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90 CHAPTER 5. THE KEY-ORIENTED XML INDEX KEYX
The postprocessing raises additional costs but if the set of elements referenced by
v j is small the total costs of the query evaluation with the index will still outperform
the exhaustive evaluation over the whole document.
If there is a wildcard operator in the path expressions of the query the index
may not be suitable anymore because it does not cover all requested elements.
For instance, the index defined by /site/regions/asia/item[name=’x’] indexes
all items located in asia. The query /site/regions/*/item[name=’Sinus
MP3 Player’] does not have a regional restriction. We could calculate the path
expression p △ = ../∗ that navigates from the item elements in asia to all children
of its parent but this would not lead to success because the index does not cover
the values of name elements that do not belong to asia. The decision whether
we can use an index or not relies on the subset relationship (containment) of the
corresponding keys.
5.4.3 Containment Problem
In general, a selective index covers all queries with a result set being a subset of
the query that defined the index. For instance, an index that is designed to accelerate
queries of the form /dblp/ ∗ [author = ′ x ′ ] is also capable of evaluating queries
like /dblp/book[author = ′ x ′ ] or /dblp/article[author = ′ x ′ ] because the selected keys
are a subset of the keys of the index.
When using an index that covers a superset of the elements that are selected by
the query an additional postprocessing step has to filter wrong hits: A simple
node test checks if the selected nodes are of the requested element type (e.g. an
element selected by ∗ is checked if it has the label book). Like in the previous case
the postprocessing requires linear complexity in the size of the elements that are
returned by the index.
If an index is defined with a non-empty set of qualifiers (e.g. only books with an
isbn child) it cannot be used to process a query that ignores the qualifier because
the index does not cover all requested elements. In contrast, a query that poses
more qualifiers than the index can be processed by the index with additional postprocessing
because the query’s result is a subset of the elements that are indexed.
The decision whether the selected nodes of one XPath expression p are a subset
of the result set of a second expression p ′ (p ⊆ p ′ ) can be solved using the containment
algorithm presented by Miklau and Suciu [82]. This algorithm constructs
tree patterns for the path expressions p and p ′ and creates two (alternating) tree
automata A and A ′ accepting XML data that can be queried by p and p ′ . Containment
holds (p ⊆ p ′ ) if lang(A) ⊆ lang(A ′ ). A third automaton A ′′ accepting the
complement of lang(A ′ ) (lang(A ′ )) is built on the base of A ′ by exchanging all accepting
states with the non-accepting states. If the product automaton B = A x A ′′
has no reachable accepting state it holds, that lang(A)∩lang(A ′ ) = ∅. This is equiv-