Carme Àlvarez

A very hard log-space counting class

Abstract We consider the logarithmic-space counting and optimization classes #L, span-L, and opt-L, which are defined analogously to their polynomial-time counterparts. We obtain complete functions for these three classes in terms of graphs and finite automata. We show that #L and opt-L are both included in NC 2 , but that, surprisingly, span-L seems to be a much harder class than #L and opt-L. We demonstrate that span-L functions can be computed in polynomial time if and only if P (#P) and all the classes of the polynomial-time hierarchy are included in P. This result follows from the fact that span-L and #P are very similar: span-L ⊆ #P, and any function in #P can be represented as the difference of a function in FL and a function in span-L. Nevertheless, the inclusion #P ⊆ span-L would imply NL = P = NP . We, furthermore, investigate restrictions of the classes opt-L and span-L.

A Characterization of Universal Stability in the Adversarial Queuing Model

We study universal stability of directed and undirected graphs in the adversarial queuing model for static packet routing. In this setting, packets are injected in some edge and have to traverse a predefined path before leaving the system. Restrictions on the allowed packet trajectory provide a way to analyze stability under different packet trajectories. We consider five packet trajectories, two for directed graphs and three for undirected graphs, and provide polynomial time algorithms for testing universal stability when considering each of them. In each case we obtain a different characterization of the universal stability property in terms of a set of forbidden subgraphs. Thus we show that variations of the allowed packet trajectory lead to nonequivalent characterizations. Using those characterizations we are also able to provide polynomial time algorithms for testing stability under the \NTGLIS (Nearest To Go-Longest In System) protocol.

A compendium of problems complete for symmetric logarithmic space

Abstract. The papers main contributions are a compendium of problems that are complete for symmetric logarithmic space (SL), a collection of material relating to SL, a list of open problems, and an extension to the number of problems known to be SL-complete. Complete problems are one method of studying SL, a class for which programming is non-intuitive. Our exposition helps make the class SL less mysterious and more accessible to other researchers.

A very hard log space counting class

Consideration is given to the logarithmic space counting classes Hash L, opt-L, and span-L, which are defined analogously to their polynomial-time counterparts. Complete functions are obtained for these three classes in terms of graphs and finite automata. It is shown that Hash L and opt-L are both contained in NC/sup 2/, but that, surprisingly, span-L seems to be much harder counting class than Hash L and opt-L. It is demonstrated that span-L-functions can be computed in polynomial time if and only if P=NP=PH=P( Hash P), i.e if the class P( Hash P) and all the classes of the polynomial-time hierarchy are contained in P. This result follows from the fact that span-L and Hash P are very similar: span-L contained in Hash P, and any function in Hash P can be represented as a subtraction of two functions in span-L. Nevertheless, Hash P contained in span-L would imply NL=P=NP. An investigation is also conducted of various restrictions of the classes opt-L and span-L, and it is shown, e.g that if opt-L coincides with one of its restricted versions, then L=NL follows.<<ETX>>

Efficient and reliable high level communication in randomly deployed wireless sensor networks

In this paper we present a way to establish a reliable and efficient high level communication system in a randomly deployed network of sensors equipped with directional antennas. Such high level communication system will enable the programming of the sensor network using high level communication functionalities without the burden of taking care of their physical capacities (low range, unidirectional links, single frequency, presence of collisions...). The high level communication functionalities we offer include point-to-point communication, point-to-area communication, and one-to-all communication. The basic idea to implement this system is to simulate a virtual network that emerges from the ad-hoc network using self-organization, self-discovery and collaborative methods. The analysis of the protocols we present shows their reasonable efficiency, scalability and robustness.

The impact of failure management on the stability of communication networks

In this work we deal with communication networks in which links may fail. We propose an adversarial model for describing the traffic pattern occurring in this type of faulty systems and study properties concerning their stability, especially under (nontrivial) underloaded worse-case scenarios. We show that, depending on how the system is organized and prepared to deal with failures, the dynamics of the system change and thus the conditions for stability. We propose three different ways of failure management and study how they influence on the stability of faulty communication networks under the adversarial model proposed. We show that some failure managements can provoke the instability of even very simple networks.

Adaptive logspace reducibility and parallel time

We discuss two notions of functional oracle for logarithmic space-bounded machines, which differ in whether there is only one oracle tape for both the query and the answer or a separate tape for the answer, which can still be read while the next query is already being constructed. The first notion turns out to be basically nonadaptive, behaving like access to an oracle set. The second notion, on the other hand, is adaptive. By imposing appropriate bounds on the number of functional oracle queries made in this computation model, we obtain new characterizations of the NC and AC hierarchies; thus the number of oracle queries can be considered as a measure of parallel time. Using this characterization of parallel classes, we solve open questions of Wilson.

Complexity Classes with Complete Problems Between P and NP-C

We study certain language classes located between P and NP that are defined by polynomial time machines with bounded amount of nondeterminism. We observe that these classes have complete problems, and find characterizations of the classes using robust machines with bounded access to the oracle, and in terms of nondeterministic complexity classes with polylog running time. We also study the relationship of these classes to P and NP.

The hardness of intervalizing four colored caterpillars

Abstract The problem of intervalizing colored graphs (ICG) has received a lot of attention due to their use as a model for DNA physical mapping with ambiguous data. If k is the number of colors, the problem is known to be NP-complete for general graphs for k ⩾4 and has polynomial time algorithms for k =2 and 3. In this paper we show that the ICG problem is NP-complete when the graph is a caterpillar tree, colored with k ⩾4 colors, strengthen the cases for which the problem remains difficult.

The HOM problem is decidable

We provide an algorithm that, given a tree homomorphism H and a regular tree language L represented by a tree automaton, determines whether H(L) is regular. This settles a question that has been open for a long time. Along the way, we develop new constructions and techniques which are interesting by themselves, and provide several significant intermediate results. For example, we prove that the universality problem is decidable for languages represented by tree automata with equality constraints, and that the equivalence and inclusion problems are decidable for images of regular tree languages through tree homomorphisms. Our algorithms are based on the following constructions. We describe a simple transformation for converting a tree automaton with equality constraints into a tree automaton with inequality constraints recognizing the complementary language. We also define a new class of automata with arbitrary inequality constraints and a particular kind of equality constraints. An automaton of this new class essentially recognizes the intersection of a tree automaton with inequality constraints and the image of a regular tree language through a tree homomorphism. We prove decidability of emptiness and finiteness for this class by a pumping mechanism.