On an empty Board

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Single sliding piece attacks on the otherwise empty board or their disjoint subsets on lines or rays are that simple than none sliding pieces. We simply use pre-calculated tables for each piece-type, line or ray, indexed by square-index. To initialize those tables, one may use a fill approach with single populated from-sets, if availably anyway since used elsewhere. While the proposed line-routines here are quite small and cheap, incremental update during an initialization loop has some merits.

The various ray-,line- and piece sets are foundation of further attack calculation considering blocking pieces, for instance to mask the occupancy of relevant rays. Of course the piece attacks are union-sets of the disjoint line attacks, while the line attacks are unions of the disjoint ray attacks.

=Ray Attacks= northwest   north   northeast noWe        nort         noEa +7   +8    +9              \  |  /  west    -1 <-  0 -> +1    east / |  \          -9    -8    -7  soWe         sout         soEa southwest   south   southeast

Rays by Line
Ray-Attacks may be conducted from Line-Attacks by intersection with "positive" and "negative" squares: positiveRay[sq] = lineAttacks[sq] & (0 - 2*singleBit[sq]); negativeRay[sq] = lineAttacks[sq] & (singleBit[sq] - 1); or with shifts instead of lookups positiveRay[sq] = lineAttacks[sq] & (C64(-2) << sq); negativeRay[sq] = lineAttacks[sq] & ((C64(1) << sq) - 1);

Positive Rays
Remember Square Mapping Considerations.

By Lookup
East (+1)          Nort (+8)            NoEa (+9)           NoWe (+7) . . . . . . . .    . . . 1 . . . .      . . . . . . . 1     . . . . . . . . . . . . . . . .     . . . 1 . . . .      . . . . . . 1 .     1 . . . . . . . . . . . . . . .     . . . 1 . . . .      . . . . . 1 . .     . 1 . . . . . . . . . . . . . .     . . . 1 . . . .      . . . . 1 . . .     . . 1 . . . . . . . . R 1 1 1 1. . . R. . . .     . . . B. . . .    . . . B. . . . . . . . . . . .    . . . . . . . .      . . . . . . . .     . . . . . . . . . . . . . . . .     . . . . . . . .      . . . . . . . .     . . . . . . . . . . . . . . . .     . . . . . . . .      . . . . . . . .     . . . . . . ..

Initialization
North attacks are simple to initialize inside a loop, starting from a1, shifting left: U64 nort = C64(0x0101010101010100); for (int sq=0; sq < 64; sq++, nort <<= 1) rayAttacks[sq][Nort] = nort;

Similar, but tad trickier for ranks and diagonals, due to the wraps. For instance the north-east direction: U64 noea = C64(0x8040201008040200); for (int f=0; f < 8; f++, noea = eastOne(noea) {  U64 ne = noea;   for (int r8 = 0; r8 < 8*8; r8 += 8, ne <<= 8)      rayAttacks[r8+f][NoEa] = ne; }

By Calculation
Orthogonal positive rays are quite cheap to calculate on the fly. For diagonal rays split the lines as mentioned. U64 eastMaskEx(int sq) { const U64 one = 1; return 2*( (one << (sq|7)) - (one << sq) ); }

U64 nortMaskEx(int sq) { return C64(0x0101010101010100) << sq; }

Negative Rays
Remember Square Mapping Considerations.

By Lookup
West (-1)          Sout (-8)            SoWe (-9)           SoEa (-7) . . . . . . . .    . . . . . . . .      . . . . . . . .     . . . . . . . . . . . . . . . .     . . . . . . . .      . . . . . . . .     . . . . . . . . . . . . . . . .     . . . . . . . .      . . . . . . . .     . . . . . . . . . . . . . . . .     . . . . . . . .      . . . . . . . .     . . . . . . . . 1 1 1 R. . . .    . . . R. . . .     . . . B. . . .    . . . B. . . . . . . . . . . .    . . . 1 . . . .      . . 1 . . . . .     . . . . 1 . . . . . . . . . . .     . . . 1 . . . .      . 1 . . . . . .     . . . . . 1 . . . . . . . . . .     . . . 1 . . . .      1 . . . . . . .     . . . . . . 1.

Initialization
South attacks are simple to initialize inside a loop, starting from h8, shifting right: U64 sout = C64(0x0080808080808080); for (int sq=63; sq >= 0; sq--, sout >>= 1) rayAttacks[sq][Sout] = sout; Similar, but tad trickier for ranks and diagonals, due to the wraps.

