6,930 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
Dynamic and Multi-functional Labeling Schemes
We investigate labeling schemes supporting adjacency, ancestry, sibling, and
connectivity queries in forests. In the course of more than 20 years, the
existence of labeling schemes supporting each of these
functions was proven, with the most recent being ancestry [Fraigniaud and
Korman, STOC '10]. Several multi-functional labeling schemes also enjoy lower
or upper bounds of or
respectively. Notably an upper bound of for
adjacency+siblings and a lower bound of for each of the
functions siblings, ancestry, and connectivity [Alstrup et al., SODA '03]. We
improve the constants hidden in the -notation. In particular we show a lower bound for connectivity+ancestry and
connectivity+siblings, as well as an upper bound of for connectivity+adjacency+siblings by altering existing
methods.
In the context of dynamic labeling schemes it is known that ancestry requires
bits [Cohen, et al. PODS '02]. In contrast, we show upper and lower
bounds on the label size for adjacency, siblings, and connectivity of
bits, and to support all three functions. There exist efficient
adjacency labeling schemes for planar, bounded treewidth, bounded arboricity
and interval graphs. In a dynamic setting, we show a lower bound of
for each of those families.Comment: 17 pages, 5 figure
Near-optimal adjacency labeling scheme for power-law graphs
An adjacency labeling scheme is a method that assigns labels to the vertices
of a graph such that adjacency between vertices can be inferred directly from
the assigned label, without using a centralized data structure. We devise
adjacency labeling schemes for the family of power-law graphs. This family that
has been used to model many types of networks, e.g. the Internet AS-level
graph. Furthermore, we prove an almost matching lower bound for this family. We
also provide an asymptotically near- optimal labeling scheme for sparse graphs.
Finally, we validate the efficiency of our labeling scheme by an experimental
evaluation using both synthetic data and real-world networks of up to hundreds
of thousands of vertices
Simpler, faster and shorter labels for distances in graphs
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 label size (log
3)/2 + o(n) (logarithms are in base 2) and O(1) 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(loglog 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 label sizes. For r-additive approximation schemes, where distances can be
off by an additive constant r, we give 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 scheme: n/2. We also give results for bipartite graphs and for
exact and 1-additive distance oracles
Labeling Workflow Views with Fine-Grained Dependencies
This paper considers the problem of efficiently answering reachability
queries over views of provenance graphs, derived from executions of workflows
that may include recursion. Such views include composite modules and model
fine-grained dependencies between module inputs and outputs. A novel
view-adaptive dynamic labeling scheme is developed for efficient query
evaluation, in which view specifications are labeled statically (i.e. as they
are created) and data items are labeled dynamically as they are produced during
a workflow execution. Although the combination of fine-grained dependencies and
recursive workflows entail, in general, long (linear-size) data labels, we show
that for a large natural class of workflows and views, labels are compact
(logarithmic-size) and reachability queries can be evaluated in constant time.
Experimental results demonstrate the benefit of this approach over the
state-of-the-art technique when applied for labeling multiple views.Comment: VLDB201
A simple and optimal ancestry labeling scheme for trees
We present a ancestry labeling scheme for trees. The
problem was first presented by Kannan et al. [STOC 88'] along with a simple solution. Motivated by applications to XML files, the label size was
improved incrementally over the course of more than 20 years by a series of
papers. The last, due to Fraigniaud and Korman [STOC 10'], presented an
asymptotically optimal labeling scheme using
non-trivial tree-decomposition techniques. By providing a framework
generalizing interval based labeling schemes, we obtain a simple, yet
asymptotically optimal solution to the problem. Furthermore, our labeling
scheme is attained by a small modification of the original solution.Comment: 12 pages, 1 figure. To appear at ICALP'1
Sublinear Distance Labeling
A distance labeling scheme labels the nodes of a graph with binary
strings such that, given the labels of any two nodes, one can determine the
distance in the graph between the two nodes by looking only at the labels. A
-preserving distance labeling scheme only returns precise distances between
pairs of nodes that are at distance at least from each other. In this paper
we consider distance labeling schemes for the classical case of unweighted
graphs with both directed and undirected edges.
We present a bit -preserving distance labeling
scheme, improving the previous bound by Bollob\'as et. al. [SIAM J. Discrete
Math. 2005]. We also give an almost matching lower bound of
. With our -preserving distance labeling scheme as a
building block, we additionally achieve the following results:
1. We present the first distance labeling scheme of size for sparse
graphs (and hence bounded degree graphs). This addresses an open problem by
Gavoille et. al. [J. Algo. 2004], hereby separating the complexity from
distance labeling in general graphs which require bits, Moon [Proc.
of Glasgow Math. Association 1965].
2. For approximate -additive labeling schemes, that return distances
within an additive error of we show a scheme of size for .
This improves on the current best bound of by
Alstrup et. al. [SODA 2016] for sub-polynomial , and is a generalization of
a result by Gawrychowski et al. [arXiv preprint 2015] who showed this for
.Comment: A preliminary version of this paper appeared at ESA'1
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