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Labeling Schemes for Vertex Connectivity
"... This paper studies labeling schemes for the vertex connectivity function on general graphs. We consider the problem of labeling the nodes of any nnode graph is such a way that given the labels of two nodes u and v, one can decide whether u and v are kvertex connected in G, i.e., whether there exis ..."
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Cited by 8 (7 self)
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This paper studies labeling schemes for the vertex connectivity function on general graphs. We consider the problem of labeling the nodes of any nnode graph is such a way that given the labels of two nodes u and v, one can decide whether u and v are kvertex connected in G, i.e., whether there exist k vertex disjoint paths connecting u and v. The paper establishes an upper bound of k 2 log n on the number of bits used in a label. The best previous upper bound for the label size of such labeling scheme is 2 k log n.
Compact Ancestry Labeling Schemes for XML Trees
 In Proc. 21st ACMSIAM Symp. on Discrete Algorithms (SODA
, 2010
"... An ancestry labeling 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 ..."
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Cited by 4 (2 self)
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An ancestry labeling 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 nnode 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. As a result, following the proposal of a simple interval based ancestry scheme with label size 2 log n bits (Kannan et al., STOC 88), a considerable amount of work was devoted to improve the bound on the label size. The current state of the art upper bound is log n + O ( √ log n) bits (Abiteboul et al., SICOMP 06) which is still far from the known log n + Ω(log log n) lower bound (Alstrup et al., SODA 03). Motivated by the fact that typical XML trees have extremely small depth, this paper 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 nnode trees of depth d with labels of size log n + 2 log d + O(1). In addition to our main result, we prove a result that may be of independent interest concerning the existence of a small universal graph for the family of trees with bounded depth. 1
Optimal Distance Labeling for Interval and Circulararc Graphs
"... In this paper we design a distance labeling scheme with O(log n) bit labels for interval graphs and circulararc graphs with n vertices. The set of all the labels is constructible in O(n) time if the interval representation of the graph is given and sorted. As a byproduct we give a new and simpl ..."
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Cited by 1 (0 self)
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In this paper we design a distance labeling scheme with O(log n) bit labels for interval graphs and circulararc graphs with n vertices. The set of all the labels is constructible in O(n) time if the interval representation of the graph is given and sorted. As a byproduct we give a new and simpler O(n) space datastructure computable after O(n) preprocessing time, and supporting constant worstcase time distance queries for interval and circulararc graphs. These optimal bounds improve the previous scheme of Katz, Katz, and Peleg (STACS '00) by a log n factor. To the best of our knowledge, the interval graph family is the rst hereditary family having 2 unlabeled nvertex graphs and supporting a o(log n) bit distance labeling scheme.
Compact Ancestry Labeling Schemes for Trees of Small Depth
, 2009
"... An ancestry labeling scheme labels the nodes of any tree in such a way that ancestry queries between any two nodes in a tree can be answered just by looking at their corresponding labels. The common measure to evaluate the quality of an ancestry labeling scheme is by its label size, that is the maxi ..."
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Cited by 1 (1 self)
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An ancestry labeling scheme labels the nodes of any tree in such a way that ancestry queries between any two nodes in a tree can be answered just by looking at their corresponding labels. The common measure to evaluate the quality of an ancestry labeling scheme is by its label size, that is the maximal number of bits stored in a label, taken over all nnode trees. The design of ancestry labeling schemes finds applications in XML search engines. In the context of these applications, 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 an original scheme of size 2 log n bits, a considerable amount of work was devoted to improve the bound on the label size. The current state of the art upper bound is log n + O ( √ log n) bits which is still far from the known log n + Ω(log log n) lower bound. Moreover, the hidden constant factor in the additive O ( √ log n) term is large, which makes this term dominate the label size for typical current XML trees. In attempt to provide good performances for real XML data, we rely on the observation that the depth of a typical XML tree is bounded from above by a small constant. Having this in mind, we present an ancestry labeling scheme of size log n + 2 log d + O(1), for the family of trees with at most n nodes and depth at most d. In addition to our main result, we prove a result that may be of independent interest concerning the existence of a linear universal graph for the family of forests with trees of bounded depth.
A Note on Models for Graph Representations ∗
"... This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. ..."
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This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. ACCEPTED MANUSCRIPT
An Optimal Ancestry Scheme and Small Universal Posets ∗
"... In this paper, we solve the ancestry problem, which was introduced more than twenty years ago by Kannan et al. [STOC ’88], and is among the most wellstudied problems in the field of informative labeling schemes. Specifically, we construct an ancestry labeling scheme for nnode trees with label size ..."
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In this paper, we solve the ancestry problem, which was introduced more than twenty years ago by Kannan et al. [STOC ’88], and is among the most wellstudied problems in the field of informative labeling schemes. Specifically, we construct an ancestry labeling scheme for nnode trees with label size log 2 n + O(log log n) bits, thus matching the log 2 n + Ω(log log n) bits lower bound given by Alstrup et al. [SODA ’03]. Besides its optimal label size, our scheme assigns the labels in linear time, and guarantees that any ancestry query can be answered in constant time. In addition to its potential impact in terms of improving the performances of XML search engines, our ancestry scheme is also useful in the context of partially ordered sets. Specifically, for any fixed integer k, our scheme enables the construction of a universal poset of size O(n k log 4k n) for the family of nelement posets with treedimension at most k. This bound is almost tight thanks to a lower bound of n k−o(1) due to Alon and Scheinerman [Order ’88].
Small Universal Distance Matrices
, 2003
"... In this paper, we propose to associate to any graph family F the following complexity measure: the dimension of the smallest universal distance matrix for F. Given n ..."
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In this paper, we propose to associate to any graph family F the following complexity measure: the dimension of the smallest universal distance matrix for F. Given n
An Optimal Labeling Scheme for Ancestry Queries ∗
, 909
"... An ancestry labeling scheme assigns labels (bit strings) to the nodes of rooted trees such that ancestry queries between any two nodes in a tree can be answered merely by looking at their corresponding labels. The quality of an ancestry labeling scheme is measured by its label size, that is the maxi ..."
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An ancestry labeling scheme assigns labels (bit strings) to the nodes of rooted trees such that ancestry queries between any two nodes in a tree can be answered merely by looking at their corresponding labels. The quality of an ancestry labeling scheme is measured by its label size, that is the maximal number of bits in a label of a tree node. In addition to its theoretical appeal, the design of efficient ancestry labeling schemes is motivated by applications in web search engines. For this purpose, 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 intervalbased ancestry scheme with label size 2 log2 n bits (Kannan et al., STOC ’88), a considerable amount of work was devoted to improve the bound on the size of a label. The current state of the art upper bound is log2 n + O ( √ log n) bits (Abiteboul et al., SODA ’02) which is still far from the known log2 n+Ω(log log n) bits lower bound (Alstrup et al., SODA’03). In this paper we close the gap between the known lower and upper bounds, by constructing an ancestry labeling scheme with label size log2 n + O(log log n) bits. In addition to the optimal label size, our scheme assigns the labels in linear time and can support any ancestry query in constant time.