By Calculation
Orthogonal negative rays are quite cheap to calculate on the fly. For diagonal rays split the lines as mentioned. U64 westMaskEx(int sq) { const U64 one = 1; return (one << sq) - (one << (sq&56)); }

U64 soutMaskEx(int sq) { return C64(0x0080808080808080) >> (sq ^ 63); } =Line Attacks= RankAttacks[sq]        = EastAttacks[sq] | WestAttacks[sq]; FileAttacks[sq]        = NortAttacks[sq] | SoutAttacks[sq]; DiagonalAttacks[sq]    = NoEaAttacks[sq] | SoWeAttacks[sq]; AntiDiagonalAttacks[sq] = NoWeAttacks[sq] | SoEaAttacks[sq];

By Lookup
Rank               File                 Diagonal            Anti-Diagonal . . . . . . . .    . . . 1 . . . .      . . . . . . . 1     . . . . . . . . . . . . . . . .     . . . 1 . . . .      . . . . . . 1 .     1 . . . . . . . . . . . . . . .     . . . 1 . . . .      . . . . . 1 . .     . 1 . . . . . . . . . . . . . .     . . . 1 . . . .      . . . . 1 . . .     . . 1 . . . . . 1 1 1 R 1 1 1 1. . . R. . . .     . . . B. . . .    . . . B. . . . . . . . . . . .    . . . 1 . . . .      . . 1 . . . . .     . . . . 1 . . . . . . . . . . .     . . . 1 . . . .      . 1 . . . . . .     . . . . . 1 . . . . . . . . . .     . . . 1 . . . .      1 . . . . . . .     . . . . . . 1.

By Calculation
To calculate them on the fly, including... U64 rankMask(int sq) {return C64(0xff) << (sq & 56);}

U64 fileMask(int sq) {return C64(0x0101010101010101) << (sq & 7);}

U64 diagonalMask(int sq) { const U64 maindia = C64(0x8040201008040201); int diag =8*(sq & 7) - (sq & 56); int nort = -diag & ( diag >> 31); int sout = diag & (-diag >> 31); return (maindia >> sout) << nort; }

U64 antiDiagMask(int sq) { const U64 maindia = C64(0x0102040810204080); int diag =56- 8*(sq&7) - (sq&56); int nort = -diag & ( diag >> 31); int sout = diag & (-diag >> 31); return (maindia >> sout) << nort; } ... or excluding the square bit: U64 rankMaskEx   (int sq) {return (C64(1) << sq) ^ rankMask(sq);} U64 fileMaskEx   (int sq) {return (C64(1) << sq) ^ fileMask(sq);} U64 diagonalMaskEx(int sq) {return (C64(1) << sq) ^ diagonalMask(sq);} U64 antiDiagMaskEx(int sq) {return (C64(1) << sq) ^ antiDiagMask(sq);} =Piece Attacks= RookAttacks[sq]  = RankAttacks[sq]     | FileAttacks[sq]; BishopAttacks[sq] = DiagonalAttacks[sq] | AntiDiagonalAttacks[sq]; QueenAttacks[sq] = RookAttacks[sq] | BishopAttacks[sq];

By Lookup
Queen . . . 1 . . . 1                              1 . . 1 . . 1 .                               . 1 . 1 . 1 . .               Rook. . 1 1 1 . . .        Bishop . . . 1 . . . .    1 1 1 Q 1 1 1 1. . . . . . . 1          . . . 1 . . . .     . . 1 1 1 . . .     1 . . . . . 1 .           . . . 1 . . . .     . 1 . 1 . 1 . .     . 1 . . . 1 . .           . . . 1 . . . .     1 . . 1 . . 1 .     . . 1 . 1 . . .           1 1 1 R 1 1 1 1. . . B. . . .          . . . 1 . . . .                         . . 1 . 1 . . .           . . . 1 . . . .                         . 1 . . . 1 . .           . . . 1 . . . .                         1 . . . . . 1.

By Calculation
U64 rookMask   (int sq) {return rankMask(sq)     | fileMask(sq);} U64 bishopMask (int sq) {return diagonalMask(sq) | antiDiagMask(sq);}

U64 rookMaskEx (int sq) {return rankMask(sq)     ^ fileMask(sq);} U64 bishopMaskEx(int sq) {return diagonalMask(sq) ^ antiDiagMask(sq);}

U64 queenMask  (int sq) {return rookMask(sq)     | bishopMask(sq);} U64 queenMaskEx (int sq) {return rookMask(sq)    ^ bishopMask(sq);} =See also=
 * Blockers and Beyond
 * Fill on an empty Board with Kogge-Stone Algorithm

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