Better results can be achieved by compensating for the bias of the raw
approximation just after 2.5m (when LINEARCOUNTING is no longer used) by
using a polynomial that approximates the bias at a given cardinality.
The curve used was found using this web page:
http://www.xuru.org/rt/PR.asp
That performs polynomial regression given a set of values.
The following form is given:
HLLADD myhll
No element is provided in the above case so if 'myhll' var does not
exist the result is to just create an empty HLL structure, and no update
will be performed on the registers.
In this case, the DB should still be set dirty and the command
propagated.
The HyperLogLog original paper suggests using LINEARCOUNTING for
cardinalities < 2.5m, however for P=14 the median / max error
curves show that a value of '3' is the best pick for m = 16384.
The more we add elements to an HyperLogLog counter, the smaller is
the probability that we actually update some register.
From this observation it is easy to see how it is possible to use
caching of a previously computed cardinality and reuse it to serve
HLLCOUNT queries as long as no register was updated in the data
structure.
This commit does exactly this by using just additional 8 bytes for the
data structure to store a 64 bit unsigned integer value cached
cardinality. When the most significant bit of the 64 bit integer is set,
it means that the value computed is no longer usable since at least a
single register was modified and we need to recompute it at the next
call of HLLCOUNT.
The value is always stored in little endian format regardless of the
actual CPU endianess.
All the Redis functions that need to modify the string value of a key in
a destructive way (APPEND, SETBIT, SETRANGE, ...) require to make the
object unshared (if refcount > 1) and encoded in raw format (if encoding
is not already REDIS_ENCODING_RAW).
This was cut & pasted many times in multiple places of the code. This
commit puts the small logic needed into a function called
dbUnshareStringValue().
We need to be sure that you can save a dataset in a Redis instance,
reload it in a different architecture, and continue to count in the same
HyperLogLog structure.
So 32 and 64 bit, little or bit endian, must all guarantee to output the
same hash for the same element.
The new representation is more obvious, starting from the LSB of the
first byte and using bits going to MSB, and passing to next byte as
needed.
There was also a subtle error: first two bits were unused, everything
was carried over on the right of two bits, even if it worked because of
the code requirement of always having a byte more at the end.
During the rewrite the code was made safer trying to avoid undefined
behavior due to shifting an uint8_t for more than 8 bits.
To test the bitfield array of counters set/get macros from the Redis Tcl
suite is hard, so a specialized command that is able to test the
internals was developed.
