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1 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 Δ 1 can be translated into an equivalent database in the data model of Δ 2

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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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3 House2 IDXYT 1XYT2<=x, x<=6, 3<=y, y<=6, 100<=t, t<=200 2XYT8<=x, x<=11, 3<=y, y<=7, 150<=t, t<=300 3XYT2<=x, x<=4, 5<=y, y<=10, 250<=t, t<=400 3XYT2<=x, x<=10, 8<=y, y<=10, 250<=t, t<=400

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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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5 Example: IDXYT Fountain x y tx = 10, y = 4, 1980 <= t, t <= 1986 Road x y t5 <= x, x <= 9, y = -x+15, 1995 <= t, t <=1996 Road x y tx = 9, 3 <= y, y <= 6, 1995 <= t, t <=1996 Tulip x y t2 <= x, x <= 6,y <= 9-x, 3 <= y, y <= 7, 1975 <= t, t <= 1990 Park x y t1 <= x, x <= 12, 2 <= y, y <= 11, 1974 <= t, t <= 1996 Pond x y tx >= 3, y >= 5, y >= x-1, y <= x+5, y <= -x+13, 1991 <= t, t <= 1996

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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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7 Bomb2 X Y T x y tt <= x, x <= t+1, t <= y, y <= t+1, t 2 <= z, z <= t 2, 0 <= t, t <= 3.19

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8 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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9 Example: Net2 X Y T x y t y <= x - t, y (t+2) >= x t - t 2 - 2t + 6, y (t+2) >= x (t-2) – t

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10 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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11 Example : Tide2 X Y T x y t 1 = x - t’ + 3 x y t 1 = x + t’ - 8.5

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Parametric and Geometric Transformation Models Theorem: Let [ai, bi] for 1<=i<=d be any set of d intervals with ai < bi, Let R=(П i=1 d [X i [, X i ] ], [from, to]) be any normal form parametric rectangle. Let G=(П i=1 d [a i, b i ], [from, to], f) be any normal form geometric transformation object where f is definable as the system of equations x i =g i x i + h i where g i and h i are functions of t for 1<=i<=d. Then R and G are equivalent if:

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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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14 Constraint(Parametric) Extreme Point (Parametric) Geometric Transformation InequalityRectanglesIdentity transformation rectangle reference object x, y linear t inequalityWorboysIdentity transformation polygon reference object Each x i bounded by a function of t Parametric rectanglesParametric scaling +translation rectangle reference object x, y linear for each tParametric 2-spaghettiParametric affine motion polygon reference object Constraint(Parametric) Extreme Point (Parametric) Geometric Transformation InequalityRectanglesIdentity transformation rectangle reference object x, y linear t inequalityWorboysIdentity transformation polygon reference object Each x i bounded by a function of t Parametric rectanglesParametric scaling +translation rectangle reference object x, y linear for each tParametric 2-spaghettiParametric affine motion polygon reference object

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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 (d+1) dimensional constraint relation with polynomial constraints

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Query Interoperability Query interoperability via Query Translation Figure Query Interoperability via Data Translation Figure 14.5

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17 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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Query Interoperability via a common basis Figure 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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19 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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Intersection of Linear Parametric rectangles Theorem: Whether two d-dimensional linear parametric rectangles intersect can be checked in O(d) time.

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