Polynomial-Time Space-Optimal Silent Self-Stabilizing Minimum-Degree Spanning Tree Construction

Motivated by applications to sensor networks, as well as to many other areas, this paper studies the construction of minimum-degree spanning trees. We consider the classical node-register state model, with a weakly fair scheduler, and we present a space-optimal \emph{silent} self-stabilizing construction of minimum-degree spanning trees in this model. Computing a spanning tree with minimum degree is NP-hard. Therefore, we actually focus on constructing a spanning tree whose degree is within one from the optimal. Our algorithm uses registers on $O(\log n)$ bits, converges in a polynomial number of rounds, and performs polynomial-time computation at each node. Specifically, the algorithm constructs and stabilizes on a special class of spanning trees, with degree at most $OPT+1$. Indeed, we prove that, unless NP $=$ coNP, there are no proof-labeling schemes involving polynomial-time computation at each node for the whole family of spanning trees with degree at most $OPT+1$. Up to our knowledge, this is the first example of the design of a compact silent self-stabilizing algorithm constructing, and stabilizing on a subset of optimal solutions to a natural problem for which there are no time-efficient proof-labeling schemes. On our way to design our algorithm, we establish a set of independent results that may have interest on their own. In particular, we describe a new space-optimal silent self-stabilizing spanning tree construction, stabilizing on \emph{any} spanning tree, in $O(n)$ rounds, and using just \emph{one} additional bit compared to the size of the labels used to certify trees. We also design a silent loop-free self-stabilizing algorithm for transforming a tree into another tree. Last but not least, we provide a silent self-stabilizing algorithm for computing and certifying the labels of a NCA-labeling scheme

Shorter Labeling Schemes for Planar Graphs

An \emph{adjacency labeling scheme} for a given class of graphs is an algorithm that for every graph $G$ from the class, assigns bit strings (labels) to vertices of $G$ so that for any two vertices $u,v$, whether $u$ and $v$ are adjacent can be determined by a fixed procedure that examines only their labels. It is known that planar graphs with $n$ vertices admit a labeling scheme with labels of bit length $(2+o(1))\log{n}$. In this work we improve this bound by designing a labeling scheme with labels of bit length $(\frac{4}{3}+o(1))\log{n}$. In graph-theoretical terms, this implies an explicit construction of a graph on $n^{4/3+o(1)}$ vertices that contains all planar graphs on $n$ vertices as induced subgraphs, improving the previous best upper bound of $n^{2+o(1)}$. Our scheme generalizes to graphs of bounded Euler genus with the same label length up to a second-order term. All the labels of the input graph can be computed in polynomial time, while adjacency can be decided from the labels in constant time

Informative labeling schemes for graphs

AbstractThis paper introduces the notion of informative labeling schemes for arbitrary graphs. Let f(W) be a function on subsets of vertices W. An f labeling scheme labels the vertices of a weighted graph G in such a way that f(W) can be inferred (or at least approximated) efficiently for any vertex subset W of G by merely inspecting the labels of the vertices of W, without having to use any additional information sources.A number of results illustrating this notion are presented in the paper. We begin by developing f labeling schemes for three functions f over the class of n-vertex trees. The first function, SepLevel, gives the separation level of any two vertices in the tree, namely, the depth of their least common ancestor. The second, LCA, provides the least common ancestor of any two vertices. The third, Center, yields the center of any three given vertices v1,v2,v3 in the tree, namely, the unique vertex z connected to them by three edge-disjoint paths. All of these three labeling schemes use O(log2n)-bit labels, which is shown to be asymptotically optimal.Our main results concern the function Steiner(W), defined for weighted graphs. For any vertex subset W in the weighted graph G, Steiner(W) represents the weight of the Steiner tree spanning the vertices of W in G. Considering the class of n-vertex trees with M-bit edge weights, it is shown that for this class there exists a Steiner labeling scheme using O((M+logn)logn) bit labels, which is asymptotically optimal. It is then shown that for the class of arbitrary n-vertex graphs with M-bit edge weights, there exists an approximate-Steiner labeling scheme, providing an estimate (up to a factor of O(logn)) for the Steiner weight Steiner(W) of a given set of vertices W, using O((M+logn)log2n) bit labels

