Tamás Lukovszki

New Results of Fault Tolerant Geometric Spanners

We investigate the problem of constructing spanners for a given set of points that are tolerant for edge/vertex faults. Let S ⊂ IRd be a set of n points and let k be an integer number. A k-edge/vertex fault tolerant spanner for S has the property that after the deletion of k arbitrary edges/vertices each pair of points in the remaining graph is still connected by a short path. Recently it was shown that for each set S of n points there exists a k-edge/vertex fault tolerant spannerwith O(k2n) edges which can be constructed in O(nlogn+ k2n) time. Furthermore, it was shown that for each set S of n points there exists a k-edge/vertex fault tolerant spannerwhose degree is bouned by O(ck+1) for some constant c. Our first contribution is a construction of a k-vertex fault tolerant spanner with O(kn) edges which is a tight bound. The computation takes O(nlogd-1n + knloglogn) time. Then we show that the same k-vertex fault tolerant spanner is also k-edge fault tolerant. Thereafter, we construct a k-vertex fault tolerant spanner with O(k2n) edges whose degree is bounded by O(k2). Finally, we give a more natural but stronger definition of k-edge fault tolerance which not necessarily can be satisfied if one allows only simple edges between the points of S. We investigate the question whether Steiner points help. We answer this question affirmatively and prove ⊖(kn) bounds on the number of Steiner points and on the number of edges in such spanners.

Distributed Maintenance of Resource Efficient Wireless Network Topologies

Multiple hop routing in mobile ad hoc networks can minimize energy consumption and increase data throughput. Yet, the problem of radio interferences remains. However if the routes are restricted to a basic network based on local neighborhoods, these interferences can be reduced such that standard routing algorithms can be applied.We compare different network topologies for these basic networks with respect to degree, spanner-properties, radio interferences, energy, and congestion, i.e. the Yao-graph (aka. ?-graph) and some also known related models, which will be called the SymmY-graph (aka. YS-graph), the SparsY-graph (aka. YY-graph) and the BoundY-graph. Further, we present a promising network topology called the HL-graph (based on Hierarchical Layers).Further, we compare the ability of these topologies to handle dynamic changes of the network when radio stations appear and disappear. For this we measure the number of involved radio stations and present distributed algorithms for repairing the network structure.

Online Admission Control and Embedding of Service Chains

The virtualization and softwarization of modern computer networks enables the definition and fast deployment of novel network services called service chains: sequences of virtualized network functions e.g., firewalls, caches, traffic optimizers through which traffic is routed between source and destination. This paper attends to the problem of admitting and embedding a maximum number of service chains, i.e., a maximum number of source-destination pairs which are routed via a sequence of l to-be-allocated, capacitated network functions. We consider an Online variant of this maximum Service Chain Embedding Problem, short OSCEP, where requests arrive over time, in a worst-case manner. Our main contribution is a deterministic Ologl-competitive online algorithm, under the assumption that capacities are at least logarithmic in l. We show that this is asymptotically optimal within the class of deterministic and randomized online algorithms. We also explore lower bounds for offline approximation algorithms, and prove that the offline problem is APX-hard for unit capacities and smalli¾źli¾ź3, and even Poly-APX-hard in general, when there is no bound oni¾źl. These approximation lower bounds may be of independent interest, as they also extend to other problems such as Virtual Circuit Routing. Finally, we present an exact algorithm based on 0-1 programming, implying that the general offline SCEP is in NP and, by the above hardness results, it is NP-complete for constant l.

Worst case mobility in ad hoc networks

We investigate distributed algorithms for mobile ad hoc networks for moving radio stations with adjustable transmission power in a worst case scenario. We consider two models to find a reasonable restriction on the worst-case mobility. In the pedestrian model we assume a maximum speed vmax of the radio stations, while in the vehicular model we assume a maximum acceleration amax of the points.Our goal is to maintain persistent routes with nice communication network properties like hop distance, energy-consumption, congestion and number of interferences. A route is persistent, if we can guarantee that all edges of this route can be uphold for a given time span Δ, which is a parameter denoting the minimum time the mobile network needs to adopt changes, i.e. update routing tables, change directory entries, etc. This Δ can be used as the length of an update interval for a proactive routing scheme.We extend some known notions such as transmission range, interferences, spanner, power spanner and congestion to both mobility models and introduce a new parameter called crowdedness that states a lower bound on the number of radio interferences. Then we prove that a mobile spanner hosts a path system that polylogarithmically approximates the optimal congestion.We present distributed algorithms based on a grid clustering technique and a high-dimensional representation of the dynamical start situation which construct mobile spanners with low congestion, low interference number, low energy-consumption, and low degree. We measure the optimality of the output of our algorithm by comparing it with the optimal choice of persistent routes under the same circumstances with respect to pedestrian or vehicular worst-case movements. Finally, we present solutions for dynamic position information management under our mobility models.

