1. Aggregation Algorithms and Instance Optimality
Moni Naor
Weizmann Institute
Joint work with Ron Fagin Amnon Lotem

2. Aggregating information from several lists/sources
Define the problem
Ways to evaluate algorithms
New algorithms
Further Research

3. The problem Database D of N objects
An object R has m fields - (x1, x2, , xm)
Each xi  0,1
The objects are given in m lists L1, L2, , Lm
list Li all objects sorted by xi value.
An aggregation function t(x1,x2,…xm)
t(x1,x2,…xm) - a monotone increasing function
Wanted: top k objects according to t

4. Goal Touch as few objects as possible Access to object? List L1
r2= 0.75
r1= 0.5
b2= 0.3
a1= 0.4
c2= 0.2

5. Where?
Problem arises when combining information from several sources/criteria
Concentrate on middleware complexity without changing subsystems

6. Example: Combining Fuzzy Information
Lists are results of query: ``find object with
color `red’ and shape `round’”
Subsystems for color and for shape.
Each returns a score in [0,1] for each object
Aggregating function t is how the middleware system should combine the two criteria
Example: t(R=(x1,x2 )) could be min(x1,x2 )

7. Example: scheduling pages
Each object - page in a data broadcast system
1st field -  of users requesting the page
2nd field - longest time user is waiting
Combining function t - product of the two fields (geometric mean)
Goal: find the page with the largest product

8. Example: Information Retrieval
Documents D1 D2 Dk T2 T1 Tn
Terms
W12
Query T1, T2, T3: find documents with largest sum of entries
Aggregation function t is  xi

9. Modes of Access to the Lists
Sequential/sorted access: obtain next object in list Li
cost cS
Random access: for object R and i m obtain xi
cost cR
Cost of an execution:
cS  ( of seq. access)  cR  (  of random access)

10. Interesting Cases cR /cS is small cS  cR or cR >> cS
Number of lists m - small

11. Fagin’s Algorithm - FA
For all lists L1, L2, , Lm get next object in sorted order.
Stop when there is set of k objects that appeared in all lists.
For every object R encountered
retrieve all fields x1, x2, , xm.
Compute t(x1,x2,…xm)
Return top k objects

12. Correctness of FA...
For any monotone t and any database D of objects, FA finds the top k objects.
Proof: any object in the real top k is better in at least one field than the objects in intersection.

13. Performance of FA
Performance : assuming that the fields are independent (N(m-1)/m).
Better performance - correlation between fields
Worse performance - negative correlation
Bad aggregating function: max

14. Goals of this work
Improve complexity and analysis - worst case not meaningful
Instead consider Instance Optimality
Expand the range of functions want to handle all monotone aggregating functions
Simplify implementation

15. cost(A,D)  c1 cost(A’,D)  c2.
Instance Optimality
A = class of algorithms,
D = class of legal inputs.
For AA and DD measure cost(A,D) 0.
An algorithm AA is instance optimal over A and D if there are constants c1 and c2 s.t.
For every A’A and DD
cost(A,D)  c1 cost(A’,D)  c2.
c1 is called the optimality ratio

16. Offline  Nondeterminism
…Instance Optimality
Common in competitive online analysis
Compare an online decision making algorithm to the best offline one.
Approximation Algorithms
Compare the size that the best algorithm can find to the one the approx. algorithm finds
In our case
Offline  Nondeterminism

17. …Instance Optimality
We show algorithms that are instance optimal for a variety of
Classes of algorithms
deterministic, Probabilistic, Approximate
Databases
access cost functions

18. Guidelines for Design of Algorithms
Format: do sequential/sorted access (with random access on other fields) until you know that you have seen the top k.
In general: greedy gathering of information; If a query might allow you to know top k objects do it.
Works in all considered scenarios

19. The Threshold Algorithm - TA
For all lists L1, L2, , Lm get next object in sorted order.
For each object R returned
Retrieve all fields x1,x2,,xm.
Compute t(x1,x2,…xm)
If one of top k answers so far - remember it.
1im let xi be bottom value seen in Li (so far)
Define the threshold value  to be t(x1,x2,…xm)
Stop when found k objects with t value  .
Return top k objects

20. Maintained Information
Example: m=2, k=1, t is min
Top object (so far) =
Bottom values x1 = x2 =
Threshold t =
c , t(c) = 1/12
r , t(r) =1/8
b , t(b) = 1/11
0.9
0.7
0.4
Maintained Information
0.1
3/4
1/2
2/3
1/4
0.4
2/3
3/4
0.1
c1= 0.9
s2= 3/4
c = (0.9, 1/12)
s = (0.05,3/4)
b1= 0.7
w2= 2/3
w = (0.07, 2/3)
b = (0.7, 1/11)
r1= 0.4
z2= 1/2
r = (0.4, 1/8)
z = (0.09, 1/2)
a1= 0.1
q2= 1/4
q = (0.08, 1/4)
a = (0.1, 1/13)

21. Correctness of TA  t(z1, z2,…zm)  t(x1,x2,…xm)  
For any monotone t and any database D of objects, TA finds the top k objects.
Proof: If object z was not seen
 1im zi  xi
 t(z1, z2,…zm)  t(x1,x2,…xm)  

22. Implementation of TA Requires only bounded buffers: Top k objects
Bottom m values x1,x2,…xm

23. Robustness of TA Approximation: Suppose want an (1) approx.
- for any R returned and R’ not returned
t(R’)  (1) t(R)
Modified stopping condition:
Stop when found k objects with t value at least /(1).
Early Stopping: can modify TA so that at any point user is
Given current view of top k list
Given a guarantee about  approximation

24. Instance Optimality
Intuition: Cannot stop any sooner, since the next object to be explored might have the threshold value.
But, life is a bit more delicate...

