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

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

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A Theoretical Model for Routing Complexity

This paper introduces a formal model for studying the complexity of routing in networks. The aim of this model is to capture both time complexity and space complexity. In particular, the model takes into account the input and output facilities of routers. A routing program is a RAM-program with five additional instructions that allow to handle incoming and outgoing headers, and input and output ports. One of these five additional instructions, called release, captures the possible use of hardware facilities to speed up routing. Using our model, we show..

Labeling Schemes for Dynamic Tree Networks

Distance labeling schemes are composed of a marker algorithm for labeling the vertices of a graph with short labels, coupled with a decoder algorithm allowing one to compute the distance between any two vertices directly from their labels (without using any additional information). As applications for distance labeling schemes concern mainly large and dynamically changing networks, it is of interest to study distributed dynamic labeling schemes. The current paper considers the problem on dynamic trees, and proposes efficient distributed schemes for it. The paper first presents a labeling scheme for distances in the dynamic tree model, with amortized message complexity O(log 2 n) per operation, where n is the size of the tree at the time the operation takes place. The protocol maintains O(log 2 n) bit labels. This label size is known to be optimal even in the static scenario. A more general labeling scheme is then introduced for the dynamic tree model, based on extending an existing static tree labeling scheme to the dynamic setting. The approach fits a number of natural tree functions, such as distance, separation level and flow. The main resulting scheme incurs an overhead of a O(log n) multiplicative factor in both the label size and amortized message complexity in the case of dynamically growing trees (with no vertex deletions). If an upper bound on n is known in advance, this method yields a different tradeoff, with an O(log 2 n / log log n) multiplicative over-head on the label size but only an O(log n / log log n) overhead on the amortized message complexity. In the fully-dynamic model the scheme incurs also an increased additive overhead in amortized communication, of O(log 2 n) messages per operation

Labeling Schemes for Weighted Dynamic Trees

A Distance labeling scheme is a type of localized network representation in which short labels are assigned to the vertices, allowing one to infer the distance between any two vertices directly from their labels, without using any additional information sources. As most applications for network representations in general, and distance labeling schemes in particular, concern large and dynamically changing networks, it is of in-terest to focus on distributed dynamic labeling schemes. The paper considers dynamic weighted trees where the vertices of the trees are fixed but the (positive integral) weights of the edges may change. The two models considered are the edge-dynamic model, where from time to time some edge changes its weight by a fixed quanta, and the increasing-dynamic model in which edge weights can only grow. The paper presents distributed approximate distance labeling schemes for the two dynamic models, which are efficient in terms of the required label size and communication complexity involved in updating the labels following the weight changes

A Space Lower Bound for Routing in Trees

The design of compact routing schemes in trees form the kernel of sophisticated strategies for compact routing in arbitrary graphs. This paper focuses on the space complexity for routing messages along shortest paths in trees. It was recently shown that the family of n-node trees supports routing schemes using addresses and routing tables of size O(log n= log log n) bits per node, if the output port numbers of each node are chosen by an adversary. This paper shows that this result is tight, that is the sum of the sizes of the address and of the local routing table is at least n= log log n) bits for some node of some tree

Improved Compact Routing Scheme for Chordal Graphs

This paper concerns routing with succinct tables in chordal graphs. We show how to construct in polynomial time, for every n-node chordal graph, a routing scheme using routing tables and addresses of O(log³ n/ log log n) bits per node, and O(log² n/ log log n) bit not alterable headers such that the length of the route between any two nodes is at most the distance between the nodes in the graph plus two