7 research outputs found
Labeling Schemes with Queries
We study the question of ``how robust are the known lower bounds of labeling
schemes when one increases the number of consulted labels''. Let be a
function on pairs of vertices. An -labeling scheme for a family of graphs
\cF labels the vertices of all graphs in \cF such that for every graph
G\in\cF and every two vertices , the value can be inferred
by merely inspecting the labels of and .
This paper introduces a natural generalization: the notion of -labeling
schemes with queries, in which the value can be inferred by inspecting
not only the labels of and but possibly the labels of some additional
vertices. We show that inspecting the label of a single additional vertex (one
{\em query}) enables us to reduce the label size of many labeling schemes
significantly
Distance labeling schemes for trees
We consider distance labeling schemes for trees: given a tree with nodes,
label the nodes with binary strings such that, given the labels of any two
nodes, one can determine, by looking only at the labels, the distance in the
tree between the two nodes.
A lower bound by Gavoille et. al. (J. Alg. 2004) and an upper bound by Peleg
(J. Graph Theory 2000) establish that labels must use
bits\footnote{Throughout this paper we use for .}. Gavoille et.
al. (ESA 2001) show that for very small approximate stretch, labels use
bits. Several other papers investigate various
variants such as, for example, small distances in trees (Alstrup et. al.,
SODA'03).
We improve the known upper and lower bounds of exact distance labeling by
showing that bits are needed and that bits are sufficient. We also give ()-stretch labeling
schemes using bits for constant .
()-stretch labeling schemes with polylogarithmic label size have
previously been established for doubling dimension graphs by Talwar (STOC
2004).
In addition, we present matching upper and lower bounds for distance labeling
for caterpillars, showing that labels must have size . For simple paths with nodes and edge weights in , we show that
labels must have size
A note on models for graph representations
AbstractThis paper is intended more to ask questions than 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 a 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
Near-optimal labeling schemes for nearest common ancestors
We consider NCA labeling schemes: given a rooted tree , label the nodes of
with binary strings such that, given the labels of any two nodes, one can
determine, by looking only at the labels, the label of their nearest common
ancestor.
For trees with nodes we present upper and lower bounds establishing that
labels of size , are both sufficient and
necessary. (All logarithms in this paper are in base 2.)
Alstrup, Bille, and Rauhe (SIDMA'05) showed that ancestor and NCA labeling
schemes have labels of size . Our lower bound
increases this to for NCA labeling schemes. Since
Fraigniaud and Korman (STOC'10) established that labels in ancestor labeling
schemes have size , our new lower bound separates
ancestor and NCA labeling schemes. Our upper bound improves the
upper bound by Alstrup, Gavoille, Kaplan and Rauhe (TOCS'04), and our
theoretical result even outperforms some recent experimental studies by Fischer
(ESA'09) where variants of the same NCA labeling scheme are shown to all have
labels of size approximately
Near Optimal Adjacency Labeling Schemes for Power-Law Graphs
An adjacency labeling scheme labels the n nodes of a graph with bit strings in a way that allows, given the labels of two nodes, to determine adjacency based only on those bit strings. Though many graph families have been meticulously studied for this problem, a non-trivial labeling scheme for the important family of power-law graphs has yet to be obtained. This family is particularly useful for social and web networks as their underlying graphs are typically modelled as power-law graphs. Using simple strategies and a careful selection of a parameter, we show upper bounds for such labeling schemes of ~O(sqrt^{alpha}(n)) for power law graphs with coefficient alpha;, as well as nearly matching lower bounds. We also show two relaxations that allow for a label of logarithmic size, and extend the upper-bound technique to produce an improved distance labeling scheme for power-law graphs
Local Maps: New Insights into Mobile Agent Algorithms
In this paper, we study the complexity of computing with mobile agents having small local knowledge. In particular, we show that the number of mobile agents and the amount of local information given initially to agents can significantly influence the time complexity of resolving a distributed problem. Our results are based on a generic scheme allowing to transform a message passing algorithm, running on an -node graph , into a mobile agent one. By generic, we mean that the scheme is independent of both the message passing algorithm and the graph . Our scheme, coupled with a well-chosen clustered representation of the graph, induces \widetilde{O}(n)kkO(n/\sqrt{k})Gnn^{\epsilon}\widetilde{O}(D)D\epsilon\widetilde{O}(1)\widetilde{O}(n)\widetilde{O}(D)$ time algorithms