Self-Stabilizing Balancing Algorithm for Containment-Based Trees

Containment-based trees encompass various handy structures such as B+-trees, R-trees and M-trees. They are widely used to build data indexes, range-queryable overlays, publish/subscribe systems both in centralized and distributed contexts. In addition to their versatility, their balanced shape ensures an overall satisfactory performance. Re- cently, it has been shown that their distributed implementations can be fault-resilient. However, this robustness is achieved at the cost of un-balancing the structure. While the structure remains correct in terms of searchability, its performance can be significantly decreased. In this paper, we propose a distributed self-stabilizing algorithm to balance containment-based trees

Robustness of the Rotor-Router Mechanism

International audienceThe rotor-router model, also called the Propp machine, was first considered as a deter-ministic alternative to the random walk. The edges adjacent to each node v (or equivalently, the exit ports at v) are arranged in a fixed cyclic order, which does not change during the exploration. Each node v maintains a port pointer π(v) which indicates the exit port to be adopted by an agent on the conclusion of the next visit to this node (the "next exit port"). The rotor-router mechanism guarantees that after each consecutive visit at the same node, the pointer at this node is moved to the next port in the cyclic order. It is known that, in an undirected graph G with m edges, the route adopted by an agent controlled by the rotor-router mechanism forms eventually an Euler tour based on arcs obtained via replacing each edge in G by two arcs with opposite direction. The process of ushering the agent to an Euler tour is referred to as the lock-in problem. In [Yanovski et al., Algorithmica 37(3), 165–186 (2003)], it was proved that, independently of the initial configuration of the rotor-router mechanism in G, the agent locks-in in time bounded by 2mD, where D is the diameter of G. In this paper we examine the dependence of the lock-in time on the initial configuration of the rotor-router mechanism. Our analysis is performed in the form of a game between a player P intending to lock-in the agent in an Euler tour as quickly as possible and its adversary A with the counter objective. We consider all cases of who decides the initial cyclic orders and the initial values π(v). We show, for example, that if A provides its own port numbering after the initial setup of pointers by P, the complexity of the lock-in problem is O(m·min{log m, D}). We also investigate the robustness of the rotor-router graph exploration in presence of faults in the pointers π(v) or dynamic changes in the graph. We show, for example, that after the exploration establishes an Eulerian cycle, if k edges are added to the graph, then a new Eulerian cycle is established within O(km) steps

Euler Tour Lock-in Problem in the Rotor-Router Model

International audienceThe rotor-router model, also called the Propp machine, was first considered as a deterministic alternative to the random walk. It is known that the route in an undirected graph G=(V,E), where |V|=n and |E|=m, adopted by an agent controlled by the rotor-router mechanism forms eventually an Euler tour based on arcs obtained via replacing each edge in G by two arcs with opposite direction. The process of ushering the agent to an Euler tour is referred to as the lock-in problem. In recent work [Yan03] Yanovski et al. proved that independently of the initial configuration of the rotor-router mechanism in G the agent locks-in in time bounded by 2m diam, where diam is the diameter of G. This upper bound can be matched asymptotically in lollipop graphs. In this paper we examine the dependence of the lock-in time on the initial configuration of the rotor-router mechanism. The case study is performed in the form of a game between a player pl intending to lock-in the agent in an Euler tour as quickly as possible and its adversary ad with the counter objective. First, we observe that in certain (easy) cases the lock-in can be achieved in time O(m). On the other hand we show that if adversary ad is solely responsible for the assignment of ports and pointers, the lock-in time Omega(m diam) can be enforced in any graph with m edges and diameter diam. Furthermore, we show that if ad provides its own port numbering after the initial setup of pointers by pl, the complexity of the lock-in problem is bounded by O(m min{log m,diam}). We also propose a class of graphs in which the lock-in requires time Omega(m log m). In the remaining two cases we show that the lock-in requires time Omega(m diam) in graphs with the worst-case topology. In addition, however, we present non-trivial classes of graphs with a large diameter in which the lock-in time is O(m)

