Project-Gang:Understanding graph representations

Inria / Raweb 2009
Presentation of the Project Gang

Section: New Results

Understanding graph representations

Graph labeling

Informative labeling schemes for trees

Participants : Pierre Fraigniaud, Amos Korman.

Ancestry labeling

In [14] , we consider ancestry labeling schemes . Such a scheme labels the nodes of any tree in such a way that ancestry queries between any two nodes can be answered just by looking at their corresponding labels. The common measure to evaluate the quality of an ancestry scheme is by its label size , that is the maximum number of bits stored in a label, taken over all n -node trees. The design of ancestry labeling schemes finds applications in XML search engines. In these contexts, even small improvements in the label size are important. In fact, the literature about this topic is interested in the exact label size rather than just its order of magnitude. As a result, following the proposal of a simple interval based ancestry scheme with label size 2logn bits (Kannan et al., SIDMA 92), a considerable amount of work was devoted to improve the bound on the label size. The current state of the art upper bound is $Im13 ${logn+O(\sqrt {logn})}$$ bits (Abiteboul et al., SICOMP 06) which is still far from the known logn + $\upper_omega$ (loglogn) lower bound (Alstrup et al., SODA 03). Moreover, the hidden constant factor in the additive $Im14 ${O(\sqrt {logn})}$$ term is large, which makes that scheme less efficient than the simple interval based scheme for typical current XML trees. It is for this reason that Kaplan et al. suggested other ancestry schemes which, on average, use labels whose size, for typical XML data, is smaller by about 10%-30% than the ones used for the interval based scheme (SODA 02). Motivated by the fact that typical XML trees have extremely small depth, we parameterizes the quality measure of an ancestry scheme not only by the number of nodes in the given tree but also by its depth. Our main result is the construction of an ancestry scheme that labels n -node trees of depth d with labels of size logn + 2logd + O(1) . This result is essentially optimal for trees of constant depth which is in fact the typical case in real XML data. In addition to our main result, we prove a result that may be of independent interest concerning the existence of a universal graph of linear size for the family of constant depth trees. This result improves the best known bound of 2^{O(log^*n)}n (Alstrup and Rauhe, FOCS 02) on the size of such a universal graph (the latter bound, however, holds for a universal graph for the family of all n -node trees).

On randomized representations of graphs using short labels

In [13] , we consider Informative labeling schemes which consist in labeling the nodes of graphs so that queries regarding any two nodes (e.g., are the two nodes adjacent?) can be answered by inspecting merely the labels of the corresponding nodes. Typically, the main goal of such schemes is to minimize the label size , that is, the maximum number of bits stored in a label. This concept was introduced by Kannan et al. [STOC'88] and was illustrated by giving very simple and elegant labeling schemes, for supporting adjacency and ancestry queries in n -node trees; both these schemes have label size 2logn . Motivated by relations between such schemes and other important notions such as universal graphs, extensive research has been made by the community to further reduce the label sizes of such schemes as much as possible. The current state of the art adjacency labeling scheme for trees has label size logn + O(log^*n) by Alstrup and Rauhe [FOCS'02], and the best known ancestry scheme for (rooted) trees has label size $Im15 ${logn+O(\sqrt {logn)}}$$ by Abiteboul et al., [SICOMP 2006]. We investigate the above notions from a probabilistic point of view. Informally, the goal is to investigate whether the label sizes can be improved if one allows for some probability of mistake when answering a query, and, if so, by how much. For that, we first present a model for probabilistic labeling schemes, and then construct various probabilistic one-sided error schemes for the adjacency and ancestry problems on trees. Some of our schemes significantly improve the bound on the label size of the corresponding deterministic schemes, while the others are matched with appropriate lower bounds showing that, for the resulting guarantees of success, one cannot expect to do much better in term of label size.

Compact labeling for spanning trees

Participants : Reuven Cohen [ Weizmann Institute of Science, Israel ] , Pierre Fraigniaud, David Ilcinkas [ CNRS LABRI, University of Bordeaux, France ] , Amos Korman, David Peleg [ Weizmann Institute of Science, Israel ] .

