Labeling Schemes with Queries

We study the question of ``how robust are the known lower bounds of labeling schemes when one increases the number of consulted labels''. Let $f$ be a function on pairs of vertices. An $f$-labeling scheme for a family of graphs \cF labels the vertices of all graphs in \cF such that for every graph G\in\cF and every two vertices $u,v\in G$, the value $f(u,v)$ can be inferred by merely inspecting the labels of $u$ and $v$. This paper introduces a natural generalization: the notion of $f$-labeling schemes with queries, in which the value $f(u,v)$ can be inferred by inspecting not only the labels of $u$ and $v$ but possibly the labels of some additional vertices. We show that inspecting the label of a single additional vertex (one {\em query}) enables us to reduce the label size of many labeling schemes significantly

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

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

An Optimal Ancestry Labeling Scheme with Applications to XML Trees and Universal Posets

International audienceIn this paper we solve the ancestry-labeling scheme problem which aims at assigning the shortest possible labels (bit strings) to nodes of rooted trees, so that ancestry queries between any two nodes can be answered by inspecting their assigned labels only. This problem was introduced more than twenty years ago by Kannan et al. [STOC '88], and is among the most well-studied problems in the field of informative labeling schemes. We construct an ancestry-labeling scheme for n-node trees with label size log 2 n + O(log log n) bits, thus matching the log 2 n + Ω(log log n) bits lower bound given by Alstrup et al. [SODA '03]. Our scheme is based on a simplified ancestry scheme that operates extremely well on a restricted set of trees. In particular, for the set of n-node trees with depth at most d, the simplified ancestry scheme enjoys label size of log 2 n + 2 log 2 d + O(1) bits. Since the depth of most XML trees is at most some small constant, such an ancestry scheme may be of practical use. In addition, we also obtain an adjacency-labeling scheme that labels n-node trees of depth d with labels of size log 2 n + 3 log 2 d + O(1) bits. All our schemes assign the labels in linear time, and guarantee that any query can be answered in constant time. Finally, our ancestry scheme finds applications to the construction of small universal partially ordered sets (posets). Specifically, for any fixed integer k, it enables the construction of a universal poset of size˜Osize˜ size˜O(n k) for the family of n-element posets with tree-dimension at most k. Up to lower order terms, this bound is tight thanks to a lower bound of n k−o(1) due to Alon and Scheinerman [Order '88]

Short and Simple Labels for Small Distances and Other Functions

We present a labeling scheme for rooted trees which allows to compute, from the label of v alone, unique identi ers for the ancestors of v that are at distance at most d from v. For any constant d our labeling scheme produce labels of length log n + O( log n), and for d 2 O( log n) the labels are still of length O(log n)

Short and Simple Labels for Small Distances and Other Functions

We present a labeling scheme for rooted trees which allows to compute, from the label of v alone, unique identi ers for the ancestors of v that are at distance at most d from v. For any constant d our labeling scheme produce labels of length log n+O( log n), and for d 2 O( the labels are still of length O(log n). In particular

Die XXL—Suchmaschine zur ontologiebasierten Ähnlichkeitssuche in XML—Dokumenten

Die effektive und effiziente Informationssuche in großen Mengen semistrukturierter Daten im XML-Format stehen im Mittelpunkt dieser Arbeit. In dieser Arbeit wird die XXL-Suchmaschine vorgestellt. Sie wertet Anfragen aus, die in der XML-Anfragesprache XXL formuliert sind. Eine XXL-Anfrage umfasst dabei Suchbedingungen an die Struktur und an den Inhalt von XML-Dokumenten. Als Ergebnis wird eine nach ihrer Relevanz absteigend sortierte Liste von Treffern produziert, wobei ein Treffer ein relevantes XML-Dokument oder nur der relevante Teil eines XML-Dokuments sein kann. Die relevanzorientierte Auswertung von gegebenen Suchbedingungen beruht zum einen auf Verfahren aus dem Vektorraummodell und zum anderen wird semantisches Wissen einer quantifizierten Ontologie hinzugezogen. Zu diesem Zweck werden Datenbank-Technologien und Verfahren aus dem Information Retrieval kombiniert, um die Qualität der Suchergebnisse im Vergleich zur traditionellen Stichwortsuche in Textdokumenten zu verbessern. Die hier vorgestellten Konzepte wurden in einem Prototypen implementiert und umfangreich evaluiert.The effective and efficient information retrieval in large sets of semistructured data using the XML format is the main theme of this thesis. This thesis presents the XXL search engine, which executes queries formulated in the XML query language XXL. An XXL query consists of search conditions on the structure and search conditions on the content of XML documents. The result is a ranked result list in descending order of relevance, where a result can be a relevant XML document or only the relevant part of an XML document. The relevance-based query evaluation uses methods from the vector space model and semantic knowledge from a quantified ontology. For this purpose, we combine database technologies and methods from information retrieval to improve the quality of search results in comparison to traditional keyword-based text retrieval. The presented concepts have been implemented and exhaustively evaluated