Labeling Schemes for Bounded Degree Graphs

We investigate adjacency labeling schemes for graphs of bounded degree $\Delta = O(1)$. In particular, we present an optimal (up to an additive constant) $\log n + O(1)$ adjacency labeling scheme for bounded degree trees. The latter scheme is derived from a labeling scheme for bounded degree outerplanar graphs. Our results complement a similar bound recently obtained for bounded depth trees [Fraigniaud and Korman, SODA 10], and may provide new insights for closing the long standing gap for adjacency in trees [Alstrup and Rauhe, FOCS 02]. We also provide improved labeling schemes for bounded degree planar graphs. Finally, we use combinatorial number systems and present an improved adjacency labeling schemes for graphs of bounded degree $\Delta$ with $(e+1)\sqrt{n} < \Delta \leq n/5$

Dynamic and Multi-functional Labeling Schemes

We investigate labeling schemes supporting adjacency, ancestry, sibling, and connectivity queries in forests. In the course of more than 20 years, the existence of $\log n + O(\log \log)$ labeling schemes supporting each of these functions was proven, with the most recent being ancestry [Fraigniaud and Korman, STOC '10]. Several multi-functional labeling schemes also enjoy lower or upper bounds of $\log n + \Omega(\log \log n)$ or $\log n + O(\log \log n)$ respectively. Notably an upper bound of $\log n + 5\log \log n$ for adjacency+siblings and a lower bound of $\log n + \log \log n$ for each of the functions siblings, ancestry, and connectivity [Alstrup et al., SODA '03]. We improve the constants hidden in the $O$-notation. In particular we show a $\log n + 2\log \log n$ lower bound for connectivity+ancestry and connectivity+siblings, as well as an upper bound of $\log n + 3\log \log n + O(\log \log \log n)$ for connectivity+adjacency+siblings by altering existing methods. In the context of dynamic labeling schemes it is known that ancestry requires $\Omega(n)$ bits [Cohen, et al. PODS '02]. In contrast, we show upper and lower bounds on the label size for adjacency, siblings, and connectivity of $2\log n$ bits, and $3 \log n$ to support all three functions. There exist efficient adjacency labeling schemes for planar, bounded treewidth, bounded arboricity and interval graphs. In a dynamic setting, we show a lower bound of $\Omega(n)$ for each of those families.Comment: 17 pages, 5 figure

Lower Bounds in the Preprocessing and Query Phases of Routing Algorithms

In the last decade, there has been a substantial amount of research in finding routing algorithms designed specifically to run on real-world graphs. In 2010, Abraham et al. showed upper bounds on the query time in terms of a graph's highway dimension and diameter for the current fastest routing algorithms, including contraction hierarchies, transit node routing, and hub labeling. In this paper, we show corresponding lower bounds for the same three algorithms. We also show how to improve a result by Milosavljevic which lower bounds the number of shortcuts added in the preprocessing stage for contraction hierarchies. We relax the assumption of an optimal contraction order (which is NP-hard to compute), allowing the result to be applicable to real-world instances. Finally, we give a proof that optimal preprocessing for hub labeling is NP-hard. Hardness of optimal preprocessing is known for most routing algorithms, and was suspected to be true for hub labeling

Stochastic Analysis of a Churn-Tolerant Structured Peer-to-Peer Scheme

We present and analyze a simple and general scheme to build a churn (fault)-tolerant structured Peer-to-Peer (P2P) network. Our scheme shows how to "convert" a static network into a dynamic distributed hash table(DHT)-based P2P network such that all the good properties of the static network are guaranteed with high probability (w.h.p). Applying our scheme to a cube-connected cycles network, for example, yields a $O(\log N)$ degree connected network, in which every search succeeds in $O(\log N)$ hops w.h.p., using $O(\log N)$ messages, where $N$ is the expected stable network size. Our scheme has an constant storage overhead (the number of nodes responsible for servicing a data item) and an $O(\log N)$ overhead (messages and time) per insertion and essentially no overhead for deletions. All these bounds are essentially optimal. While DHT schemes with similar guarantees are already known in the literature, this work is new in the following aspects: (1) It presents a rigorous mathematical analysis of the scheme under a general stochastic model of churn and shows the above guarantees; (2) The theoretical analysis is complemented by a simulation-based analysis that validates the asymptotic bounds even in moderately sized networks and also studies performance under changing stable network size; (3) The presented scheme seems especially suitable for maintaining dynamic structures under churn efficiently. In particular, we show that a spanning tree of low diameter can be efficiently maintained in constant time and logarithmic number of messages per insertion or deletion w.h.p. Keywords: P2P Network, DHT Scheme, Churn, Dynamic Spanning Tree, Stochastic Analysis