The article [17] (awarded the best conference paper) deals with compact label-based representations for trees. Consider an n -node undirected connected graph G with a predefined numbering on the ports of each node. The all-ports tree labeling $Im16 $\mtext cL_{all}$$ gives each node v of G a label containing the port numbers of all the tree edges incident to v . The upward tree labeling $Im17 $\mtext cL_{up}$$ labels each node v by the number of the port leading from v to its parent in the tree. Our measure of interest is the worst case and total length of the labels used by the scheme, denoted M_up(T) and S_up(T) for $Im17 $\mtext cL_{up}$$ and M_all(T) and S_all(T) for $Im16 $\mtext cL_{all}$$ . The problem studied in this paper is the following: Given a graph G and a predefined port labeling for it, with the ports of each node v numbered by 0, ..., deg(v)-1 , select a rooted spanning tree for G minimizing (one of) these measures. We show that the problem is polynomial for M_up(T) , S_up(T) and S_all(T) but NP-hard for M_all(T) (even for 3-regular planar graphs). We show that for every graph G and port numbering there exists a spanning tree T for which S_up(T) = O(nloglogn) . We give a tight bound of O(n) in the cases of complete graphs with arbitrary labeling and arbitrary graphs with symmetric port assignments. We conclude by discussing some applications for our tree representation schemes.

Labeling schemes for trees with queries

Participants : Amos Korman, Shay Kutten [ Technion, Israel ] .

On a more prospective way, [18] is intended more to ask questions than to give answers. Specifically, we consider models for labeling schemes, and discuss issues regarding the number of labels consulted vs. the sizes of the labels.

Recently, quite a few papers studied methods for representing network properties by assigning informative labels to the vertices of a network. Consider some graph function f on pairs of vertices (for example, f can be the distance function). In an f -labeling scheme, the labels are constructed in such a way so that given the labels of any two vertices u and v , one can compute the function f(u, v) (e.g. the graph distance between u and v ) just by looking at these two labels. Some very involved lower bounds for the sizes of the labels were proven. Also, some highly sophisticated labeling schemes were developed to ensure short labels.

In this paper, we demonstrate that such lower bounds are very sensitive to the number of vertices consulted. That is, we show several constructions of such labeling schemes that beat the lower bounds by large margins. Moreover, as opposed to the strong technical skills that were needed to develop the traditional labeling schemes, most of our schemes are almost trivial. The catch is that in our model, one needs to consult the labels of three vertices instead of two. That is, a query about vertices u and v can access also the label of some third vertex w (w is determined by the labels of u and v ). More generally, we address the model in which a query about vertices u and v can access also the labels of c other vertices. We term our generalized model labeling schemes with queries .

The main importance of this model is theoretical. Specifically, this paper may serve as a first step towards investigating different tradeoffs between the amount of labels consulted and the amount of information stored at each vertex. As we show, if all vertices can be consulted then the problem almost reduces to the corresponding sequential problem. On the other hand, consulting just the labels of u and v (or even just the label of u ) reduces the problem to a purely distributed one. Therefore, in a sense, our model spans a range of intermediate notions between the sequential and the distributed settings.

In addition to the theoretical interest, we also show cases that schemes constructed for our model can be translated to the traditional model or to the sequential model, thus, simplifying the construction for those models as well. For implementing query labeling schemes in a distributed environment directly, we point at a potential usage for some new paradigms that became common recently, such as P2P and overlay networks.

Efficient graph spanners

Distributed Computation of Sparse Spanner

Participants : Bilel Derbel [ CNRS LIFL, University of Lille, France ] , Cyril Gavoille [ CNRS LABRI, University of Bordeaux, France ] , David Peleg [ Weizmann Institute of Science, Israel ] , Laurent Viennot.

An ( $\alpha$ , $\beta$ ) -spanner of a graph G is a subgraph H that approximates distances in G within a multiplicative factor $\alpha$ and an additive error $\beta$ . More precisely, for any two nodes u, v of G , d_H(u, v) $\le$ $\alpha$ ·d_G(u, v) + $\beta$ . Computing sparse spanners is a fundamental problem of distributed computing and compact routing.

In [9] we provide distributed algorithms for computing sparse ( $\alpha$ , $\beta$ ) -spanners with $\alpha$ close to one. We present a generic distributed algorithm that in constant number of rounds computes, for every n -node graph and integer k $\ge$ 1 , an ( $\alpha$ , $\beta$ ) -spanner of O( $\beta$ n^{1 + 1/k}) edges, where $\alpha$ and $\beta$ are constants depending on k . For suitable parameters, this algorithm provides a (2k-1, 0) -spanner of at most kn^{1 + 1/k} edges in k rounds. For k = 2 and constant $\varepsilon$ >0 , it can also produce a (1 + $\varepsilon$ , 2- $\varepsilon$ ) -spanner of O(n^3/2) edges in constant time. More interestingly, for every integer k>1 , it can construct in constant time a (1 + $\varepsilon$ , O(1/ $\varepsilon$ )^k-2) -spanner of O( $\varepsilon$ ^{-k + 1}n^{1 + 1/k}) edges. Such deterministic construction was not previously known.

