Localized and compact data-structure for comparability graphs

Bazzaro, Fabrice; Gavoille, Cyril

doi:10.1016/j.disc.2007.12.091

Cited by 25 publications

(41 citation statements)

References 27 publications

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“…We say that a vertex v of G is identified if one can decide whether v = a i,j or v = b i or v = c j (and then give for each case the corresponding indices i and j). So identifying all the vertices of G completes step (2). As we will see later, identifying a vertex v also allows the recovery of its interval representation and eventually its associated integer s i,j , completing step (1) as well.…”

Section: On the Number Of Interval Graphsmentioning

confidence: 95%

“…This paper positively answers the above questions by considering the family of interval graphs. Interestingly, using some ideas from this paper, [2] positively answers the case of permutation graphs.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, up to now, none of the hereditary graph families is known to support an o(log 2 n)-distance labeling scheme. 2 It should be noticed that among the graph families listed above, the Ω(log 2 n) lower bound on label length for n-vertex trees applies to neither interval nor permutation graphs. So the second question is: (2) Can we improve the O(log 2 n) upper bound [20] for either interval or permutation graphs?…”

Section: Introductionmentioning

confidence: 99%

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Optimal Distance Labeling for Interval Graphs and Related Graph Families

Gavoille¹,

Paul²

2008

SIAM J. Discrete Math.

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Abstract.A distance labeling scheme is a distributed graph representation that assigns labels to the vertices and enables answering distance queries between any pair (x, y) of vertices by using only the labels of x and y. This paper presents an optimal distance labeling scheme with labels of O(log n) bits for the n-vertex interval graphs family. It improves by log n factor the best known upper bound of [M. Katz, N. A. Katz, and D. Peleg, Distance labeling schemes for well-separated graph classes, in Proceedings of the 17th Annual Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Comput. Sci. 1770, Springer-Verlag, Berlin, 2000. Moreover, the scheme supports constant time distance queries, and if the interval representation of the input graph is given and the intervals are sorted, then the set of labels can be computed in O(n) time. Our result is tight as we show that the length of any label is at least 3 log n − O(log log n) bits. This lower bound derives from a new estimator of the number of unlabeled n-vertex interval graphs, that is, 2 Ω(n log n) . To our knowledge, interval graphs are thereby the first known nontrivial hereditary family with 2 Ω(nf (n)) unlabeled elements and with a distance labeling scheme with f (n) bit labels.

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Section: On the Number Of Interval Graphsmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Optimal Distance Labeling for Interval Graphs and Related Graph Families

Gavoille¹,

Paul²

2008

SIAM J. Discrete Math.

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“…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

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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.

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“…Theorem E (Bazzaro and Gavoille [2]). The number of unlabeled permutation graphs with N vertices is 2 Ω (N log N) .…”

Section: Permutation Graphsmentioning

confidence: 99%

OBDD Representation of Intersection Graphs

Takaoka

Tayu

Ueno

2015

IEICE Trans. Inf. & Syst.

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Asahi TAKAOKA†a) , Student Member, Satoshi TAYU †b) , Member, and Shuichi UENO †c) , Fellow SUMMARY Ordered Binary Decision Diagrams (OBDDs for short) are popular dynamic data structures for Boolean functions. In some modern applications, we have to handle such huge graphs that the usual explicit representations by adjacency lists or adjacency matrices are infeasible. To deal with such huge graphs, OBDD-based graph representations and algorithms have been investigated. Although the size of OBDD representations may be large in general, it is known to be small for some special classes of graphs. In this paper, we show upper bounds and lower bounds of the size of OBDDs representing some intersection graphs such as bipartite permu-

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Localized and compact data-structure for comparability graphs

Cited by 25 publications

References 27 publications

Optimal Distance Labeling for Interval Graphs and Related Graph Families

Optimal Distance Labeling for Interval Graphs and Related Graph Families

Simpler, faster and shorter labels for distances in graphs

OBDD Representation of Intersection Graphs

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