25. Wild Guesses
Wild guesses: random access for a field i of object R that has not been sequentially accessed before
Neither FA nor TA use wild guesses
Subsystem might not allow wild guesses
More exotic queries: jth position in ith list...

26. Instance Optimality- No Wild Guesses
Theorem: For any monotone t let
A be the class of algorithms that
correctly find top k answers for every database with aggregation function t.
Do not make wild guesses
D be the class of all databases.
Then TA is instance optimal over A and D
Optimality ratio is m+m2 ·cR/cS - best possible!

27. Proof of Optimality
Claim: If TA gets to iteration d, then any (correct) algorithm A’ must get to depth d-1
Proof: let Rmax be top object returned by TA
(d)  t(Rmax)  (d-1)
There exists D’ with R’ at level d-1
R’ (x1(d-1), x2 (d-1),…xm(d-1) )
Where A’ fails

28. Do wild guesses help? Aggregation function - min, k=1
Database … n n1 … 2n1
1 1 … …0
0 0 … …1
L1 : 1 2 … n n1 … 2n1
L2 : 2n1 … n1 n …1
Wild guess: access object n1 and top elements

29. t(x1, x2,…xm)  t(x’1,x’2,…x’m)
Strict Monotonicity
An aggregation function t is strictly monotone if
when 1im xi  x’i
Then
t(x1, x2,…xm)  t(x’1,x’2,…x’m)
Examples: min, max, avg...

30. Instance Optimality - Wild Guesses
Theorem: For any strictly monotone t let
A be the class of algorithms that
correctly find top k answers for every database.
D be the class of all databases with distinct values in each field.
Then TA is instance optimal over A and D
Optimality Ratio is c · m where c=max{cR /cS ,cS /cR }

31. Related Work
An algorithm similar to TA was discovered independently by two other groups
Nepal and Ramakrishna
Gntzer, Balke and Kiessling
No instance optimality analysis
Hence proposed modifications that are not instance optimal algorithm
Power of Abstraction?

32. Dealing with the Cost of Random Access
In some scenarios random access may be impossible
Cannot ask a major search engine for it internal score on some document
In some scenarios random access may be expensive
Cost corresponds to disk access (seq. vs. random)
Need algorithms to deal with these scenarios
NRA - No Random Access
CA - Combined Algorithm

33. No Random Access - NRA March down the lists getting the next object
Maintain:
For any object R with discovered fields S1,..,m:
W(R)  t(x1,x2,…,x|S|,,0…0)
Worst (smallest) value t(R) can obtain
B(R)  t(x1,x2,…,x|S|, x|S|+1,, …, xm)
Best (largest) value t(R) can obtain

34. …maintained information (NRA)
Top k list, based on k largest W(R) seen so far
Ties broken according to B values
Define Mk to be the kth largest W(R) in top k list
An object R is viable if B(R)  Mk
Stop when there are no viable elements left I.e. B(R)  Mk for all R top list
Return the top k list

35. Correctness  no other objects with t(R)  Ck
For any monotone t and any database D of objects, NRA finds the top k objects.
Proof: At any point, for all objects t(R)B(R)
Once B(R)  Ck for all but top list
 no other objects with t(R)  Ck

36. Optimality Theorem: For any monotone t let
A be the class of algorithms that
correctly find top k answers for every database.
make only sequential access
D be the class of all databases.
Then NRA is instance optimal over A and D
Optimality Ratio is m !

37. Implementation of NRA
Not so simple - need to update B(R) for all existing R when x1,x2,…xm changes
For specific aggregation functions (min) good data structures
Open Problem: Which aggregation function have good data structures?

38. Combined Algorithm CA Maintain information as in NRA
Can combine TA and NRA
Let h = cR /cS
Maintain information as in NRA
For every h sequential accesses:
Do m random access on an objects from each list. Choose top viable for which not all fields are known

39. Instance Optimality Instance optimality statement a bit more complex
Under certain assumptions (including t = min, sum) CA is instance optimal with optimality ratio ~ 2m

40. Further Research Middleware Scenario:
Better implementations of NRA
Is large storage essential
Additional useful information in each list?
How widely applicable is instance optimality?
String Matching, Stable Marriage...
Aggregation functions and methods in other scenarios
Rank Aggregation of Search Engines
P=NP?

41. More Details
See