1. Wavelet-based histograms for selectivity estimation
Paper: Yossi Matias, Jeffrey Scott Vitter, and Min Wang
Presentation: Michael Ernst

2. Executive Summary Histograms aid query optimization
Problem: histograms are bulky
Solution: compress histograms
Technique: wavelet-based compression
Claim: it works
Estimate sizes of relations and query results
Surajit Chaudhuri mentioned histograms as “a substantial [space] cost”, especially if there are very many attributes.
Compression is not a new idea. We’ll use lossy compression.
Wavelets are a compact, computationally efficient approximation to a function.
Despite pages of numbers, they provide precious little evidence to back up their claim.
The talk will follow this outline.

3. Histograms for query optimization
Estimate sizes of relations, range query results
join ordering
join implementation
placement in operator tree
Some other techniques, like sampling, are good for arbitrary queries, not just range queries.
Join ordering: keep relations as small as possible, as long as possible
Join implementation: e.g., hash-join if one relation fits in memory
Operator tree placement: filters, group-by, other operations (eager vs. lazy)

4. Types of histogram Sampling Equi-width Equi-depth Maxdiff
Cumulative frequency ( frequency):
splines
wavelets 5 2 3 6 4 4 4 4 7 2 1 6
(16 values in [1..20])
All the data is itself a histogram: the perfect histogram. We must approximate to reduce the storage costs.
Equi-width: buckets have same width
Equi-depth: buckets have same number of elements (must store bucket boundaries instead of sizes).
Maxdiff: split buckets at gaps in data (must store both bucket boundaries and sizes).
Cumulative frequency is much smoother (easier to approximate) than frequency, but suffers no information loss.

5. The wavelet transform Lossless function representation (like Fourier)
Haar wavelets: made up of , ,
Result: sequence of coefficients
Reconstruction: find and add the relevant components
Just as Fourier transform decomposes a waveform into its constituent sine waves, the wavelet transform decomposes a waveform into its constituent finite square waves.
Example: show Fourier transform for square wave.
Example: show Haar wavelet transform for an exponential decay.

6. Wavelet compression Lossless: no compression
Thresholding: discard some coefficients. Keep:
first m coefficients
biggest m/2 coefficients
greedy: select some coefficients, then iterate adding/deleting coefficients
Further compression may be possible
As many coefficients as original function values (exact if power of 2).
Biggest m/2 coefficients is best for 2-norm error (also known as least-squares fit).
Further compression (has nothing to do with wavelets, generally applicable):
quantization, entropy encoding.

7. Multidimensional histograms
Avoid assumption of independence
To estimate selectivity from 2-D cumulative frequency:
(0,0) 1 -1 -1 1

8. Empirical evaluation Synthetic benchmarks Compare:
smooth cumulative frequency
no “equals” queries
Compare:
sampling
maxdiff
Haar wavelets
linear wavelets
Are these benchmarks representative?
Sampling: how many tries? Is this result typical?
Maxdiff: performs worst when cumulative frequency is smooth
Linear wavelets are best by far.
Haar wavelets often outperform maxdiff, sometimes worse

9. The bottom line
Wavelets improve histogram accuracy, thus improving selectivity estimation.
Does this affect query execution time?
This is the key issue. It doesn’t matter how much accuracy is improved if that additional accuracy can’t be used. Maybe current schemes are good enough.
They don’t even mention this as an issue!