Silent MST approximation for tiny memory

In network distributed computing, minimum spanning tree (MST) is one of the key problems, and silent self-stabilization one of the most demanding fault-tolerance properties. For this problem and this model, a polynomial-time algorithm with $O(\log^2\!n)$ memory is known for the state model. This is memory optimal for weights in the classic $[1,\text{poly}(n)]$ range (where $n$ is the size of the network). In this paper, we go below this $O(\log^2\!n)$ memory, using approximation and parametrized complexity. More specifically, our contributions are two-fold. We introduce a second parameter~$s$, which is the space needed to encode a weight, and we design a silent polynomial-time self-stabilizing algorithm, with space $O(\log n \cdot s)$. In turn, this allows us to get an approximation algorithm for the problem, with a trade-off between the approximation ratio of the solution and the space used. For polynomial weights, this trade-off goes smoothly from memory $O(\log n)$ for an $n$-approximation, to memory $O(\log^2\!n)$ for exact solutions, with for example memory $O(\log n\log\log n)$ for a 2-approximation

Compact Labelings For Efficient First-Order Model-Checking

We consider graph properties that can be checked from labels, i.e., bit sequences, of logarithmic length attached to vertices. We prove that there exists such a labeling for checking a first-order formula with free set variables in the graphs of every class that is \emph{nicely locally cwd-decomposable}. This notion generalizes that of a \emph{nicely locally tree-decomposable} class. The graphs of such classes can be covered by graphs of bounded \emph{clique-width} with limited overlaps. We also consider such labelings for \emph{bounded} first-order formulas on graph classes of \emph{bounded expansion}. Some of these results are extended to counting queries

Separating Hierarchical and General Hub Labelings

In the context of distance oracles, a labeling algorithm computes vertex labels during preprocessing. An $s,t$ query computes the corresponding distance from the labels of $s$ and $t$ only, without looking at the input graph. Hub labels is a class of labels that has been extensively studied. Performance of the hub label query depends on the label size. Hierarchical labels are a natural special kind of hub labels. These labels are related to other problems and can be computed more efficiently. This brings up a natural question of the quality of hierarchical labels. We show that there is a gap: optimal hierarchical labels can be polynomially bigger than the general hub labels. To prove this result, we give tight upper and lower bounds on the size of hierarchical and general labels for hypercubes.Comment: 11 pages, minor corrections, MFCS 201

Distance-Aware Selective Online Query Processing Over Large Distributed Graphs

Performing online selective queries against graphs is a challenging problem due to the unbounded nature of graph queries which leads to poor computation locality. It becomes even difficult when a graph is too large to be fit in the memory. Although there have been emerging efforts on managing large graphs in a distributed and parallel setting, e.g., Pregel, HaLoop and etc, these computing frameworks are designed from the perspective of scalability instead of the query efficiency. In this work, we present our solution methodology for online selective graph queries based on the shortest path distance semantic, which finds various applications in practice. The essential intuition is to build a distance-aware index for online distance-based query processing and to eliminate redundant graph traversal as much as possible. We discuss how the solution can be applied to two types of research problems, distance join and vertex set bonding, which are distance-based graph pattern discovery and finding the structure-wise bonding of vertices, respectively

Space-Efficiency for Routing Schemes of Stretch Factor Three (Extended Abstract)

) Cyril Gavoille 1 , Marc Gengler 2 1 LaBRI, Universit&apos;e Bordeaux I, 351, cours de la Lib&apos;eration, 33405 Talence Cedex, France ([email protected]) 2 LIP, &apos; Ecole Normale Sup&apos;erieure de Lyon, 69364 Lyon Cedex 07, France ([email protected]). Abstract. We deal with routing algorithms on arbitrary n-node networks. A routing algorithm is a deterministic distributed algorithm which routes messages from any source to any destination. It includes not only the classical routing tables, but also the routing algorithm that generates paths with loops. Our goal is to design routing algorithms which minimize, for each router of the network, the amount of routing information that needs to be stored by the router in order to implement its own local routing algorithm. So as to simplify the implementation of a routing algorithm, names of the routers can be chosen in advance. We take also into account the efficiency of the routing, i.e., the length of the routing paths. The stretch fa..