Re: WIP: multivariate statistics / proof of concept

On 10/13/2014 01:00 AM, Tomas Vondra wrote:
> Hi,
>
> attached is a WIP patch implementing multivariate statistics.

Great! Really glad to see you working on this.

> + * FIXME This sample sizing is mostly OK when computing stats for
> + * individual columns, but when computing multi-variate stats
> + * for multivariate stats (histograms, mcv, ...) it's rather
> + * insufficient. For small number of dimensions it works, but
> + * for complex stats it'd be nice use sample proportional to
> + * the table (say, 0.5% - 1%) instead of a fixed size.

I don't think a fraction of the table is appropriate. As long as the
sample is random, the accuracy of a sample doesn't depend much on the
size of the population. For example, if you sample 1,000 rows from a
table with 100,000 rows, or 1000 rows from a table with 100,000,000
rows, the accuracy is pretty much the same. That doesn't change when you
go from a single variable to multiple variables.

You do need a bigger sample with multiple variables, however. My gut
feeling is that if you sample N rows for a single variable, with two
variables you need to sample N^2 rows to get the same accuracy. But it's
not proportional to the table size. (I have no proof for that, but I'm
sure there is literature on this.)

> + * Multivariate histograms
> + *
> + * Histograms are a collection of buckets, represented by n-dimensional
> + * rectangles. Each rectangle is delimited by an array of lower and
> + * upper boundaries, so that for for the i-th attribute
> + *
> + * min[i] <= value[i] <= max[i]
> + *
> + * Each bucket tracks frequency (fraction of tuples it contains),
> + * information about the inequalities, number of distinct values in
> + * each dimension (which is used when building the histogram) etc.
> + *
> + * The boundaries may be either inclusive or exclusive, or the whole
> + * dimension may be NULL.
> + *
> + * The buckets may overlap (assuming the build algorithm keeps the
> + * frequencies additive) or may not cover the whole space (i.e. allow
> + * gaps). This entirely depends on the algorithm used to build the
> + * histogram.

That sounds pretty exotic. These buckets are quite different from the
single-dimension buckets we currently have.

The paper you reference in partition_bucket() function, M.
Muralikrishna, David J. DeWitt: Equi-Depth Histograms For Estimating
Selectivity Factors For Multi-Dimensional Queries. SIGMOD Conference
1988: 28-36, actually doesn't mention overlapping buckets at all. I
haven't read the code in detail, but if it implements the algorithm from
that paper, there will be no overlap.

- Heikki