Morton Numbers

From WikiCoder

Interleaved bits (aka Morton numbers) are useful for linearizing 2D integer coordinates, so x and y are combined into a single number that can be compared easily and has the property that a number is usually close to another if their x and y values are close.

Interleave bits the obvious way

 unsigned short x;   // Interleave bits of x and y, so that all of the
 unsigned short y;   // bits of x are in the even positions and y in the odd;
 unsigned int z = 0; // z gets the resulting Morton Number.
 for (int i = 0; i < sizeof(x) * CHAR_BIT; i++) // unroll for more speed...
 {
   z |= (x & 1U << i) << i | (y & 1U << i) << (i + 1);
 }

Interleave bits by table lookup

 static const unsigned short MortonTable256[256] = 
 {
   0x0000, 0x0001, 0x0004, 0x0005, 0x0010, 0x0011, 0x0014, 0x0015, 
   0x0040, 0x0041, 0x0044, 0x0045, 0x0050, 0x0051, 0x0054, 0x0055, 
   0x0100, 0x0101, 0x0104, 0x0105, 0x0110, 0x0111, 0x0114, 0x0115, 
   0x0140, 0x0141, 0x0144, 0x0145, 0x0150, 0x0151, 0x0154, 0x0155, 
   0x0400, 0x0401, 0x0404, 0x0405, 0x0410, 0x0411, 0x0414, 0x0415, 
   0x0440, 0x0441, 0x0444, 0x0445, 0x0450, 0x0451, 0x0454, 0x0455, 
   0x0500, 0x0501, 0x0504, 0x0505, 0x0510, 0x0511, 0x0514, 0x0515, 
   0x0540, 0x0541, 0x0544, 0x0545, 0x0550, 0x0551, 0x0554, 0x0555, 
   0x1000, 0x1001, 0x1004, 0x1005, 0x1010, 0x1011, 0x1014, 0x1015, 
   0x1040, 0x1041, 0x1044, 0x1045, 0x1050, 0x1051, 0x1054, 0x1055, 
   0x1100, 0x1101, 0x1104, 0x1105, 0x1110, 0x1111, 0x1114, 0x1115, 
   0x1140, 0x1141, 0x1144, 0x1145, 0x1150, 0x1151, 0x1154, 0x1155, 
   0x1400, 0x1401, 0x1404, 0x1405, 0x1410, 0x1411, 0x1414, 0x1415, 
   0x1440, 0x1441, 0x1444, 0x1445, 0x1450, 0x1451, 0x1454, 0x1455, 
   0x1500, 0x1501, 0x1504, 0x1505, 0x1510, 0x1511, 0x1514, 0x1515, 
   0x1540, 0x1541, 0x1544, 0x1545, 0x1550, 0x1551, 0x1554, 0x1555, 
   0x4000, 0x4001, 0x4004, 0x4005, 0x4010, 0x4011, 0x4014, 0x4015, 
   0x4040, 0x4041, 0x4044, 0x4045, 0x4050, 0x4051, 0x4054, 0x4055, 
   0x4100, 0x4101, 0x4104, 0x4105, 0x4110, 0x4111, 0x4114, 0x4115, 
   0x4140, 0x4141, 0x4144, 0x4145, 0x4150, 0x4151, 0x4154, 0x4155, 
   0x4400, 0x4401, 0x4404, 0x4405, 0x4410, 0x4411, 0x4414, 0x4415, 
   0x4440, 0x4441, 0x4444, 0x4445, 0x4450, 0x4451, 0x4454, 0x4455, 
   0x4500, 0x4501, 0x4504, 0x4505, 0x4510, 0x4511, 0x4514, 0x4515, 
   0x4540, 0x4541, 0x4544, 0x4545, 0x4550, 0x4551, 0x4554, 0x4555, 
   0x5000, 0x5001, 0x5004, 0x5005, 0x5010, 0x5011, 0x5014, 0x5015, 
   0x5040, 0x5041, 0x5044, 0x5045, 0x5050, 0x5051, 0x5054, 0x5055, 
   0x5100, 0x5101, 0x5104, 0x5105, 0x5110, 0x5111, 0x5114, 0x5115, 
   0x5140, 0x5141, 0x5144, 0x5145, 0x5150, 0x5151, 0x5154, 0x5155, 
   0x5400, 0x5401, 0x5404, 0x5405, 0x5410, 0x5411, 0x5414, 0x5415, 
   0x5440, 0x5441, 0x5444, 0x5445, 0x5450, 0x5451, 0x5454, 0x5455, 
   0x5500, 0x5501, 0x5504, 0x5505, 0x5510, 0x5511, 0x5514, 0x5515, 
   0x5540, 0x5541, 0x5544, 0x5545, 0x5550, 0x5551, 0x5554, 0x5555
 };
 unsigned short x; // Interleave bits of x and y, so that all of the
 unsigned short y; // bits of x are in the even positions and y in the odd;
 unsigned int z;   // z gets the resulting 32-bit Morton Number.
 z = MortonTable256[y >> 8]   << 17 | 
     MortonTable256[x >> 8]   << 16 |
     MortonTable256[y & 0xFF] <<  1 | 
     MortonTable256[x & 0xFF];

For more speed, use an additional table with values that are MortonTable256 pre-shifted one bit to the left. This second table could then be used for the y lookups, thus reducing the operations by two, but almost doubling the memory required. Extending this same idea, four tables could be used, with two of them pre-shifted by 16 to the left of the previous two, so that we would only need 11 operations total.

Interleave bits with 64-bit multiply

In 11 operations, this version interleaves bits of two bytes (rather than shorts, as in the other versions), but many of the operations are 64-bit multiplies so it isn't appropriate for all machines. The input parameters, x and y, should be less than 256.

 unsigned char x;  // Interleave bits of (8-bit) x and y, so that all of the
 unsigned char y;  // bits of x are in the even positions and y in the odd;
 unsigned short z; // z gets the resulting 16-bit Morton Number.
 z = ((x * 0x0101010101010101ULL & 0x8040201008040201ULL) * 
      0x0102040810204081ULL >> 49) & 0x5555 |
     ((y * 0x0101010101010101ULL & 0x8040201008040201ULL) * 
      0x0102040810204081ULL >> 48) & 0xAAAA;

Interleave bits by Binary Magic Numbers

 static const unsigned int B[] = {0x55555555, 0x33333333, 0x0F0F0F0F, 0x00FF00FF};
 static const unsigned int S[] = {1, 2, 4, 8};
 unsigned int x; // Interleave lower 16 bits of x and y, so the bits of x
 unsigned int y; // are in the even positions and bits from y in the odd;
 unsigned int z; // z gets the resulting 32-bit Morton Number.  
                 // x and y must initially be less than 65536.
 x = (x | (x << S[3])) & B[3];
 x = (x | (x << S[2])) & B[2];
 x = (x | (x << S[1])) & B[1];
 x = (x | (x << S[0])) & B[0];
 y = (y | (y << S[3])) & B[3];
 y = (y | (y << S[2])) & B[2];
 y = (y | (y << S[1])) & B[1];
 y = (y | (y << S[0])) & B[0];
 z = x | (y << 1);

References