Optimal Vertex Fault Tolerant Spanners (for fixed stretch)

A $k$-spanner of a graph $G$ is a sparse subgraph $H$ whose shortest path distances match those of $G$ up to a multiplicative error $k$. In this paper we study spanners that are resistant to faults. A subgraph $H \subseteq G$ is an $f$ vertex fault tolerant (VFT) $k$-spanner if $H \setminus F$ is a $k$-spanner of $G \setminus F$ for any small set $F$ of $f$ vertices that might "fail." One of the main questions in the area is: what is the minimum size of an $f$ fault tolerant $k$-spanner that holds for all $n$ node graphs (as a function of $f$, $k$ and $n$)? This question was first studied in the context of geometric graphs [Levcopoulos et al. STOC '98, Czumaj and Zhao SoCG '03] and has more recently been considered in general undirected graphs [Chechik et al. STOC '09, Dinitz and Krauthgamer PODC '11]. In this paper, we settle the question of the optimal size of a VFT spanner, in the setting where the stretch factor $k$ is fixed. Specifically, we prove that every (undirected, possibly weighted) $n$-node graph $G$ has a $(2k-1)$-spanner resilient to $f$ vertex faults with $O_k(f^{1 - 1/k} n^{1 + 1/k})$ edges, and this is fully optimal (unless the famous Erdos Girth Conjecture is false). Our lower bound even generalizes to imply that no data structure capable of approximating $dist_{G \setminus F}(s, t)$ similarly can beat the space usage of our spanner in the worst case. We also consider the edge fault tolerant (EFT) model, defined analogously with edge failures rather than vertex failures. We show that the same spanner upper bound applies in this setting. Our data structure lower bound extends to the case $k=2$ (and hence we close the EFT problem for $3$-approximations), but it falls to $\Omega(f^{1/2 - 1/(2k)} \cdot n^{1 + 1/k})$ for $k \ge 3$. We leave it as an open problem to close this gap.Comment: To appear in SODA 201

Design and analysis of sequential and parallel single-source shortest-paths algorithms

We study the performance of algorithms for the Single-Source Shortest-Paths (SSSP) problem on graphs with n nodes and m edges with nonnegative random weights. All previously known SSSP algorithms for directed graphs required superlinear time. Wie give the first SSSP algorithms that provably achieve linear O(n-m)average-case execution time on arbitrary directed graphs with random edge weights. For independent edge weights, the linear-time bound holds with high probability, too. Additionally, our result implies improved average-case bounds for the All-Pairs Shortest-Paths (APSP) problem on sparse graphs, and it yields the first theoretical average-case analysis for the "Approximate Bucket Implementation" of Dijkstra\u27s SSSP algorithm (ABI-Dijkstra). Futhermore, we give constructive proofs for the existence of graph classes with random edge weights on which ABI-Dijkstra and several other well-known SSSP algorithms require superlinear average-case time. Besides the classical sequential (single processor) model of computation we also consider parallel computing: we give the currently fastest average-case linear-work parallel SSSP algorithms for large graph classes with random edge weights, e.g., sparse rondom graphs and graphs modeling the WWW, telephone calls or social networks.In dieser Arbeit untersuchen wir die Laufzeiten von Algorithmen für das Kürzeste-Wege Problem (Single-Source Shortest-Paths, SSSP) auf Graphen mit n Knoten, M Kanten und nichtnegativen zufälligen Kantengewichten. Alle bisherigen SSSP Algorithmen benötigen auf gerichteten Graphen superlineare Zeit. Wir stellen den ersten SSSP Algorithmus vor, der auf beliebigen gerichteten Graphen mit zufälligen Kantengewichten eine beweisbar lineare average-case-Komplexität O(n+m)aufweist. Sind die Kantengewichte unabhängig, so wird die lineare Zeitschranke auch mit hoher Wahrscheinlichkeit eingehalten. Außerdem impliziert unser Ergebnis verbesserte average-case-Schranken für das All-Pairs Shortest-Paths (APSP) Problem auf dünnen Graphen und liefert die erste theoretische average-case-Analyse für die "Approximate Bucket Implementierung" von Dijkstras SSSP Algorithmus (ABI-Dijkstra). Weiterhin führen wir konstruktive Existenzbeweise für Graphklassen mit zufälligen Kantengewichten, auf denen ABI-Dijkstra und mehrere andere bekannte SSSP Algorithmen durchschnittlich superlineare Zeit benötigen. Neben dem klassischen seriellen (Ein-Prozessor) Berechnungsmodell betrachten wir auch Parallelverarbeitung; für umfangreiche Graphklassen mit zufälligen Kantengewichten wie z.B. dünne Zufallsgraphen oder Modelle für das WWW, Telefonanrufe oder soziale Netzwerke stellen wir die derzeit schnellsten parallelen SSSP Algorithmen mit durchschnittlich linearer Arbeit vor

