9,879 research outputs found
The vertex-transitive TLF-planar graphs
We consider the class of the topologically locally finite (in short TLF)
planar vertex-transitive graphs, a class containing in particular all the
one-ended planar Cayley graphs and the normal transitive tilings. We
characterize these graphs with a finite local representation and a special kind
of finite state automaton named labeling scheme. As a result, we are able to
enumerate and describe all TLF-planar vertex-transitive graphs of any given
degree. Also, we are able decide to whether any TLF-planar transitive graph is
Cayley or not.Comment: Article : 23 pages, 15 figures Appendix : 13 pages, 72 figures
Submitted to Discrete Mathematics The appendix is accessible at
http://www.labri.fr/~renault/research/research.htm
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
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
Polynomial-Time Space-Optimal Silent Self-Stabilizing Minimum-Degree Spanning Tree Construction
Motivated by applications to sensor networks, as well as to many other areas,
this paper studies the construction of minimum-degree spanning trees. We
consider the classical node-register state model, with a weakly fair scheduler,
and we present a space-optimal \emph{silent} self-stabilizing construction of
minimum-degree spanning trees in this model. Computing a spanning tree with
minimum degree is NP-hard. Therefore, we actually focus on constructing a
spanning tree whose degree is within one from the optimal. Our algorithm uses
registers on bits, converges in a polynomial number of rounds, and
performs polynomial-time computation at each node. Specifically, the algorithm
constructs and stabilizes on a special class of spanning trees, with degree at
most . Indeed, we prove that, unless NP coNP, there are no
proof-labeling schemes involving polynomial-time computation at each node for
the whole family of spanning trees with degree at most . Up to our
knowledge, this is the first example of the design of a compact silent
self-stabilizing algorithm constructing, and stabilizing on a subset of optimal
solutions to a natural problem for which there are no time-efficient
proof-labeling schemes. On our way to design our algorithm, we establish a set
of independent results that may have interest on their own. In particular, we
describe a new space-optimal silent self-stabilizing spanning tree
construction, stabilizing on \emph{any} spanning tree, in rounds, and
using just \emph{one} additional bit compared to the size of the labels used to
certify trees. We also design a silent loop-free self-stabilizing algorithm for
transforming a tree into another tree. Last but not least, we provide a silent
self-stabilizing algorithm for computing and certifying the labels of a
NCA-labeling scheme
Adjacency labeling schemes and induced-universal graphs
We describe a way of assigning labels to the vertices of any undirected graph
on up to vertices, each composed of bits, such that given the
labels of two vertices, and no other information regarding the graph, it is
possible to decide whether or not the vertices are adjacent in the graph. This
is optimal, up to an additive constant, and constitutes the first improvement
in almost 50 years of an bound of Moon. As a consequence, we
obtain an induced-universal graph for -vertex graphs containing only
vertices, which is optimal up to a multiplicative constant,
solving an open problem of Vizing from 1968. We obtain similar tight results
for directed graphs, tournaments and bipartite graphs
Answering Regular Path Queries on Workflow Provenance
This paper proposes a novel approach for efficiently evaluating regular path
queries over provenance graphs of workflows that may include recursion. The
approach assumes that an execution g of a workflow G is labeled with
query-agnostic reachability labels using an existing technique. At query time,
given g, G and a regular path query R, the approach decomposes R into a set of
subqueries R1, ..., Rk that are safe for G. For each safe subquery Ri, G is
rewritten so that, using the reachability labels of nodes in g, whether or not
there is a path which matches Ri between two nodes can be decided in constant
time. The results of each safe subquery are then composed, possibly with some
small unsafe remainder, to produce an answer to R. The approach results in an
algorithm that significantly reduces the number of subqueries k over existing
techniques by increasing their size and complexity, and that evaluates each
subquery in time bounded by its input and output size. Experimental results
demonstrate the benefit of this approach
Near-Optimal Induced Universal Graphs for Bounded Degree Graphs
A graph is an induced universal graph for a family of graphs if every
graph in is a vertex-induced subgraph of . For the family of all
undirected graphs on vertices Alstrup, Kaplan, Thorup, and Zwick [STOC
2015] give an induced universal graph with vertices,
matching a lower bound by Moon [Proc. Glasgow Math. Assoc. 1965].
Let . Improving asymptotically on previous results by
Butler [Graphs and Combinatorics 2009] and Esperet, Arnaud and Ochem [IPL
2008], we give an induced universal graph with vertices for the family of graphs with vertices of maximum degree
. For constant , Butler gives a lower bound of
. For an odd constant , Esperet et al.
and Alon and Capalbo [SODA 2008] give a graph with
vertices. Using their techniques for any
(including constant) even values of gives asymptotically worse bounds than
we present.
For large , i.e. when , the previous best
upper bound was due to Adjiashvili and
Rotbart [ICALP 2014]. We give upper and lower bounds showing that the size is
. Hence the optimal size is
and our construction is within a factor of
from this. The previous results were
larger by at least a factor of .
As a part of the above, proving a conjecture by Esperet et al., we construct
an induced universal graph with vertices for the family of graphs with
max degree . In addition, we give results for acyclic graphs with max degree
and cycle graphs. Our results imply the first labeling schemes that for any
are at most bits from optimal
Route Planning in Transportation Networks
We survey recent advances in algorithms for route planning in transportation
networks. For road networks, we show that one can compute driving directions in
milliseconds or less even at continental scale. A variety of techniques provide
different trade-offs between preprocessing effort, space requirements, and
query time. Some algorithms can answer queries in a fraction of a microsecond,
while others can deal efficiently with real-time traffic. Journey planning on
public transportation systems, although conceptually similar, is a
significantly harder problem due to its inherent time-dependent and
multicriteria nature. Although exact algorithms are fast enough for interactive
queries on metropolitan transit systems, dealing with continent-sized instances
requires simplifications or heavy preprocessing. The multimodal route planning
problem, which seeks journeys combining schedule-based transportation (buses,
trains) with unrestricted modes (walking, driving), is even harder, relying on
approximate solutions even for metropolitan inputs.Comment: This is an updated version of the technical report MSR-TR-2014-4,
previously published by Microsoft Research. This work was mostly done while
the authors Daniel Delling, Andrew Goldberg, and Renato F. Werneck were at
Microsoft Research Silicon Valle
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