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This bachelor thesis is concerned with arrangements of hyperplanes, that
is, finite collections of hyperplanes in a finite-dimensional vector
space. Such arrangements can be studied using methods from
combinatorics, topology or algebraic geometry. Our focus lies on an
algebraic object associated to an arrangement \(\mathcal{A}\), the module \(\mathcal{D(A)}\) of
logarithmic derivations along \(\mathcal{A}\). It was introduced by K. Saito in the
context of singularity theory, and intensively studied by Terao and
others. If \(\mathcal{D(A)}\) admits a basis, the arrangement \(\mathcal{A}\) is called free.
Ziegler generalized the concept of freeness to so-called
multiarrangements, where each hyperplane carries a multiplicity. Terao
conjectured that freeness of arrangements can be decided based on the
combinatorics. We pursue the analogous question for multiarrangements in
special cases. Firstly, we give a new proof of a result of Ziegler
stating that generic multiarrangements are totally non-free, that is,
non-free for any multiplicity. Our proof relies on the new concept of
unbalanced multiplicities. Secondly, we consider freeness asymptotically
for increasing multiplicity of a fixed hyperplane. We give an explicit
bound for the multiplicity where the freeness property has stabilized.

In 2003, a dictionary data structure called jumplist has been introduced by Brönnimann, Cazals and Durand. It is based on a circularly closed (singly) linked list, but additional jump-pointers are added to provide shortcuts to parts further ahead in the list.
The original jump-and-walk data structure by Brönnimann, Cazals and Durand only introduces one jump-pointer per node. In this thesis, I add one more-jump pointer to each node and present algorithms for generation, insertion and search for the resulting data structure.
Furthermore, I try to evaluate the effects on the expected search costs and the complexity of the generation and insertion.
It turns out that the two-jump-pointer variant of the jumplist has a slightly better prefactor (1.2 vs. 2) in the leading term of the expected internal path length than the original version and despite the more complex structure of the two-jump-pointer variant compared to the regular jumplist, the complexity of generation and insertion remains linearithmic.