url - Universität zu Lübeck

136 CHAPTER 7. THE XML INDEX UPDATE PROBLEM
1. if (!intersection(linearize(p), linearize(p) ′ )) return false;
2. NEG = getNegV ars(p) ∪ getNegV ars(p ′ );
3. P OS = getP osV ars(p) ∪ getP osV ars(p ′ );
4. NonT rivialV = NEG ∩ P OS;
5. forEach permutation of vars ∈ NonT rivialV
if (is satisfiable(p ∧ p ′ ,vars)) return true;
6. return false;
Figure 7.4: Pseudo code of the intersection algorithm for XIP NOT
and checks their satisfiability. If we find no satisfying configuration the algorithm
returns false because the expressions are mutually exclusive.
The optimization steps lead to a substantial acceleration as we show in the next
section. Anyhow, the algorithm still has an exponential complexity leading to
exhaustive computations in general. In the context of real world’s XPath based
database operations with a limited number of variables and qualifiers the satisfiability
of path expressions can be decided in acceptable time: With an average case
simulation (see next section) we determined an expected value of 155 milliseconds
for testing the satisfiability of XPath expressions with 100 qualifiers and up to 150
variables.
7.4 Evaluation
In this section we present performance measurements for the introduced algorithms
that decide the intersection and satisfiability of two path expressions. The
implementations were done in Java; the tests were performed on a Pentium 4 with
2.66 GHz and 1 GB main memory.
7.4.1 Evaluation of Intersection
The first evaluation tests the intersection algorithm of figure 7.3 for path expressions
p, p ′ ∈ XP {[],∗,//} . We increased the length of the path expressions p and p ′
from 1 to 20; the location steps of the expressions were created randomly from
a finite alphabet Σ. The time measurements include the parsing of the path expressions,
the creation of the three automata and the analysis whether the final
state is reachable. In order to obtain stable values we executed the algorithm
some thousand times with path expressions of the same length. The measured
total time is divided afterwards to get a stable average execution time for one run.
As the diagram in figure 7.5 shows the execution time of the optimized algorithm
increases almost linearly.
The optimized algorithm that avoids the construction of the full product automaton
has an average runtime of less than 1 millisecond for the largest path expression.