Integer values are typically stored using 32 bits. Yet if you are given an array of integers between 0 and 131 072, you could store these numbers using as little as 17 bits each—a net saving of almost 50%.
Programmers nearly never store integers in this manner despite the obvious compression benefits. Indeed, bit packing and unpacking is expensive. How expensive? Intuitively, you might think that recovering 32-bit integers from a stream of packed integers must be at least as expensive as copying the 32-bit integers, and possibly much more expensive. If that is your intuition, then you might be wrong. It can be cheaper to recover 32-bit integers from packed 4-bit integers because you only need to load one 32-bit word to unpack 8 integers.
Clearly, packing integers in units of 17 bits is not especially convenient. Indeed, 17 and 32 are coprime. We expect that it would be much faster to pack and unpack integers in units of 4, 8 or 16 bits, than in units of 17 bits. Indeed it is but the difference is maybe not as large as you might think.
I have implemented efficient packing and unpacking routines in C++. To simplify the implementation, we pack and unpack integers in sets of 32 numbers. I have optimized the code using the GNU GCC 4.6.2 compiler.
On my macbook air (Intel core i7), I get that the unpacking speed is not very sensitive to the specific number of bits: generally, the smaller the bit width, the faster the unpacking. The packing speed is much faster when the bit width is 8 or 16. Even so, the difference is only by a factor of two or so. The results are presented in the next figure. On the y axis, you have the time (smaller is better). On the the x axis, we have the number of bits we packed to. For example, when bit is 1, we pack 32 integers into a single 32-bit word. When the number of bits is set to 32 bits, we have a regular copy.
I also provide the raw numbers behind the figure in the next table.
| bits | pack (ms) | unpack (ms) |
|---|---|---|
| 1 | 219 | 211 |
| 2 | 215 | 216 |
| 3 | 210 | 205 |
| 4 | 198 | 194 |
| 5 | 222 | 214 |
| 6 | 229 | 218 |
| 7 | 242 | 222 |
| 8 | 167 | 202 |
| 9 | 252 | 240 |
| 10 | 243 | 225 |
| 11 | 255 | 235 |
| 12 | 246 | 231 |
| 13 | 276 | 244 |
| 14 | 279 | 245 |
| 15 | 304 | 255 |
| 16 | 183 | 223 |
| 17 | 292 | 252 |
| 18 | 297 | 256 |
| 19 | 316 | 266 |
| 20 | 300 | 256 |
| 21 | 329 | 280 |
| 22 | 321 | 274 |
| 23 | 332 | 278 |
| 24 | 299 | 257 |
| 25 | 341 | 289 |
| 26 | 340 | 298 |
| 27 | 352 | 295 |
| 28 | 336 | 284 |
| 29 | 367 | 311 |
| 30 | 357 | 299 |
| 31 | 384 | 319 |
| 32 | 256 | 261 |
Conclusion: Bit packing and unpacking can be quite fast. In particular, it can be cheaper to unpack integers from a small number of bits to 32-bit integers than to copy the same 32-bit integers. Exact results will vary depending on your compiler and CPU.
Note: Strictly speaking my implementation packs the first bits of each integer: it is not assumed that the integers are between 0 and 2bit. By adding this assumption, you can improve the packing speed somewhat (at least when the number of bits is not 8 or 16).
Further reading : We have written recent research papers that survey related schemes for this problem. Please see:
- Daniel Lemire and Leonid Boytsov, Decoding billions of integers per second through vectorization, Software: Practice & Experience 45 (1), 2015.
- Daniel Lemire, Nathan Kurz, Leonid Boytsov, SIMD Compression and the Intersection of Sorted Integers, Software: Practice and Experience (to appear)
They include an extensive experimental evaluation. You can find a complete implementation of all techniques in C++11 online:
FastPFor and SIMDCompressionAndIntersection. There are also C libraries: simdcomp and MaskedVByte. If you prefer Java, please see JavaFastPFOR.