  & (gen << 1); pro &=        (pro << 1); gen |= pro  & (gen << 2); pro &=        (pro << 2); gen |= pro  & (gen << 4); gen =~aFile & (gen << 1); return gen; } code The mentioned routines also demonstrate, that this east attacking direction may cheaper implemented with Fill by Subtraction based on SWAR-wise techniques and subtracting a rook from a blocking piece: code format="cpp" U64 byteAdd(U64 a, U64 b) {return ((a & ~hFile) + (b & ~hFile)) ^ ((a ^ b) & hFile);} U64 byteSub(U64 a, U64 b) {return ((a | hFile) - (b & ~hFile)) ^ ((a ^ ~b) & hFile);} code However, due to various left and right shifts, Kogge-Stone can deal with all other seven sliding directions as well.

 =See also=
 * Parallel prefix MSB
 * SWAR population count of the too slow loop approach
 * Flipping and Mirroring
 * Pawn Fills
 * Occupancy of any Line
 * Kogge-Stone Algorithm

=Publications=
 * [|Peter M. Kogge], Harold S. Stone (**1973**). //A Parallel Algorithm for the Efficient Solution of a General Class of Recurrence Equations//. IEEE Transactions on Computers, 1973, C-22, 783-791, [|pdf]
 * [|Richard E. Ladner], [|Michael J. Fischer] (**1980**). //Parallel Prefix Computation//. [|pdf] via [|CiteSeerX]
 * Clyde Kruskal, [|Larry Rudolph], [|Marc Snir] (**1985**). //The power of parallel prefix//. [|IEEE Transactions on Computers], Vol. C-34
 * Peter Sanders, [|Jesper Larsson Träff] (**2006**). //Parallel Prefix (Scan) Algorithms for MPI//. in EuroPVM/MPI 2006, LNCS, [|pdf]
 * [|Carl Burch] (**2009**). //[|Introduction to parallel & distributed algorithms]//. On-line Book

=Forum Posts=
 * [|flood fill attack bitboards] by Steffan Westcott from CCC, September 15, 2002
 * [|Re: Hyperbola Quiesscene: hardly any improvement] by Karlo Bala Jr., CCC, January 14, 2009 » Hyperbola Quintessence

=External Links= > media type="custom" key="24650284"
 * [|Hardware algorithms for arithmetic modules] from the ARITH research group, Aoki lab., [|Tohoku University]
 * [|Prefix sum from Wikipedia]
 * [|parallel prefix computation] from [|National Institute of Standards and Technology]
 * [|parallel prefix algorithm] from [|Everything2.com]
 * [|Charm++ Tutorial - Parallel Prefix Program]
 * Joe Zawinul, Trilok Gurtu - Orient Express Part1, [|YouTube] Video

=References= =What links here?= include page="Parallel Prefix Algorithms" component="backlinks" limit="40"
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