Growing phonological and morphological knowledge and improving spelling outcomes in Year 2 primary school children through Explicit Instruction and contextualised dictation

Using Explicit Instruction (EI) to teach spelling is controversial because teaching approaches vary considerably in the contemporary classroom. Teachers may privilege visual over linguistic strategies and include target words based around themes, rather than the phono-morphological structures of words. There is also little current research about the benefits of using sentence dictation to practise taught spelling skills and thus to increase the likelihood of developing spelling automaticity. Spelling automaticity is important because it complements crucial reading and writing skills. Developing fluent spelling through EI, followed by sentence dictation, was a specific focus of this study. Two primary schools in rural NSW and a total of 30 teachers were involved in this mixed methods research. One of the schools was used as a comparison school and the other was the intervention school. All 30 teachers involved in the study completed a knowledge survey about the components of the English spelling system considered essential to teach spelling explicitly. From this data, the specific knowledge of the teachers involved in the Year 2 intervention, the Learning Support Teacher and the Acting Principal, was extracted. The two Year 2 teachers in the comparison school received professional development on meaning-based approaches to spelling, whereas the five teachers at the intervention school received professional development on EI techniques and word level components of the English spelling system. Mid-intervention teacher interviews gathered data about their feelings on implementing EI techniques in practice. Post-intervention quantitative tests and interviews allowed in-depth and rich understandings of aspects that either enabled or hindered implementation of the intervention. The spelling competence of 60 students at the two schools was also assessed before any intervention took place. The 35 Year 2 students in the two classes at the intervention school received EI in the phonological and morphological aspects of words, editing, and contextualised sentence dictation during Term 3. The 25 students in the Year 2 class at the comparison school continued their established literacy routine. Interviews with randomly selected students from both schools facilitated an exploration of their feelings about spelling approaches used during the term. The findings showed that spelling results in both schools improved as expected. However, overall the intervention school had superior results to the comparison school; one class in the intervention school consistently outperformed all other classes in word spelling and dictation assessments with moderate to large effects. Many of the teachers demonstrated an increase in morpheme knowledge, but not in word structure. In this study the EI spelling Lesson elements were reinforced by teaching strategies that included contextualised editing tasks and daily sentence dictations. These tasks were embedded in the term science theme of Insects, which was chosen in collaboration with the intervention teachers. The dictation component, a previously underutilised tool, involved students writing two lines from a contextualised poem, each day. In Australia, current methods of teaching spelling remain varied and contentious. Teachers who are engaged in improving spelling knowledge may find that using EI strategies reinforced by contextualised dictation can improve outcomes for all students