We also present a second generic deterministic and distributed algorithm based on the construction of small dominating sets and maximal independent sets. After computing such sets in sub-polynomial time, it constructs at its best a (1 + $\varepsilon$ , $\beta$ ) -spanner with O( $\beta$ n^{1 + 1/k}) edges, where $Im18 ${\#946 =k^{log({logk}/\#949 )+O(1)}}$$ . For k = 3 , it provides a (1 + $\varepsilon$ , 6- $\varepsilon$ ) -spanner with O( $\varepsilon$ ^-1n^4/3) edges.

Spanners for Ad Hoc Networks

Participants : Philippe Jacquet [ INRIA Hipercom, CRI Paris Rocquencourt, France ] , Laurent Viennot.

In [16] , we study a variation of spanners where a node is supposed to know the list of its neighbors. This is particularly suited to optimize the set of links advertized in a practical link state algorithm running in a possibly dense network such as an ad hoc network. Given an unweighted graph G , a sub-graph H with vertex set V(H) = V(G) is an (a, b) -remote-spanner if for each pair of points u and v the distance between u and v in H_u , the graph H augmented by all the edges between u and its neighbors in G , is at most a times the distance between u and v in G plus b . We extend this definition to k -connected graphs by considering the minimum length sum over k disjoint paths as a distance. We then say that an (a, b) -remote-spanner is k -connecting .

We give distributed algorithms for computing (1 + $\varepsilon$ , 1-2 $\varepsilon$ ) -remote-spanners for any $\varepsilon$ >0 , k -connecting (1, 0) -remote-spanners for any k $\ge$ 1 (yielding (1, 0) -remote-spanners for k = 1 ) and 2-connecting (2, -1) -remote-spanners. All these algorithms run in constant time for any unweighted input graph. The number of edges obtained for k -connecting (1, 0) -remote-spanner is within a logarithmic factor from optimal (compared to the best k -connecting (1, 0) -remote-spanner of the input graph). Interestingly, sparse (1, 0) -remote-spanners (i.e. preserving exact distances) with O(n^4/3) edges exist in random unit disk graphs. The number of edges obtained for (1 + $\varepsilon$ , 1-2 $\varepsilon$ ) -remote-spanners and 2-connecting (2, -1) -remote-spanners is linear if the input graph is the unit ball graph of a doubling metric (even if distances between nodes are unknown). Our methodology consists in characterizing remote-spanners as sub-graphs containing the union of small depth tree sub-graphs dominating nearby nodes. This leads to simple local distributed algorithms.

In [15] , we analyse the size of k -connecting (1,0)-remote- spanners in classical random graph models. Interestingly, the expected compression ratio in number of edges is $Im19 ${O(\mfrac knlogn)}$$ in the Erdös-Rényi graph model and $Im20 ${O({(\mfrac kn)}^\mfrac 23)}$$ in the unit disk graph model with a uniform Poisson distribution of nodes.

This work gives a theoretical foundation to the OLSR routing protocol (RFC 3626). In particular, it shows that multipoint relays (which are the basis of OLSR functionning) are an inherent structure for providing (1, 0) -remote-spanners, i.e. optimal routes.

Graph decompositions

Participants : Binh-Minh Bui-Xuan [ University of Bergen, Norway ] , Michel Habib, Vincent Limouzy, Fabien de Montgolfier, Michael Rao [ CNRS LABRI, University of Bordeaux, France ] .

NLC decomposition

Many width graph decompositions have been proposed. Thanks to Courcelle theorem, they allow to efficiently solve many hard (NP-complete) problems for graph classes, provided the decomposition width is bounded. NLC decomposition is a variation of cliquewidth, where the decomposition is a labelled tree. In [48] , the recognition of graphs of NLC 2 is addressed. The previous time complexity is improved to O(n²m) , and the algorithm is robust.

Umodular decomposition

A new decomposition of combinatorial structures is presented in [33] . Is is based on a generalisation of the modular decomposition. When applied to undirected graph, it gives the bijoin decomposition, and when applied to tournaments, it gives a new decomposition. We present proofs of existence and uniqueness of a decomposition tree, and polynomial-time algorithms.

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