Routing and wavelength assignment in optical networks

We study models for routing and wavelength assignment in optical networks, aiming at showing properties of these models that must be taken into consideration when optical networks are deployed in practice. More specifically, we propose approximation algorithms for maximizing the number of satisfied requests in optical ring networks where the number of available wavelengths per fiber is given as part of the input. The proposed algorithms, which all possess a bounded approximation ratio, are also compared experimentally with other algorithms already known from the literature. From the comparison, we conclude that the algorithm with the theoretically best approximation ratio produces the best solutions but consumes too much running time. On the contrary, one of the proposed algorithms produces very satisfactory solutions with a running time several orders of magnitude faster than the time of the better algorithm. Moreover, we study a generalization of the problem where every communication request is associated with a given profit, and we seek to maximize the total profit of satisfied requests. We propose an extremely fast, purely combinatorial, and easily implemented algorithm for this problem, which has worse approximation ratio them an already known algorithm, but manages to produce competitive solutions—in some cases, it produces better solutions than all the other algorithms included in the study. From the experimental comparison, we conclude that the proposed algorithm is a decent choice whenever we require decent solutions in limited running time. We also study game-theoretic models for routing and wavelength assignment in multifiber optical networks. We present a full analysis of the price of anarchy when players selfishly choose the wavelength of already routed communication requests, they are charged according to the maximum fiber multiplicity incurred by their choice of wavelength, and the social cost is determined by the maximum wavelength multiplicity that appears at any edge of the network. We prove that the game thus defined always converges to a Nash equilibrium in a finite number of moves, and also propose algorithms for efficiently computing socially optimal and approximate Nash equilibria in specific network topologies. The price of anarchy can grow unbounded even in tree networks with maximum degree three. However, in the case of chains and rings, the price of anarchy is bounded by a constant when the number of available wavelengths is not too large compared to the load of the network—an assumption which covers most cases that can appear in practice. Extending the previous model, we propose a general framework for studying selfish routing and wavelength assignment games in multifiber optical networks, under player cost and social cost functions. We prove upper and lower bounds on the price of anarchy of these games. Finally, we study the complexity of scheduling a set of routes that must be executed periodically in a transportation network with a given period, so that the safety distance between successive vehicles that use the same portion of the network is maximized. For solving this problem, we prove and utilize its connection with a path coloring problem which has been used extensively for modeling routing and wavelength assignment problems in optical networks. Thus, we show the generality of the graph theoretic path coloring models which we studied in the thesis.