Processing Ranked Queries with the Minimum Space

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Processing Ranked Queries with the Minimum Space

• Yufei Tao City University of Hong Kong
• Marios Hadjieleftheriou Boston University

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Outline

• Top-k
• Top-(k, K)

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Top-k definition

• A top-k query specifies two non-negative weights
  w1, w2.
• If a point has coordinates (x, y), its score
  equals w1 ? x w2 ? y.
• The goal is to report the k objects with the
  largest scores.
• The definition extends to higher dimensionalities
  in a straightforward manner.

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Top-k applications

• Any application where multi-dimensional points
  need to be ranked in various ways, according to
  different weights on the dimensions.
• Student ranking
• Dimensions scores in courses algorithm,
  database,
• Property sales
• Price, size, security, distance to the town
  center,
• Stock marketing
• Price, expected growth percentage, risk,

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An observation

• In practice, users are typically interested in
  top-k queries with small k.
• E.g., most often, people talk about top-10.
• The largest k that may be queried is usually
  orders of magnitude smaller than the dataset
  cardinality.

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Goal

• Let c be the largest k that will be supported by
  our top-k system.
• We aim at computing the union of the objects
  retrieved by all top-k queries, when k is at most
  c.
• Note that, even if k is fixed, there are still an
  infinite number of queries, which differ in
  weights.
• For example, if c 1, the unionconstitutes the
  convex hull.

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Why the goal?

• Goal compute the union of the objects retrieved
  by all top-k queries, when k is at most c.
• We refer to the union as the minimum covering
  subset.
• Facts
• The subset contains the data that must be
  retained by any algorithm that correctly answers
  all top-k queries, for k c.
• Once the subset is computed, we immediately
  improve the asymptotical performance of any
  algorithm.
• E.g., if the algorithm answers a top-k query in
  log(n) time, now it takes log(m) time, where m is
  the size of the subset.

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Previous results

• Heuristic results?
• Many!
• A very well studied problem in, for example,
  databases.
• These solutions focus on deriving heuristics that
  help solve the problem with small cost in most
  cases, but not in the worst case.
• Theoretical results?
• See next.

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Onion Chang et al. SIGMOD 2000
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Ranking Index Tsaparas et al. ICDE 2003
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Our result 1

• The minimum covering subset
• The result holds in any dimensionality.

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Example

• c 2
• PCH(D) p2, p3, p4, p5
• Add the above result into the 2-minimumcovering
  subset.
• Let us remove p2 from D.
• PCH(D - p2) p3, p4, p5
• Union the above result into the
  2-minimumcovering subset, which incurs no
  change.
• Then, compute PCH(D - p3)
• Eventually, the 2-minimum covering subset p2,
  p3, p4, p5

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Our result 2

• Time for computing the minimum covering subset
• where ? is the cost of computing the convex
  hull, and ? the size of the convex hull.
• Remember this works for any dimensionality.

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An alternative approach

• Next, lets try to do better in 2D space.
• The following algorithm has an interesting
  property It is incremental.
• It first finds the 1-minimum covering subset,
  then the 2-, 3-,

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Slope space

• Weights 10, 20 retrieve the same top-k results as
  weights 1, 2.

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Top-k decomposition

• An example with k 1.
• 0, slope of l1), slope of l1, slope of l2),
  slope of l1, slope of l3), slope of l3, 8)
• All slopes in the same interval have the same
  top-1 result.
• In the same way, we can also define top-k
  decomposition.

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Tail set

• Illustration of tail set.

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Our result 3
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Our result 4

• The 2D minimum covering subset can be computed in
  
• O(? ? m ? c)
• time, where m is the size of the subset.
• Compare it with our earlier result
• A side result the total number of top-c results
  is O(m).

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A 2D ranked index
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Our result 5

• Any top-k query with k c can be answered in
• O(logB(m / B) c / B)
• I/Os, using
• O(m / B)
• space, where m is the size of the c-minimum
  covering subset.

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Outline

• Top-k
• Top-(k, K)

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Approximate top-k retrieval

• In practice, slight (bounded) imprecision in the
  top-k result can often be tolerated
• especially if retrieval of such results
  requires less space and computation overhead.
• We propose top-(k, K) search.
• For a top-(k, K) query, a correct result contains
  any k objects among the top-K objects.
• For example, for a top-(1, 10) query, we may
  return any of the top-1, top-2, , and top-10
  objects.
• Quality guarantee even in the worst case, the
  returned object is the top-10.

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Our goal

• Find a small subset of the database that is
  sufficient for answering all top-(k, C) queries
  correctly, for any k c, where c and C are
  system constants, and c C.
• c controls the largest number of objects
  returned, and C determines the worst-case quality
  of the returned objects.
• For instance, if c 5 and C 10, then
  regardless of the query weights, the subset
  should allow retrieval of 5 objects, each of
  which is no worse than a top-10 object.
• In general, we refer to a subset satisfying our
  requirement as (c, C)-covering subset.

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Obvious results

• Result 1
• is a subset that can answer any top-(k, C) query
  with k c.
• But the subset may not be the minimum (c,
  C)-covering subset.
• Result 2 the minimum subset must be a subset of

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Our result

• We do not have a solution for finding the minimum
  (c, C)-covering subset
• but, in 2D space, we can find a covering subset
  that is larger than the minimum one by a factor
  of
  ln(? 1), where ? is the size of the minimum
  subset.

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Idea of our algorithm

• Assume c 1, C 2.
• Compute the top-2 decomposition.
• Region 1 p2, p4
• Region 2 p2, p1
• Region 3 p1, p3
• Let us consider region 1.
• Both p2 and p4 can be a legal answerfor a
  top-(1, 2) query whose queryvector falls in this
  region.

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Idea of our algorithm (cont.)

• Region 1 p2, p4
• Region 2 p2, p1
• Region 3 p1, p3
• We associate p1 with a legal set Region 3
• p2 with Regions 1, 2
• p3 with Region 3
• p4 with Region 1.
• Goal identify the smallest number ofpoints such
  that the union of theirlegal sets contains all
  regions.
• E.g., p2, p3.
• A minimal set cover problem.

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Query processing
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Conclusions

• top-(k, K) retrieval