On an empty Board

Home * Board Representation * Bitboards * Sliding Piece Attacks * On an empty Board

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) );
}

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));
}

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 line masks for ranks, files, diagonals and antidiagonals on the fly:

```U64 rankMask(int sq) {return  C64(0xff) << (sq & 56);}

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

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;
}

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;
}
```

The generalized shift version for diagonals and antidiagonals as introduced by Thomas Jahn [1] produces shorter and faster code on modern x86-64 processors due to BMI2 shift instructions not affecting the flags [2]:

```U64 diagonalMask(int sq) {
const U64( maindia = C64(0x8040201008040201);
int diag  = (sq&7) - (sq>>3);
return diag >= 0 ? maindia >> diag*8 : maindia << -diag*8;
}

const U64( maindia = C64(0x0102040810204080);
int diag  = 7 - (sq&7) - (sq>>3);
return diag >= 0 ? maindia >> diag*8 : maindia << -diag*8;
}
```

Excluding the square bit:

```U64 rankMaskEx    (int sq) {return (C64(1) << sq) ^ rankMask(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);}