1. SPACE EFFICENCY OF SYNOPSIS CONSTRUCTION ALGORITHMS
Sudipto Guha
UPENN

2. Space efficiency in synopsis construction algorithms
Synopses
Given n input numbers, summarize the input using B numbers, minimizing some error.
Examples
Histograms – piecewise constant repn.
Wavelets – uses the wavelet basis
Fourier, Bessel, SVD, what have you…
VLDB 2005
Space efficiency in synopsis construction algorithms

3. Space efficiency in synopsis construction algorithms
Why space efficiency
“Interestingly, according to modern astronomers, space is finite. This is a very comforting thought – particularly for people who can never remember where they left things.”
Woody Allen.
From a computational viewpoint however…
VLDB 2005
Space efficiency in synopsis construction algorithms

4. Space is the cruelest resource
Resources
Time : tweedle thumbs
Access (stream): make more passes
Program simply will not run – or if data is shifted to disk, will run quite slow(er).
Further, if we had more space, maybe we can compute a better (more accurate) synopsis
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Space efficiency in synopsis construction algorithms

5. Space efficiency in synopsis construction algorithms
Examples - I
Histograms
Many error measures
V-OPT, Jagadish etal, 1998
O(n2B) time O(nB) space
Only O(n) space at a time (working space)
O(n2B2) time and O(n) space
Is that the best ?
Here: O(n2B) time O(n) space.
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Space efficiency in synopsis construction algorithms

6. Space efficiency in synopsis construction algorithms
Example - II
(Haar) Wavelets
Orthonormal systems
For l2 error store the
largest B coeffs of input
Does not work for non l2
Find the best B coeffs to retain (note, restricted).
Garofalakis & Kumar, 04 O(n2B log B) time O(n2B) space, but O(nB) needed at a time (for l1 )
Here O(n) space, and O(n2) time
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Space efficiency in synopsis construction algorithms

7. Space efficiency in synopsis construction algorithms
Example - III
Extended Wavelets
Multiple measures
Optimization is similar to Knapsack with choices.
Previous best –
Deligiannakis and Rossopoulos, 04, O(Mn(B+ log n)) time and space O(MnB), but needing O(nM+MB) at a time
Guha, Kim, Shim, 04, reduced space to O(BM+min {nM,B2})
Here, O(BM) space
VLDB 2005
Space efficiency in synopsis construction algorithms

8. What we will not talk about
Approximation algorithms for histograms
Range Query Histograms
Basically improvement of a factor B in space across the board.
B is not always small, specially when n is large
VLDB 2005
Space efficiency in synopsis construction algorithms

9. Space efficiency in synopsis construction algorithms
The main idea
Can we solve using a non DP paradigm ?
Well, divide & conquer …
Small details – how do we divide ?
Interaction
Does a small interaction partitioning exist ?
How (much size) to represent it ?
Ease of finding it (in the given representation) ?
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Space efficiency in synopsis construction algorithms

10. A case study - Histograms
Formally, given a signal X find a piecewise constant representation H with at most B pieces minimizing ||X-H||2
Consider one bucket.
The mean is the best value.
A natural DP …
VLDB 2005
Space efficiency in synopsis construction algorithms

11. Space efficiency in synopsis construction algorithms
The DP for histograms
Err[i,b] = Error of approximating x1,…,xi using b buckets
For i=1 to n do
For 2 to B do
For j=1 to i-1 do
Err[i,b] = min Err[i,b], Err[j,b-1] + error(j+1,i) B n
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Space efficiency in synopsis construction algorithms

12. Space efficiency in synopsis construction algorithms
What if
We could figure out what was the story at the middlepoint !
Two questions
So what ?
How ? (use a DP)
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Space efficiency in synopsis construction algorithms

13. Space efficiency in synopsis construction algorithms
Wait a minute …
We just replaced a DP by another and claimed something … !!!
Exactly. The second DP needs only O(n) space. So as the conquer steps re-use/share the same space; the total space is O(n) too.
The idea is to use divide and conquer; and use a (small) DP to find the divide step.
Is it really that simple ?
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Space efficiency in synopsis construction algorithms

14. Space efficiency in synopsis construction algorithms
The code
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Space efficiency in synopsis construction algorithms

15. The end of working space
If you can partition a problem using the working space – you can recompute the solution of the parts at a little extra cost.
Working space = total space.
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Space efficiency in synopsis construction algorithms

16. Space efficiency in synopsis construction algorithms
How much is little ?
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Space efficiency in synopsis construction algorithms

17. Space efficiency in synopsis construction algorithms
Wavelets
A set of vectors
{1,-1,0,0,0,0…}, {0,0,1,-1,0,0,…},{0,0,0,0,1,-1,0,0},{0,0,0,0,0,0,1-1}
{1,1,-1,-1,0,0,0,0},{0,0,0,0,1,1,-1,-1}
{1,1,1,1,-1,-1,-1,-1},{1,1,1,1,1,1,1,1}
A natural multi-resolution
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Space efficiency in synopsis construction algorithms

18. Wavelet Synopsis Construction
Formally, given a signal X and the Haar basis {i} find a representation F=i zi i with at most B non-zero zi minimizing some error which a fn of X-F
Restriction. Zi is either 0 or h X,i i
Debate. Unrestricted or restricted. Omit.
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Space efficiency in synopsis construction algorithms

19. Space efficiency in synopsis construction algorithms
Wavelets
||X-F||1
Long history
Matias, Vitter Wang ’98
Garofalakis, Gibbons, ’02
Garofalakis, Kumar, ’04
State of the Art
O(n2B log B) time
O(n2B) space
O(nB) working space
Here O(n2log B) time O(n) space
SEE ALSO NEXT TALK …
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Space efficiency in synopsis construction algorithms

20. What happens to wavelets [GK04] ?
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Space efficiency in synopsis construction algorithms

21. Space efficiency in synopsis construction algorithms
Extensions
Approximation Algorithms
Range Query Histograms
Extended Wavelets
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Space efficiency in synopsis construction algorithms

22. Space efficiency in synopsis construction algorithms
Histograms
Saves space across all algorithms except algorithms which extend to general error measure over streams
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Space efficiency in synopsis construction algorithms

23. Space efficiency in synopsis construction algorithms
Range Query
Same story
Open Q:
faster algorithm obeying synopsis size
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Space efficiency in synopsis construction algorithms

24. Space efficiency in synopsis construction algorithms
That’s all folks
VLDB 2005
Space efficiency in synopsis construction algorithms