Concise Description of Structured Sets

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Concise Description of Structured Sets

• Alberto O. Mendelzon
• Ken Q. Pu
• University of Toronto

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Motivation A true story

• An OLAP report writing software

GUI Range Selection
SQL Translation
RDBMS
Report Writer / Plotting
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The GUI
User selects the region of interest via
GUI CTRL-A selects ALL, Clickn-drag selects
LOTS.
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An Engineering Problem and Solution

• The Problem
• SQL statement has a length limit.
• User selection is LARGE, especially when the
  visualization is plotting.
• The Solution
• Break a large selection to a collection of SQL
  queries.
• The Complaint
• Kind of slow!

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A formalization

• A tree-structured categorical set (dimensions)
• A selected subset (user selection)
• Find the shortest possible description of the
  selected subset (translated SQL query)

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Outline

• Formal definition of structured sets, expressions
  and the Minimal Descriptive Length (MDL) problem
• Partition structures
• Hierarchical structures
• Multidimensional product structures

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The structured set

• A finite universe U
• A finite alphabet ?
• An interpretation function ? ? ? P(U)
• E.g. a 1, 2, 3 1 1
• U/? is the set cover naturally induced by ? on U.

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Expressions

• A language L
• An evaluation function ? L ? P(U)
• The Propositional Language L (U, ?) L ? L
  L L L L L
• The L (U, ?) L ? L L
• is union, is difference, is
  intersection.

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The L -MDL Problem

• Let V ? U, its descriptive lengthVL min
  s s V and s ? L
• The L -MDL decision problem VL ? k ?
• The L -MDL problemFind a (non-unique) compact
  (shortest) expression for V in L.

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Complexity Issues

• The L (U, ?)-MDL problem is NP-complete.
• The L-MDL problem is NP-complete. reduction
  from set-cover

Back to Engineering

• Is there anything good we can do?

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Tractable MDL-Problems

• Strategy reduce the complexity of the MDL
  problem by restricting to a class of structures.
• Two tractable cases
• Partition ? ?1 ? U, U/?1 is a partition
• Hierarchy ? ?N ? ?N-1 ? ?1 ? U, and U/?i ?
  U/?i1 for all i gt 0.

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Partition
Ottawa
San Jose
Toronto
Windsor
San Francisco
San Diego
Kingston
U Ottawa, Toronto, Windsor, Kingston
San Jose, San Francisco, San Diego
?1 Ontario, California
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Efficient Symbols

• Efficient symbols for V ?(V) ? ??1 ?
  ? V ? ? V 1

• V Ottawa, Toronto, Kingston, San Francisco
  ?(V) Ontario
• California is not efficient.

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Normal Form for Partitions

• N is a sub-language of L t (?1 ?2 ?3
  ) (a1 a2 a3 ) (b1 b2 b3 )
• t is N -compact for V if
• S ?(V)
• A V S
• B S V

S
A
B
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L-MDL for Partitions

• Every expression in L can be reduced to an
  equivalent expression in N !
• To compute a compact expression for V
• ? (V) ?(V)
• ?(V) (V ? (V) , ? (V) V )

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An Example
Ottawa
San Jose
Toronto
Windsor
San Francisco
San Diego
Kingston

• V Ottawa, Toronto, Kingston, San Francisco
  
• ? (V) Ontario , ?(V) (San Francisco,
  Windsor)
• t Ontario San Francisco Windsor

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A Normal Form for Hierarchy

• Levels (?0U) ? ?1 ? ?2 ? ?3 ? ?N
• A recursive definition for N The expression t
  is normal w.r.t level i ift t A B, with
  A, B ? ?i and t is normal w.r.t. level (i1).
• Examplet (( Canada California Ontario )
  Toronto San Jose)

t
A
B
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L-MDL for Hierarchies

• Every expression in L can be reduced to an
  equivalent expression in N !
• t t A B is a N -compact expression for V
  if
• t 1 ?1(V) and is N -compact
• (A, B) ?(V)

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An iterative algorithm
V
( VN AN - BN )
( s3 A3 - B3 )
( s1 A1 - B1 ) is compact
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Back to Engineering
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Multiple dimensions

• What about multi-dimensional sets
• V ? U1 ? U2
• U1 and U2 are hierarchically structured sets
• U1 has ? ?0 ? ?1 ? ?2 ?
• U2 has ? ?0 ? ?1 ? ?2 ?
• Form a new (product) structured set
• U U1 ? U2
• ? ? ? ?
• Interpretation

.
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Example
U1 -- Time Jan, Feb, Mar U2 --
Geography Ontario, California
Jan ? Ontario (Jan, Ontario)
L-MDL for product structure is NP-complete.
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Reduction by example

• Reduction from 3-set cover

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Related Work

• The generalized MDL approach for summarization,
  Lakshmanan, Ng, Wang, Zhou, Johnson, VLDB, 1999
• limited expressiveness for multidimensional data
  sets optimal expression is computed in
  polynomial time
• Multidimensional MDL problem resembles
  rectilinear polygon packing and covering problem
• Covering rectilinear polygons with axis-parallel
  rectangles, Kumar, Ramesh, ACM STOC, 1999

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Whats next
?

• More complex structures -- ordering
• More complex operators -- range operator
• Conjectured and known NP-hardness of the MDL
  problems
• L (?, ) is NP-complete , polynomial if ? is
  total.

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The End.