14. Interoperability

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14. Interoperability

• Database interoperability ---
• Is the problem of making the data and queries of
  one database system usable to the users of
  another database system.
• Requires that the data models used in them have
  the same data expressiveness.
• Data expressiveness ---
• Database written in the data model used in ?1 can
  be translated into an equivalent database in the
  data model of ?2

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14.1.1 Constraint and Extreme Point Data Models

• Each database in the rectangles data model and
  Worboys data model is equivalent to a constraint
  database with some suitable types of constraints.
• Theorem
• Any rectangle relation R is equivalent to a
  constraint relation C with only inequality
  constraints between constants and variables.

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House2
ID X Y T
1 X Y T 2ltx, xlt6, 3lty, ylt6, 100ltt, tlt200
2 X Y T 8ltx, xlt11, 3lty, ylt7, 150ltt, tlt300
3 X Y T 2ltx, xlt4, 5lty, ylt10, 250ltt, tlt400
3 X Y T 2ltx, xlt10, 8lty, ylt10, 250ltt, tlt400
4

• Theorem
• Any Worboys relation W is equivalent to a
  constraint relation C with two spatial variables
  with linear constraints and one temporal variable
  with inequality constraints.

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• Example

ID X Y T
Fountain x y t x 10, y 4, 1980 lt t, t lt 1986
Road x y t 5 lt x, x lt 9, y -x15, 1995 lt t, t lt1996
Road x y t x 9, 3 lt y, y lt 6, 1995 lt t, t lt1996
Tulip x y t 2 lt x, x lt 6 ,y lt 9-x, 3 lt y, y lt 7, 1975 lt t, t lt 1990
Park x y t 1 lt x, x lt 12, 2 lt y, y lt 11, 1974 lt t, t lt 1996
Pond x y t x gt 3, y gt 5, y gt x-1, y lt x5, y lt -x13, 1991 lt t, t lt 1996
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14.1.2 Constraint and Parametric Extreme Point
Data Models

• Theorem
• Any parametric rectangle relation R with
  m-degree polynomial parametric functions of t is
  equivalent to a constraint relation C with
  inequality constraints in which the spatial
  variables are bound from above or below by
  m-degree polynomial functions of t and t is
  bounded from above and below by constants.

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Bomb2
X Y T
x y t t lt x, x lt t1, t lt y, y lt t1, 100 - 9.8 t2 lt z, z lt 102 - 9.8t2, 0 lt t, t lt 3.19
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• Theorem
• Any parametric 2-spaghetti relation W with
  quotient of polynomial functions of t is
  equivalent to a constraint relation C with
  polynomial constraints over the variables x, y,
  and t such that for each instance of t all the
  constraints are linear.

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• Example Net2

X Y T
x y t y lt x - t, y (t2) gt x t - t2 - 2t 6, y (t2) gt x (t-2) t2 16
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• Theorem
• Any periodic parametric 2-spaghetti relation
  with periodic parametric functions of t is
  equivalent to a constraint database relation with
  periodic constraints over the variables x, y, t
  such that for each instance of t all the
  constraints are linear.

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Example Tide2
X Y T
x y t 1 lt x, x lt 3, 1 lt y, y lt 4, 0 lt t, t lt5.75, y gt x - t 3
x y t 1 lt x, x lt 3, 1 lt y, y lt 4, 5.75 lt t, tlt11.5, y gt x t - 8.5
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14.1.3 Parametric and Geometric Transformation
Models

• Theorem
• Let ai, bi for 1ltiltd be any set of d
  intervals with ai lt bi, Let
• R(?i1dXi, Xi, from, to) be any normal
  form parametric rectangle. Let G(?i1dai, bi,
  from, to, f) be any normal form geometric
  transformation object where f is definable as the
  system of equations xigixi hi where gi and hi
  are functions of t for 1ltiltd. Then R and G are
  equivalent if

13

• Theorem
• Any parametric 2-spaghetti relation W with
  m-degree polynomial functions of t is equivalent
  to a two-dimensional parametric affine
  transformation object relation G with m-degree
  polynomial functions of t and a polygonal
  reference object.

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Constraint (Parametric) Extreme Point (Parametric) Geometric Transformation
Inequality Rectangles Identity transformation rectangle reference object
x, y linear t inequality Worboys Identity transformation polygon reference object
Each xi bounded by a function of t Parametric rectangles Parametric scaling translation rectangle reference object
x, y linear for each t Parametric 2-spaghetti Parametric affine motion polygon reference object
Constraint (Parametric) Extreme Point (Parametric) Geometric Transformation
Inequality Rectangles Identity transformation rectangle reference object
x, y linear t inequality Worboys Identity transformation polygon reference object
Each xi bounded by a function of t Parametric rectangles Parametric scaling translation rectangle reference object
x, y linear for each t Parametric 2-spaghetti Parametric affine motion polygon reference object
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14.1.4 Constraint and Geometric Transformation
Models

• Theorem
• Any d-dimensional parametric affine
  transformation object relation with m-degree
  polynomial function soft t can be represented as
  a (d1) dimensional constraint relation with
  polynomial constraints

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14.2 Query Interoperability

• 14.2.1 Query interoperability via Query
  Translation
• Figure 14.4.
• 14.2.2 Query Interoperability via Data
  Translation
• Figure 14.5

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• Theorem
• All the spatiotemporal models appearing in
  Figure 14.3 are closed under intersection,
  complement, union, join, projection, and
  selection with inequality constraints that
  contain spatiotemporal variables and constants.

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14.2.3 Query Interoperability via a common
basisFigure 14.7

• Precise data translation ---
• We can translate each of the spatiotemporal data
  models of Chapter 13 into a syntactically
  restricted type of constraint database. We can
  also easily compare the expressive power of
  several different data models by translating them
  to restricted types of constraint databases

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Advantages of common basis

• Easy query translation ---
• Many spatiotemporal query languages contain
  numerous spatial operators and other special
  language features.
• Safety and complexity ---
• By knowing the allowed syntax of the constraints
  in the common basis, we can gain valuable
  information about the safety and computational
  complexity of queries.

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14.2.4 Intersection of Linear Parametric
rectangles

• Theorem
• Whether two d-dimensional linear parametric
  rectangles intersect can be checked in O(d) time.