Μελετάμε μοντέλα για δρομολόγηση και ανάθεση μήκους κύματος σε οπτικά δίκτυα, με στόχο να καταδειχθούν ιδιότητες των εν λόγω μοντέλων που πρέπει να λαμβάνονται υπόψιν κατά την υλοποίηση και ανάπτυξη οπτικών δικτύων στην πράξη. Πιο συγκεκριμένα, προτείνονται προσεγγιστικοί αλγόριθμοι για τη μεγιστοποίηση του πλήθους των ικανοποιούμενων αιτήσεων σε οπτικά δίκτυα τοπολογίας δακτυλίου όπου ο αριθμός των μηκών κύματος ανά ίνα δίδεται ως μέρος της εισόδου. Οι προτεινόμενοι αλγόριθμοι, οι οποίοι έχουν όλοι φράγμενο λόγο προσέγγισης στη χειρότερη περίπτωση, συγκρίνονται και πειραματικά με ήδη γνωστούς από τη βιβλιογραφία αλγορίθμους. Από τη σύγκριση προκύπτει ότι ο αλγόριθμος με τον θεωρητικά καλύτερο λόγο προσέγγισης αποδίδει μεν καλύτερα από τους υπόλοιπους αλλά καταναλώνει υπερβολικά πολύ χρόνο. Αντίθετα, ένας από τους προτεινόμενους αλγόριθμους παράγει πολύ ικανοποιητικές λύσεις σε χρόνο που είναι αρκετές τάξεις μεγέθους μικρότερος από τον χρόνο του καλύτερου αλγορίθμου. Επιπλέον, μελετάται μία γενίκευση του προβλήματος όπου κάθε αίτηση επικοινωνίας έχει ένα δεδομένο κέρδος, και ζητείται η μεγιστοποίηση του συνολικού κέρδους των ικανοποιούμενων αιτήσεων. Προτείνεται ένας εξαιρετικό γρήγορος, καθαρά συνδυαστικός και εύκολος στην υλοποίηση αλγόριθμος για το πρόβλημα αυτό, ο οποίος έχει χειρότερο λόγο προσέγγισης από έναν ήδη γνωστό αλγόριθμο, όμως καταφέρνει να παράγει ανταγωνιστικές λύσεις και μάλιστα σε ορισμένες περιπτώσεις καλύτερες από όλους τους άλλους αλγορίθμους που συμπεριλαμβάνονται στη μελέτη. Από την πειραματική σύγκριση προκύπτει το συμπέρασμα ότι ο προτεινόμενος αλγόριθμος αποτελεί ιδανική επιλογή όταν απαιτούνται λύσεις στο πρόβλημα σε σύντομο χρονικό διάστημα. Μελετώνται παιγνιοθεωρητικά μοντέλα για τη δρομολόγηση και την ανάθεση μηκών κύματος σε οπτικά δίκτυα πολλαπλών ινών. Ειδικότερα, παρουσιάζεται μια πλήρης ανάλυση του κόστους της αναρχίας όταν οι παίκτες επιλέγουν εγωιστικό το μήκος κύματος ήδη δρομολογημένων αιτήσεων επικοινωνίας, χρεώνονται με βάση την μέγιστη πολλαπλότητα του μήκους κύματος που επέλεξαν κατά μήκος του μονοπατιού στο οποίο έχει δρομολογηθεί η αίτηση, και το κοινωνικό κόστος καθορίζεται από την μέγιστη πολλαπλότητα μήκους κύματος που εμφανίζεται σε ολόκληρο το δίκτυο. Αποδεικνύεται ότι το παίγνιο που ορίζεται με αυτόν τον τρόπο συγκλίνει πάντοτε σε ισορροπία Nash σε πεπερασμένο αριθμό κινήσεων, ενώ προτείνονται αλγόριθμοι για τον υπολογισμό κοινωνικά βέλτιστης ισορροπίας Nash και προσεγγιστικά βέλτιστης ισορροπίας Nash σε συγκεκριμένες τοπολογίες. Αποδεικνύεται ότι το κόστος της αναρχίας μπορεί να γίνει αυθαίρετα μεγάλο ακόμη και σε δενδρικές τοπολογίες δικτύων με μέγιστο βαθμό τρία. Όμως, στην περίπτωση του δακτυλίου και της αλυσίδας, το κόστος της αναρχίας φράσσεται από μία σταθερά αν το πλήθος των διαθέσιμων μηκών κύματος δεν είναι πολύ μεγάλο σε σχέση με το φορτίο του δικτύου, υπόθεση που καλύπτει ουσιαστικά την πλειοψηφία των περιπτώσεων που μπορεί να εμφανιστούν στην πράξη. Προς επέκταση του προηγούμενου μοντέλου, προτείνεται ένα γενικότερο πλαίσιο μελέτης των παιγνίων εγωιστικής δρομολόγησης και ανάθεσης μηκών κύματος σε οπτικά δίκτυα πολλαπλών ινών, υπό διάφορες συναρτήσεις κόστους των παικτών και υπό διάφορες συναρτήσεις κοινωνικού κόστους. Αποδεικνύονται άνω και κάτω φράγματα για το κόστος της αναρχίας των εν λόγω παιγνίων. Τέλος, μελετάται η πολυπλοκότητα του προβλήματος χρονικού προγραμματισμού ενός συνόλου δρομολογίων που πρέπει να εκτελούνται περιοδικά με δοσμένη συχνότητα σε ένα δίκτυο μεταφορών, έτσι ώστε να μεγιστοποιούνται οι αποστάσεις ασφαλείας μεταξύ διαδοχικών οχημάτων που χρησιμοποιούν το ίδιο τμήμα του δικτύου. Για την επίλυση αυτού του προβλήματος αποδεικνύεται και αξιοποιείται η σύνδεσή του με ένα πρόβλημα χρωματισμού μονοπατιών που έχει χρησιμοποιηθεί κατά κόρον για τη μοντελοποίηση προβλημάτων δρομολόγησης και ανάθεσης μηκών κύματος σε οπτικά δίκτυα. Έτσι, καταδεικνύεται η γενικότητα των γραφοθεωρητικών μοντέλων χρωματισμού μονοπατιών που μελετήθηκαν στη διατριβή

