Aggregation Algorithms and Instance Optimality

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Aggregation Algorithms and Instance Optimality

• Moni Naor

Weizmann Institute
Joint work with Ron Fagin Amnon Lotem
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Aggregating information from several lists/sources

• Define the problem
• Ways to evaluate algorithms
• New algorithms
• Further Research

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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

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Goal

• Touch as few objects as possible
• Access to object?

List L2
List L1
s2 0.85
a2 0.84
r2 0.75
s1 0.65
r1 0.5
b2 0.3
a1 0.4
c2 0.2
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Where?

• Problem arises when combining information from
  several sources/criteria
• Concentrate on middleware complexity without
  changing subsystems

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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 )

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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

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Example Information Retrieval
Documents
Dk
D1
D2
W12
Query T1, T2, T3 find documents with largest sum
of entries
Aggregation function t is ? xi
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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)

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Interesting Cases

• cR /cS is small
• cS ? cR or
• cR gtgt cS
• Number of lists m - small

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Fagins 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

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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.

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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

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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

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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

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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

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Instance Optimality

• We show algorithms that are instance optimal for
  a variety of
• Classes of algorithms
• deterministic, Probabilistic, Approximate
• Databases
• access cost functions

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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

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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

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Example m2, k1, t is min
b , t(b) 1/11
c , t(c) 1/12

• Top object (so far)
• Bottom values x1 x2
• Threshold t

r , t(r) 1/8
Maintained Information
0.9
0.7
0.4
0.1
3/4
2/3
1/2
1/4
0.4
0.1
2/3
3/4
s (0.05,3/4)
c (0.9, 1/12)
w (0.07, 2/3)
b (0.7, 1/11)
z (0.09, 1/2)
r (0.4, 1/8)
q (0.08, 1/4)
a (0.1, 1/13)
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Correctness of TA

• 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) ? ?

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Implementation of TA

• Requires only bounded buffers
• Top k objects
• Bottom m values x1,x2,xm

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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

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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...

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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...

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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 mm2 cR/cS - best possible!

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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

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Do wild guesses help?

• Aggregation function - min, k1
• Database - 1 2 n n?1 2n?1
• 1 1 1 1 0 0 0
• 0 0 0 1 1 1 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

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Strict Monotonicity

• An aggregation function t is strictly monotone if
• when ?1?i?m xi ? xi
• Then
• t(x1, x2,xm) ? t(x1,x2,xm)
• Examples min, max, avg...

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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
  cmaxcR /cS ,cS /cR

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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?
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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

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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,,xS,,00)
• Worst (smallest) value t(R) can obtain
• B(R) ? t(x1,x2,,xS, xS1,, , xm)
• Best (largest) value t(R) can obtain

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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

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Correctness

• 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

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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 !

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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?

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Combined Algorithm CA

• 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

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Instance Optimality

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

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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
• PNP?

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More Details

• See
• www.wisdom.weizmann.ac.il/naor/PAPERS/middle_agg.
  html