url - Universität zu Lübeck

7.4. EVALUATION 137
Figure 7.5: Measurements of the intersection algorithm
7.4.2 Evaluation of Satisfiability
In a second scenario we evaluated the more complex algorithm checking the satisfiability
of two path expressions p, p ′ ∈ XP {[],∗,//,NOT } . The algorithm has an
exponential runtime because all variables must be checked in order to satisfy
the qualifiers. In order to show that the exponential complexity is no significant
limitation in the database context we determine the expected value for the times
expenses for typical operations with an average case simulation.
In general, the expected value EX of a random variable X : Ω → R is
∫
EX = X(ω) · p(ω) dω
ω∈Ω
with Ω = XP {[],∗,//,NOT } the set of all path expressions, X(ω) is the runtime of the
algorithm for a specific path expression ω ∈ Ω and p(ω) its probability.
There are three variables for a path expression: the length l p of the path linearize(p)
which affects the intersection algorithm. Because intersection is determined very
quickly (see previous experiment) l p has only a little effect. The second and third
variables are the number of qualifiers and element names in a path expression.
The element names correspond to the variables in 3SAT. Because Ω is infinite in
general we have to restrict it to a reasonable boundary: We think that 100 qualifiers
in a path expression provide a realistic upper value for database operations
(comparable to a SELECT statement with 100 WHERE clauses in SQL). Second,
because we have no distribution function for the probabilities of path expressions
we assume an equal distribution.