Problèmes vérifiables par agents mobiles

International audienceNous considérons les problèmes de décision qui sont résolus de manière répartie par des agents mobiles synchrones évoluant dans un réseau inconnu et anonyme. Chaque agent dispose d'un identifiant unique et d'une chaîne d'entrée, et ces agents doivent décider collectivement de la validité d'une propriété qui peut être basée sur les chaînes d'entrée, le graphe dans lequel les agents évoluent, et leurs positions de départ. Poursuivant un travail récent de Friagniaud et Pelc [LATIN 2012, LNCS 7256, p. 362–374], nous introduisons plusieurs nouvelles classes naturelles de calculabilité par agents mobiles, nous permettant d'obtenir une classification plus fine des problèmes inclus dans co-MAV ou MAV, cette dernière étant la classe des problèmes vérifiables lorsque les agents disposent d'un certificat approprié. Dans cet article, nous exhibons des résultats d'inclusion ou de séparation entre toutes ces classes. Nous déterminons également leurs propriétés de clôture vis-à-vis des opérations classiques de la théorie des ensembles. Notre principal outil technique, intéressant en soi, est un nouveau méta-protocole qui permet l'exécution essentiellement en parallèle d'un nombre potentiellement infini de protocoles d'agents mobiles, de façon similaire à la technique classique de déployeur universel (dovetailing) présente en calculabilité classique

On mobile agent verifiable problems

International audienceWe consider decision problems that are solved in a distributed fashion by synchronous mobile agents operating in an unknown, anonymous network. Each agent has a unique identifier and an input string and they have to decide collectively a property which may involve their input strings, the graph on which they are operating, and their particular starting positions. Building on recent work by Fraigniaud and Pelc [J. Parallel Distrib. Comput, vol. 109, pp. 117–128], we introduce several natural new computability classes allowing for a finer classification of problems below MAV or its complement class co-MAV, the former being the class of problems that are verifiable when the agents are provided with an appropriate certificate. We provide inclusion and separation results among all these classes. We also determine their closure properties with respect to set-theoretic operations. Our main technical tool, which is of independent interest, is a new meta-protocol that enables the execution of a possibly infinite number of mobile agent protocols essentially in parallel, similarly to the well-known dovetailing technique from classical computability theory

On Mobile Agent Verifiable Problems

International audienceWe consider decision problems that are solved in a distributed fashion by synchronous mobile agents operating in an unknown, anonymous network. Each agent has a unique identifier and an input string and they have to decide collectively a property which may involve their input strings, the graph on which they are operating, and their particular starting positions. Building on recent work by Fraigniaud and Pelc [LATIN 2012, LNCS 7256, pp. 362-374], we introduce several natural new computability classes allowing for a finer classification of problems below $\mathsf{co\text{-}MAV}$ or $\mathsf{MAV}$, the latter being the class of problems that are verifiable when the agents are provided with an appropriate certificate. We provide inclusion and separation results among all these classes. We also determine their closure properties with respect to set-theoretic operations. Our main technical tool, which is of independent interest, is a new meta-protocol that enables the execution of a possibly infinite number of mobile agent protocols essentially in parallel, similarly to the well-known dovetailing technique from classical computability theory

Maximum Request Satisfaction in WDM Rings: Algorithms and Experiments

We study the problem of satisfying a maximum number of communication requests in alloptical WDM rings in which the number of available wavelengths per fiber is limited. We investigate two variations of the problem: with or without prior routing of requests. We consider a number of new and existing algorithmic approaches for these two variations. We perform an experimental comparison of the resulting algorithms, with respect to: (a) the number of satisfied requests and (b) the running time. We end up with interesting observations that reveal merits and deficiencies of the algorithms in hand: • All algorithms almost always manage to satisfy many more requests than indicated by their worst-case analysis. • Some algorithms are considerably faster than others with comparable request satisfaction performance. In fact, there are simple, fast algorithms that achieve a competent number of satisfied requests. We anticipate that our results will prove useful in practice, especially in deciding which routing and wavelength assignment method to choose, taking into account the desired level of accuracy and the affordable time cost