Properties and Algorithms of the KCube Interconnection Networks

The KCube interconnection network was first introduced in 2010 in order to exploit the good characteristics of two well-known interconnection networks, the hypercube and the Kautz graph. KCube links up multiple processors in a communication network with high density for a fixed degree. Since the KCube network is newly proposed, much study is required to demonstrate its potential properties and algorithms that can be designed to solve parallel computation problems. In this thesis we introduce a new methodology to construct the KCube graph. Also, with regard to this new approach, we will prove its Hamiltonicity in the general KC(m; k). Moreover, we will find its connectivity followed by an optimal broadcasting scheme in which a source node containing a message is to communicate it with all other processors. In addition to KCube networks, we have studied a version of the routing problem in the traditional hypercube, investigating this problem: whether there exists a shortest path in a Qn between two nodes 0n and 1n, when the network is experiencing failed components. We first conditionally discuss this problem when there is a constraint on the number of faulty nodes, and subsequently introduce an algorithm to tackle the problem without restrictions on the number of nodes

Estudi i disseny de grans xarxes d'interconnexió: modularitat i comunicació

Normalment les grans xarxes d'interconnexió o de comunicacions estan dissenyades utilitzant tècniques de la teoria de grafs. Aquest treball presenta algunes contribucions a aquest tema. Concretament, presentem dues noves operacions: el "producte Jeràrquic" de grafs i el "producte Manhattan" de digrafs. El primer és una generalització del producte cartesià de grafs i ens permet construir algunes famílies amb un alt grau de jerarquia, com l'arbre binomial, que és una estructura de dades molt utilitzada en algorísmica. El segon dóna lloc a les conegudes Manhattan Street Networks, les quals han estat extensament estudiades i utilitzades per modelar algunes classes de xarxes òptiques. En el nostre treball, definim formalment i analitzem el cas multidimensional d'aquestes xarxes. Estudiem algunes propietats dels grafs o digrafs obtinguts mitjançant les dues operacions esmentades, especialment: els paràmetres estructurals (les propietats de l'operació, els subdigrafs induïts, la distribució de graus i l'estructura de digraf línia), els paràmetres mètrics (el diàmetre, el radi i la distància mitjana), la simetria (els grups d'automorfismes i els digrafs de Cayley), l'estructura de cicles (els cicles hamiltonians i la descomposició en cicles hamiltonians arc-disjunts) i les propietats espectrals (els valors i vectors propis). En el darrer cas, hem trobat, per exemple, que la família dels arbres binomials tenen tots els seus valors propis diferents, "omplint" tota la recta real. A més a més, mostrem la relació del seu conjunt de vectors propis amb els polinomis de Txebishev de segona espècie. També hem estudiat alguns protocols de comunicació, com els enrutaments locals i els algorismes de difusió. Finalment, presentem alguns models deterministes (com les xarxes Sierpinski i d'altres), els quals presenten algunes propietats pròpies de les xarxes complexes reals (com, per exemple, Internet).Large interconnection or communication networks are usually designed and studied by using techniques from graph theory. This work presents some contributions to this subject. With this aim, two new operations are proposed: the "hierarchical product" of graphs and the "Manhattan product" of digraphs. The former can be seen as a generalization of the Cartesian product of graphs and allows us to construct some interesting families with a high degree of hierarchy, such as the well-know binomial tree, which is a data structure very used in the context of computer science. The latter yields, in particular, the known topologies of Manhattan Street Networks, which has been widely studied and used for modelling some classes of light-wave networks. In this thesis, a multidimensional approach is analyzed. Several properties of the graphs or digraphs obtained by both operations are dealt with, but special attention is paid to the study of their structural parameters (operation properties, induced subdigraphs, degree distribution and line digraph structure), metric parameters (diameter, radius and mean distance), symmetry (automorphism groups and Cayley digraphs), cycle structure (Hamilton cycles and arc-disjoint Hamiltonian decomposition) and spectral properties (eigenvalues and eigenvectors). For instance, with respect to the last issue, it is shown that some families of hypertrees have spectra with all different eigenvalues "filling up" all the real line. Moreover, we show the relationship between its eigenvector set and Chebyshev polynomials of the second kind. Also some protocols of communication, such as local routing and broadcasting algorithms, are addressed. Finally, some deterministic models (Sierpinsky networks and others) having similar properties as some real complex networks, such as the Internet, are presented