On coding labeled trees

Trees are probably the most studied class of graphs in Computer Science. In this thesis we study bijective codes that represent labeled trees by means of string of node labels. We contribute to the understanding of their algorithmic tractability, their properties, and their applications. The thesis is divided into two parts. In the first part we focus on two types of tree codes, namely Prufer-like codes and Transformation codes. We study optimal encoding and decoding algorithms, both in a sequential and in a parallel setting. We propose a unified approach that works for all Prufer-like codes and a more generic scheme based on the transformation of a tree into a functional digraph suitable for all bijective codes. Our results in this area close a variety of open problems. We also consider possible applications of tree encodings, discussing how to exploit these codes in Genetic Algorithms and in the generation of random trees. Moreover, we introduce a modified version of a known code that, in Genetic Algorithms, outperform all the other known codes. In the second part of the thesis we focus on two possible generalizations of our work. We first take into account the classes of k-trees and k-arch graphs (both superclasses of trees): we study bijective codes for this classes of graphs and their algorithmic feasibility. Then, we shift our attention to Informative Labeling Schemes. In this context labels are no longer considered as simple unique node identifiers, they rather convey information useful to achieve efficient computations on the tree. We exploit this idea to design a concurrent data structure for the lowest common ancestor problem on dynamic trees. We also present an experimental comparison between our labeling scheme and the one proposed by Peleg for static trees

Randomized Proof-Labeling Schemes

International audienceA proof-labeling scheme, introduced by Korman, Kutten and Peleg [PODC 2005], is a mechanism enabling to certify the legality of a network configuration with respect to a boolean predicate. Such a mechanism finds applications in many frameworks, including the design of fault-tolerant distributed algorithms. In a proof-labeling scheme, the verification phase consists of exchanging labels between neighbors. The size of these labels depends on the network predicate to be checked. There are predicates requiring large labels, of poly-logarithmic size (e.g., MST), or even polynomial size (e.g., Symmetry). In this paper, we introduce the notion of randomized proof-labeling schemes. By reduction from deterministic schemes, we show that randomization enables the amount of communication to be exponentially reduced. As a consequence, we show that checking any network predicate can be done with probability of correctness as close to one as desired by exchanging just a logarithmic number of bits between neighbors. Moreover, we design a novel space lower bound technique that applies to both deterministic and randomized proof-labeling schemes. Using this technique, we establish several tight bounds on the verification complexity of classical distributed computing problems, such as MST construction, and of classical predicates such as acyclicity, connectivity, and cycle length

Controller and estimator for dynamic networks

International audienceAfek, Awerbuch, Plotkin, and Saks identified an important fundamental problem inherent to distributed networks, which they called the Resource Controller problem. Consider, first, the problem in which one node (called the 'root') is required to estimate the number of events that occurred all over the network. This counting problem can be viewed as a useful variant of the heavily studied and used task of topology update (that deals with collecting all remote information). The Resource Controller problem generalizes the counting problem: such remote events are considered as requests, and the counting node, i.e., the 'root', also issues permits for the requests. That way, the number of requests granted can be controlled (bounded). An efficient Resource Controller was constructed in the paper by Afek et al., and it can operate on a dynamic network assuming that the network is spanned by a tree that may only grow, and only by allowing leaves to join the tree. In contrast, the Resource Controller presented here can operate under a more general dynamic model, allowing the spanning tree of the network to undergo both insertions and deletions of both leaves and internal nodes. Despite the more dynamic network model we allow, the message complexity of our controller is never more than the message complexity of the more restricted controller. All the applications for the controller of Afek et al. can be used also with our controller. Moreover, with the same message complexity, our controller can handle these applications under the more general dynamic model mentioned above. In particular, the new controller can be transformed into an efficient size-estimation protocol, i.e., a protocol allowing all nodes to maintain a constant factor estimation of the number of nodes in the dynamically changing network. Informally, the resulting new size-estimation protocol uses View the MathML source amortized message complexity per topological change (assuming that the number of changes in the network size is "not too small"), where n is the current number of nodes in the network. An application of the size estimation of Afek et al. was to solve agreement in the case of initial faults (Fischer, Lynch, and Paterson) and leader election under initial faults (Bar-Yehuda and Kutten). Hence, the controllers in this paper can be useful for these applications too

Randomized proof-labeling schemes

International audienceProof-labeling schemes, introduced by Korman et al. (Distrib Comput 22(4):215–233, 2010. https://doi.org/10.1007/s00446-010-0095-3), are a mechanism to certify that a network configuration satisfies a given boolean predicate. Such mechanisms find applications in many contexts, e.g., the design of fault-tolerant distributed algorithms. In a proof-labeling scheme, predicate verification consists of neighbors exchanging labels, whose contents depends on the predicate. In this paper, we introduce the notion of randomized proof-labeling schemes where messages are randomized and correctness is probabilistic. We show that randomization reduces verification complexity exponentially while guaranteeing probability of correctness arbitrarily close to one. We also present a novel message-size lower bound technique that applies to deterministic as well as randomized proof-labeling schemes. Using this technique, we establish several tight bounds on the verification complexity of MST, acyclicity, connectivity, and longest cycle size

