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Multi-dimensional information and queries

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• Review of the indexes studied so far....
  • So far, we have studied the following 3 index structures:

    Sorted indexes: We studied the commonly used representative of sorted index -- the B+-tree index: Hashing-based indexes:

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  • Common property of these indexes:

    The search key values are values taken from a one-dimensional space/set

    Example:

    If we index the search keys that are names, the search key values can be ordered on a one-dimensional space: Search keys: AAAA < AAAB < .... < KLM < .... < ZZZZ

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• Multi-dimensional Information
  • There are information that are naturally multi-dimensional

    Examples:

    Geographic information: Circuit board:

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• Multi-dimensional queries

  • Queries can be formulated on multi-dimensional data.

    Example 1: Nearest point query

    Suppose we have a relation containing points on a map Each point is stored in the following relation: Point( x, y ) Given a point p0(10, 20) Query: Find the point that is closest to point p0(10, 20) In SQL: select * from Point P /* P is the closest point to p0 */ where not exists (select * from Point Q /* A closer point Q cannot exist ! */ where "distance(Q, p0) < distance(P, p0)") select * from Point P /* P is the closest point to p0 */ where not exists (select * from Point Q /* A closer point Q cannot exist ! */ where (Q.x-10)2 + (Q.y-20)2 < (P.x-10)2 + (P.y-20)2 )

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  • Example 2: containment query

    Suppose we have a relation that define rectanglar objects: Rectangle( id, xll, yll, xur, yur ) A rectangle can be represented (= defined) by the coordinates of the Lower Left and Upper Right points. Example of representation of a rectangle: Given a point P(10,20) Query: Find all rectangles that contains the point P(10,20) Answer in SQL: select id from rectangle where xll ≤ 10 AND yll ≤ 20 AND 10 ≤ xur AND 20 ≤ yur Graphically:

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• "Commonly used" (or "formulated") multi-dimensional queries
  • Partial Match queries

    The query specifies conditions on some dimensions but not on all dimensions

    Example:

    Find all points/objects that intersects with y = 50

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  • Range queries

    Find objects that are located either partial or wholly within a certain range

    Example:

    Find all objects that have an overlap with the green area (= a range specification): The green area can be specified as: x1 ≤ x ≤ x2 y1 ≤ y ≤ y2

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  • Nearest neighbor queries:

    Find object(s) that is located closest (nearest) to a given point:

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  • Where-am-I queries:

    Given a location (i.e., coordinate) Find the object(s) that contains the location:

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• Multi-dimensional indexes
  • Multi-dimensional index:

    Multi-dimensional index = an index on the multi-dimensional data that help us answer a (commonly used) multi-dimensional query (more) efficiently (= speedier)

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  • Before we develop such multi-dimensional index, we answer this question first:

    Can we use a one-dimensional index to help us answer (commonly used) multi-dimensional queries (more) efficiently (= speedier) ??? (If so, we don't need to do anything !!!!)