Optimal Distance Labeling for Interval Graphs and Related Graph Families

Gavoille, Cyril; Paul, Christophe

doi:10.1137/050635006

Cited by 32 publications

(30 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The O(n) bit upper bound is tight for general graphs, and a lower bound of (log 2 n) bit on the label length is known for trees [36], implying that all the results mentioned above are tight as well, since all those graph families contain trees. Later, [9,32] showed an optimal bound of O(log n) bits for interval graphs, permutation graphs, and their generalizations.…”

Section: Related Work On Distance and Routing Labeling Schemesmentioning

confidence: 98%

Additive Spanners and Distance and Routing Labeling Schemes for Hyperbolic Graphs

et al. 2010

View full text Add to dashboard Cite

International audience$\delta$-Hyperbolic metric spaces have been defined by M. Gromov in 1987 via a simple 4-point condition: for any four points $u,v,w,x$, the two larger of the distance sums $d(u,v)+d(w,x), d(u,w)+d(v,x), d(u,x)+d(v,w)$ differ by at most $2\delta$. They play an important role in geometric group theory, geometry of negatively curved spaces, and have recently become of interest in several domains of computer science, including algorithms and networking. For example, (a) it has been shown empirically that the internet topology embeds with better accuracy into a hyperbolic space than into an Euclidean space of comparable dimension, (b) every connected finite graph has an embedding in the hyperbolic plane so that the greedy routing based on the virtual coordinates obtained from this embedding is guaranteed to work. A connected graph $G=(V,E)$ equipped with standard graph metric $d_G$ is $\delta$-{\it hyperbolic} if the metric space $(V,d_G)$ is $\delta$-hyperbolic. In this paper, using our Layering Partition technique, we provide a simpler construction of distance approximating trees of $\delta$-hyperbolic graphs on $n$ vertices with an additive error $O(\delta\log n)$ and show that every $n$-vertex $\delta$-hyperbolic graph has an additive $O(\delta \log n)$-spanner with at most $O(\delta n)$ edges. As a consequence, we show that the family of $\delta$-hyperbolic graphs with $n$ vertices enjoys an $O(\delta\log n)$-additive routing labeling scheme with $O(\delta\log^2n)$ bit labels and $O(\log\delta)$ time routing protocol, and an easier constructable $O(\delta\log n)$-additive distance labeling scheme with $O(\log^2n)$ bit labels and constant time distance decoder

show abstract

Section: Related Work On Distance and Routing Labeling Schemesmentioning

confidence: 98%

Additive Spanners and Distance and Routing Labeling Schemes for Hyperbolic Graphs

et al. 2010

View full text Add to dashboard Cite

show abstract

“…Gavoille and Paul [13] have proved a lower bound of 2 Ω(N log N) for the number of interval graphs on N vertices which can easily be adapted for convex graphs on 3N vertices. We show that 2 N log N−o(N log N) is also a lower bound for the number of unlabeled convex graphs on 3N vertices.…”

Section: Theorem 4 ([27]) For the Number Of Unlabeled Trees N T (N) mentioning

confidence: 98%

On the OBDD representation of some graph classes

Bollig

Bury

2016

Discrete Applied Mathematics

View full text Add to dashboard Cite

a b s t r a c tOrdered Binary Decision Diagrams (OBDDs) are a popular data structure for Boolean functions. Given the rapid growth of application-based networks, an heuristic approach to deal with very large structured graphs are implicit OBDD-based graph algorithms. Vertices of an input graph are binary encoded and the edge set is represented by its characteristic function. Since OBDDs are able to take advantage of the presence of regular substructures, this approach leads sometimes to sublinear graph representations. By simple counting arguments it is easy to see that almost all graphs on N vertices cannot be represented by OBDDs of polylogarithmic size with respect to N. On the other hand, very simply structured graphs like grid graphs have small OBDD representations. Here, the investigation for which significant graph classes succinct OBDD representations are possible is continued. Using a new method how to use the structure of the adjacency matrix representation of a graph to count subfunctions of the characteristic Boolean function of its edge set, known upper bounds on the OBDD size of interval graphs are improved. Furthermore, the OBDD size for interval bigraphs, (bi)convex graphs, bipartite permutation graphs, chain graphs, threshold graphs, and trees is analyzed. Except for interval graphs, interval bigraphs, and convex graphs the presented bounds are tight. Finally, regular graphs are considered and examples are presented with small and large OBDD size. Moreover, a variable ordering often used in the implicit setting is investigated.

show abstract

“…Space is listed presented without second order terms. A graphical presentation of the results is given in Figure 1 Distance labeling schemes for various families of graphs exist, e.g., for trees [5,55], bounded treewidth [36], distance-hereditary [34], bounded clique-width [21], some non-positively curved plane [18], interval [35] and permutation graphs [10]. In [36] it is proved that distance labels require Θ(log 2 n) bits for trees, O( √ n log n) and Ω(n 1/3 ) bits for planar graphs, and Ω( √ n) bits for bounded degree graphs.…”

Section: Distance Labelingmentioning

confidence: 99%

Simpler, faster and shorter labels for distances in graphs

Alstrup¹,

Gavoille²,

Halvorsen³

et al. 2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

Self Cite

View full text Add to dashboard Cite

We consider how to assign labels to any undirected graph with n nodes such that, given the labels of two nodes and no other information regarding the graph, it is possible to determine the distance between the two nodes. The challenge in such a distance labeling scheme is primarily to minimize the maximum label lenght and secondarily to minimize the time needed to answer distance queries (decoding). Previous schemes have offered different trade-offs between label lengths and query time. This paper presents a simple algorithm with shorter labels and shorter query time than any previous solution, thereby improving the state-of-the-art with respect to both label length and query time in one single algorithm. Our solution addresses several open problems concerning label length and decoding time and is the first improvement of label length for more than three decades.More specifically, we present a distance labeling scheme with labels of length log 3 2 n + o(n) bits 1 and constant decoding time. This outperforms all existing results with respect to both size and decoding time, including Winkler's (Combinatorica 1983) decade-old result, which uses labels of size (log 3)n and O(n/ log n) decoding time, and Gavoille et al. (SODA'01), which uses labels of size 11n + o(n) and O(log log n) decoding time. In addition, our algorithm is simpler than the previous ones. In the case of integral edge weights of size at most W , we present almost matching upper and lower bounds for the label size ℓ: 1 2 (n − 1) log W 2 + 1 ≤ ℓ ≤ 1 2 n log (2W + 1) + O(log n · log(nW )). Furthermore, for r-additive approximation labeling schemes, where distances can be off by up to an additive constant r, we present both upper and lower bounds. In particular, we present an upper bound for 1-additive approximation schemes which, in the unweighted case, has the same size (ignoring second order terms) as an adjacency labeling scheme, namely n/2. We also give results for bipartite graphs as well as for exact and 1-additive distance oracles.

show abstract

Optimal Distance Labeling for Interval Graphs and Related Graph Families

Cited by 32 publications

References 32 publications

Additive Spanners and Distance and Routing Labeling Schemes for Hyperbolic Graphs

Additive Spanners and Distance and Routing Labeling Schemes for Hyperbolic Graphs

On the OBDD representation of some graph classes

Simpler, faster and shorter labels for distances in graphs

Contact Info

Product

Resources

About