It's a Match!: Near-Optimal and Incremental Middlebox Deployment

The virtualization and softwarization of modern computer networks offers new opportunities for the simplified management and exible placement of middleboxes as e.g. rewalls and proxies. This paper initiates the study of algorithmically exploiting the exibilities present in virtualized and software-defined networks. Particularly, we are interested in the initial as well as the incremental deployment of middleboxes. We present a deterministic O(log(min{n,k})) approximation algorithm for n-node computer networks, where k is the middlebox capacity. The algorithm is based on optimizing over a submodular function which can be computed efficiently using a fast augmenting path approach. The derived approximation bound is optimal: the underlying problem is computationally hard to approximate within sublogarithmic factors, unless P = NP holds. We additionally present an exact algorithm based on integer programming, and complement our formal analysis with simulations. In particular, we consider the number of used middleboxes and highlight the benefits of the approximation algorithm in incremental deployments. Our approach also finds interesting applications, e.g., in the context of incremental deployment of software-defined networks.

I/O-Efficient Well-Separated Pair Decomposition and Applications

We present an external-memory algorithm to compute a well-separated pair decomposition (WSPD) of a given point set S in ℝd in O(sort(N)) I/Os, where N is the number of points in S and sort(N) denotes the I/O-complexity of sorting N items. (Throughout this paper we assume that the dimension d is fixed.) As applications of the WSPD, we show how to compute a linear-size t-spanner for S within the same I/O-bound and how to solve the K-nearest-neighbour and K-closest-pair problems in O(sort(KN)) and O(sort(N+K)) I/Os, respectively.

Experiences with Phonebook-Centric Social Networks

Social networks are becoming more and more popular nowadays. The increasing capabilities of mobile phones enable them to participate in such networks. The phonebooks in the mobile devices represent social relationships, that can be integrated in the social networks. Following we refer to this solution as phonebook-centric social networks. Such networks provide a synchronization mechanism between phonebooks of the users and the social network which allows detecting network members listed in the phonebooks (semi-) automatically. The detection of those members is based on the similarity of the personal data, e.g., similar name, same phone number, address, etc... Users can mark similarities between their phonebook contacts and members in the network. After this, if one of their contacts changes her or his personal detail, it will be propagated automatically into the phonebooks, after considering privacy settings. Synchronization time and energy consumption for synchronization on the mobiles and the scalability of the system strongly depend on the number of users and similarities. We have implemented a phone book-centric social network, called Phonebookmark and investigated the structure of the network. We experienced that the distribution of similarities follows a power law, as well as the distribution of the in- and out-degrees in the social network. In this paper we propose a model for estimating the total number of similarities and we show that this estimation applies very well to the historical data of Phonebookmark.

Formal description of a distributed location service for mobile ad hoc networks

We define here a distributed abstract state machine (DASM) [7] of the network or routing layer of mobile ad hoc networks [13]. Such networks require routing strategies substantially different from those used in static communication networks, since storing and updating large routing tables at mobile hosts would congest the network with administration packets very fast. In [1], the hypercubic location service is presented, which considers a very strong definition of fault-tolerance thereby improving state-of-the-art ad hoc routing protocols in several respects. Our goal in modeling the protocols for the distributed location service and the position based routing is twofold. First, we support the definition and validation of wireless communication protocols and implementations based thereon. Second, we feel that the abstract computation model naturally reflects the layering principle of communication architectures in combination with an uncompromisingly local view of the application domain. Thus we can identify fundamental semantic concepts, such as concurrency, reactivity and asynchronism, directly with the related concepts as imposed by the given application context.

Target Surrounding Solution for Swarm Robots

In this paper we present a distributed algorithm, which enables to follow and surround moving objects by a swarm of homogenous robots that only use local sensing. We introduce the multi orbit surrounding problem and present a solution for it. We prove that our solution always guarantees that the robots enclose the target and circulate around them. We also evaluate our solution by simulations.

Similarity Distribution in Phonebook-Centric Social Networks

In this work we define the term of phonebook centric-social networks, describe a graph model and study structural properties of this. The key difference between phonebook-centric social networks and usual social networks is allowing synchronization between the phonebook of the mobile phone and the network. By the synchronization the goal is to identify the persons listed in the phonebook and the network, it means to find similar entries and keep the data consistent. In that way the contacts of a member of a phonebook-centric social network becomes independent from the current phone. The number of similarities is crucial in phonebook-centric social networks from scalability point of view. We found that the distribution of similarities can be very well approximated by power law distribution. This means that few users involve a huge amount of similarities while the most of them only few ones.