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17
Labeling Schemes for Vertex Connectivity
 JOURNAL OF DISTRIBUTED COMPUTING
, 2007
"... This paper studies labeling schemes for the vertex connectivity function on general graphs. We consider the problem of assigning short labels to the nodes of any nnode graph is such a way that given the labels of any two nodes u and v, one can decide whether u and v are kvertex connected in G, i.e ..."
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This paper studies labeling schemes for the vertex connectivity function on general graphs. We consider the problem of assigning short labels to the nodes of any nnode graph is such a way that given the labels of any 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 k2 logn on the number of bits used in a label. The best previous upper bound for the label size of such a labeling scheme is 2k 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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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
Shorter Implicit Representation for Planar Graphs and Bounded Treewidth Graphs
 IN "15TH ANNUAL EUROPEAN SYMPOSIUM ON ALGORITHMS
, 2007
"... Implicit representation of graphs is a coding of the structure of graphs using distinct labels so that adjacency between any two vertices can be decided by inspecting their labels alone. All previous implicit representations of planar graphs were based on the classical three forests decomposition ..."
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Implicit representation of graphs is a coding of the structure of graphs using distinct labels so that adjacency between any two vertices can be decided by inspecting their labels alone. All previous implicit representations of planar graphs were based on the classical three forests decomposition technique (a.k.a. Schnyder’s trees), yielding asymptotically toa3lognbit 1 label representation where n is the number of vertices of the graph. We propose a new implicit representation of planar graphs using asymptotically 2 log nbit labels. As a byproduct we have an explicit construction of a graph with n 2+o(1) vertices containing all nvertex planar graphs as induced subgraph, the best previous size of such induceduniversal graph was O(n 3). More generally, for graphs excluding a fixed minor, we construct a 2logn + O(log log n) implicit representation. For treewidthk graphs we give a log n + O(k log log(n/k)) implicit representation, improving the O(k log n) representation of Kannan, Naor and Rudich [18] (STOC ’88). Our representations for planar and treewidthk graphs are easy to implement, all the labels can be constructed in O(n log n) time, and support constant time adjacency testing.
Distributed Relationship Schemes for Trees
 In Proc. 18th Int. Symp. on Algorithms and Computation
"... Abstract. We consider a distributed representation scheme for trees, supporting some special relationships between nodes at small distance. For instance, we show that for a tree T and an integer k we can assign local information on nodes such that we can decide for any two nodes u and v if the dist ..."
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Abstract. We consider a distributed representation scheme for trees, supporting some special relationships between nodes at small distance. For instance, we show that for a tree T and an integer k we can assign local information on nodes such that we can decide for any two nodes u and v if the distance between u and v is at most k and if so, compute it only using the local information assigned. For trees with n nodes, the local information assigned by our scheme are binary labels of log n + O(k log(k log(n/k))) bits, improving the results of Alstrup, Bille, and Rauhe (SODA '03).
An Optimal Ancestry Scheme and Small Universal Posets
, 2010
"... 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].
Labeling Schemes with Queries
, 2006
"... We study the question of “how robust are the known lower bounds of labeling schemes when one increases the number of consulted labels”. Let f be a function on pairs of vertices. An flabeling scheme for a family of graphs F labels the vertices of all graphs in F such that for every graph G ∈ F and e ..."
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We study the question of “how robust are the known lower bounds of labeling schemes when one increases the number of consulted labels”. Let f be a function on pairs of vertices. An flabeling scheme for a family of graphs F labels the vertices of all graphs in F such that for every graph G ∈ F and every two vertices u, v ∈ G, the value f(u, v) can be inferred by merely inspecting the labels of u and v. This paper introduces a natural generalization: the notion of flabeling schemes with queries, in which the value f(u, v) can be inferred by inspecting not only the labels of u and v but possibly the labels of some additional vertices. We show that inspecting the label of a single additional vertex (one query) enables us to reduce the label size of many labeling schemes significantly. In particular, we show that to support the distance function on nnode trees as well as the flow function on nnode general graphs, O(log n + log W)bit labels are sufficient and necessary, where W is the maximum (integral) capacity of an edge. We note that it was shown that any labeling scheme (without queries) supporting either the flow function on general graphs or the distance function on trees, must have label size Ω(log 2 n + log n log W). Using a single query, we also show a routing
Optimal Distance Labeling for Interval and Circulararc Graphs
, 2003
"... 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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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(log² n) bit distance labeling scheme.
On Randomized Representations of Graphs Using Short Labels
 in "Proc. 21st ACM Symposium on Parallelism in Algorithms and Architectures (SPAA)", 2009, p. 131137 IL
"... Informative labeling schemes 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 ..."
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Informative labeling schemes 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 nnode trees; both these schemes have label size 2 log n. 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 log n + O(log ∗ n) by Alstrup and Rauhe [FOCS’02], and the best known ancestry scheme for (rooted) trees has label size log n + O ( p log n) by Abiteboul et al., [SICOMP 2006]. This paper aims at investigating 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 onesided 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 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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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.
Chapter 14 Labeling Schemes
"... Imagine you want to repeatedly query a huge graph, e.g., a social or a road network. For example, you might need to find out whether two nodes are connected, or what the distance between two nodes is. Since the graph is so large, you distribute it among multiple servers in your data center. 14.1 Adj ..."
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Imagine you want to repeatedly query a huge graph, e.g., a social or a road network. For example, you might need to find out whether two nodes are connected, or what the distance between two nodes is. Since the graph is so large, you distribute it among multiple servers in your data center. 14.1 Adjacency Theorem 14.1. It is possible to assign labels of size 2 log n bits to nodes in a tree so that for every pair u, v of nodes, it is easy to tell whether u is adjacent to v by just looking at u and v’s labels. Proof. Choose a root in the tree arbitrarily so that every nonroot node has a parent. The label of each node u consists of two parts: The ID of u (from 1 to n), and the ID of u’s parent (or nothing if u is the root). Remarks: • What we have constructed above is called a labeling scheme, more precisely a labeling scheme for adjacency in trees. Formally, a labeling scheme is defined as follows. Definition 14.2. A labeling scheme consists of an encoder e and a decoder d. The encoder e assigns to each node v a label e(v). The decoder d receives the labels of the nodes in question and returns an answer to some query. The largest size (in bits) of a label assigned to a node is called the label size of the labeling scheme. Remarks: • In Theorem 14.1, the decoder receives two node labels e(u) and e(v), and its answer is Yes or No, depending on whether u and v are adjacent or not. The label size is 2 log n. • The label size is the complexity measure we are going to focus on in this chapter. The runtime of the encoder and the decoder are two other complexity measures that are studied in the literature.