74,866 research outputs found
Distributed Graph Isomorphism using Quantum Walks
Graph isomorphism being an NP problem, most of the systems that solves the graph isomorphism are constrained with some classes of the graph, and do not work for all types of graphs in polynomial time. We exploited the two particle quantum walks on different classes of graphs including strongly regular graphs which are co-spectral in nature. We simulated two particle quantum walks on graph using distributed algorithm. To show the effectiveness of the technique, we applied it to the large graphs derived from images using Delauney triangulation. The results show a remarkable speedup for large data. The two-particle quantum walks is implemented in map-reduce programming technique which scales the computation as the cluster get scaled to account Big data. We checked the isomorphism of the graphs with upto 100 vertices in polynomial time. The system is scalable to accept big inputs from any other domain in graph format.
DOI: 10.17762/ijritcc2321-8169.15021
Map matching queries on realistic input graphs under the Fr\'echet distance
Map matching is a common preprocessing step for analysing vehicle
trajectories. In the theory community, the most popular approach for map
matching is to compute a path on the road network that is the most spatially
similar to the trajectory, where spatial similarity is measured using the
Fr\'echet distance. A shortcoming of existing map matching algorithms under the
Fr\'echet distance is that every time a trajectory is matched, the entire road
network needs to be reprocessed from scratch. An open problem is whether one
can preprocess the road network into a data structure, so that map matching
queries can be answered in sublinear time.
In this paper, we investigate map matching queries under the Fr\'echet
distance. We provide a negative result for geometric planar graphs. We show
that, unless SETH fails, there is no data structure that can be constructed in
polynomial time that answers map matching queries in query
time for any , where and are the complexities of the
geometric planar graph and the query trajectory, respectively. We provide a
positive result for realistic input graphs, which we regard as the main result
of this paper. We show that for -packed graphs, one can construct a data
structure of size that can answer -approximate
map matching queries in time, where hides lower-order factors and dependence of .Comment: To appear in SODA 202
Deterministic Graph Exploration with Advice
We consider the task of graph exploration. An -node graph has unlabeled
nodes, and all ports at any node of degree are arbitrarily numbered
. A mobile agent has to visit all nodes and stop. The exploration
time is the number of edge traversals. We consider the problem of how much
knowledge the agent has to have a priori, in order to explore the graph in a
given time, using a deterministic algorithm. This a priori information (advice)
is provided to the agent by an oracle, in the form of a binary string, whose
length is called the size of advice. We consider two types of oracles. The
instance oracle knows the entire instance of the exploration problem, i.e., the
port-numbered map of the graph and the starting node of the agent in this map.
The map oracle knows the port-numbered map of the graph but does not know the
starting node of the agent.
We first consider exploration in polynomial time, and determine the exact
minimum size of advice to achieve it. This size is ,
for both types of oracles.
When advice is large, there are two natural time thresholds:
for a map oracle, and for an instance oracle, that can be achieved
with sufficiently large advice. We show that, with a map oracle, time
cannot be improved in general, regardless of the size of advice.
We also show that the smallest size of advice to achieve this time is larger
than , for any .
For an instance oracle, advice of size is enough to achieve time
. We show that, with any advice of size , the time of
exploration must be at least , for any , and with any
advice of size , the time must be .
We also investigate minimum advice sufficient for fast exploration of
hamiltonian graphs
Twin-width I: tractable FO model checking
Inspired by a width invariant defined on permutations by Guillemot and Marx
[SODA '14], we introduce the notion of twin-width on graphs and on matrices.
Proper minor-closed classes, bounded rank-width graphs, map graphs, -free
unit -dimensional ball graphs, posets with antichains of bounded size, and
proper subclasses of dimension-2 posets all have bounded twin-width. On all
these classes (except map graphs without geometric embedding) we show how to
compute in polynomial time a sequence of -contractions, witness that the
twin-width is at most . We show that FO model checking, that is deciding if
a given first-order formula evaluates to true for a given binary
structure on a domain , is FPT in on classes of bounded
twin-width, provided the witness is given. More precisely, being given a
-contraction sequence for , our algorithm runs in time where is a computable but non-elementary function. We also prove that
bounded twin-width is preserved by FO interpretations and transductions
(allowing operations such as squaring or complementing a graph). This unifies
and significantly extends the knowledge on fixed-parameter tractability of FO
model checking on non-monotone classes, such as the FPT algorithm on
bounded-width posets by Gajarsk\'y et al. [FOCS '15].Comment: 49 pages, 9 figure
Trimming of Graphs, with Application to Point Labeling
For , a vertex-weighted graph of total weight is -trimmable
if it contains a vertex-induced subgraph of total weight at least
and with no simple path of more than edges. A family of graphs is trimmable
if for each constant , there is a constant such that every
vertex-weighted graph in the family is -trimmable. We show that every
family of graphs of bounded domino treewidth is trimmable. This implies that
every family of graphs of bounded degree is trimmable if the graphs in the
family have bounded treewidth or are planar. Based on this result, we derive a
polynomial-time approximation scheme for the problem of labeling weighted
points with nonoverlapping sliding labels of unit height and given lengths so
as to maximize the total weight of the labeled points. This settles one of the
last major open questions in the theory of map labeling
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