Single-Molecule Analysis of Ribosome and Initiation Factor Dynamics during the Late Stages of Translation Initiation

Protein synthesis in all organisms is catalyzed by a highly-conserved ribonucleoprotein macromolecular machine known as the ribosome. Prior to each round of protein synthesis in the cell, a functional ribosomal complex is assembled from its component parts at the start site of a messenger RNA (mRNA) template during the process of translation initiation. In bacteria, rapid and high-fidelity translation initiation is promoted by three canonical initiation factors: IF1, IF2, and IF3. In this thesis, I report the use of single-molecule fluorescence methods to study the role of the initiation factors and ribosome-factor interactions in regulating molecular events that occur during late stages of the translation initiation pathway. In Chapter 1, I provide a structural and biochemical framework for understanding one of the key events of the initiation pathway: docking of the large (50S) ribosomal subunit with the small subunit 30S initiation complex (30S IC). The 50S subunit joining reaction is catalyzed by GTP-bound IF2 and results in formation of a 70S initiation complex (70S IC) that contains an initiator transfer RNA (tRNA) and is primed for formation of the first peptide bond. During 50S subunit joining, IF2-GTP establishes interactions with RNA and protein components of the 50S subunit's GTPase-associated center (GAC), which play an important role in subunit recruitment as well as the subsequent activation of GTP hydrolysis by IF2. In Chapter 2, I describe the development of a single-molecule fluorescence resonance energy transfer (smFRET) signal to monitor the interactions between IF2 and the ribosome's GAC during real-time 50S subunit joining reactions. Specifically, the role of the L11 region, comprising ribosomal protein L11 and its associated ribosomal RNA (rRNA) helices, was investigated. The L11 region is a prominent structural component of the GAC that is believed to undergo large-scale conformational changes during protein synthesis; however, the nature and timescale of these conformational dynamics, and their role in regulating the biochemical activities of IF2 during initiation, are not known. I demonstrate that my smFRET-based 50S subunit joining assay is sensitive to conformational rearrangements between IF2 and L11 within the 70S IC and can thus be used as a tool for characterizing GAC dynamics and elucidating their function during initiation. Furthermore, my smFRET approach is shown to provide information on the rate of 50S subunit joining as well as the rate of IF2 dissociation from the 70S IC. Notably, IF2-dependent GTP hydrolysis was found to influence the extent of 70S IC conformational dynamics as well as the dissociation rate of IF2. The role of IF3 in regulating 50S-subunit joining dynamics is discussed in Chapter 3. IF3 plays an important role in ensuring the fidelity of translation initiation by preventing the formation of initiation complexes containing a non-initiator tRNA and/or a non-canonical mRNA start codon. Inclusion of IF3 within the 30S IC in the smFRET experiments was found to render the IF2-catalyzed 50S subunit joining reaction highly reversible. Direct observation of repetitive docking and undocking of the 50S subunit with the 30S IC indicates that IF3 may modulate translation initiation efficiency by influencing the stability of the 70S IC. The individual 50S subunit docking events were found to result in the formation of very different classes of 70S IC, characterized by different stabilities and unique patterns of IF2-L11 interactions. I propose that these dynamics reflect an underlying conformational equilibrium of the IF3-bound 30S IC that is read out during 50S subunit joining, and that this equilibrium could be modulated in order to regulate the efficiency of translation initiation. Following initiation-factor mediated assembly of the 70S IC, the first aminoacyl-tRNA is delivered to the ribosome in ternary complex with elongation factor Tu (EF-Tu) and GTP. Accommodation of aminoacyl-tRNA into the ribosome's peptidyl transferase center leads to formation of the first peptide bond, which signals the end of initiation and entry into the elongation phase of protein synthesis. The ternary complex binding site on the ribosome overlaps with that of IF2 at the GAC; a question of key mechanistic importance in understanding how the ribosome coordinates the transition from initiation to elongation thus concerns the relative timing of ternary complex binding with respect to IF2 dissociation from the 70S IC. In Chapter 4, I present preliminary results from two- and three-color fluorescence co-localization experiments aimed at characterizing the timing of these events at the single-molecule level. The data strongly suggest the occurrence of simultaneous occupancy of the ribosome by IF2 and ternary complex, implying that the ribosome is structurally capable of recruiting ternary complex prior to IF2 release from the 70S IC. The observation that the ribosome can accommodate more than one translation factor at a time may have important implications for understanding how it efficiently coordinates factor binding and release throughout protein synthesis, and opens the door to mechanistic studies of the ribosomal L7/L12 stalk, presumed to play a prominent role in these processes