12,570 research outputs found
Efficient Fastest-Path Computations in Road Maps
In the age of real-time online traffic information and GPS-enabled devices,
fastest-path computations between two points in a road network modeled as a
directed graph, where each directed edge is weighted by a "travel time" value,
are becoming a standard feature of many navigation-related applications. To
support this, very efficient computation of these paths in very large road
networks is critical. Fastest paths may be computed as minimal-cost paths in a
weighted directed graph, but traditional minimal-cost path algorithms based on
variants of the classic Dijkstra algorithm do not scale well, as in the worst
case they may traverse the entire graph. A common improvement, which can
dramatically reduce the number of traversed graph vertices, is the A*
algorithm, which requires a good heuristic lower bound on the minimal cost. We
introduce a simple, but very effective, heuristic function based on a small
number of values assigned to each graph vertex. The values are based on graph
separators and computed efficiently in a preprocessing stage. We present
experimental results demonstrating that our heuristic provides estimates of the
minimal cost which are superior to those of other heuristics. Our experiments
show that when used in the A* algorithm, this heuristic can reduce the number
of vertices traversed by an order of magnitude compared to other heuristics
Efficient motion planning for problems lacking optimal substructure
We consider the motion-planning problem of planning a collision-free path of
a robot in the presence of risk zones. The robot is allowed to travel in these
zones but is penalized in a super-linear fashion for consecutive accumulative
time spent there. We suggest a natural cost function that balances path length
and risk-exposure time. Specifically, we consider the discrete setting where we
are given a graph, or a roadmap, and we wish to compute the minimal-cost path
under this cost function. Interestingly, paths defined using our cost function
do not have an optimal substructure. Namely, subpaths of an optimal path are
not necessarily optimal. Thus, the Bellman condition is not satisfied and
standard graph-search algorithms such as Dijkstra cannot be used. We present a
path-finding algorithm, which can be seen as a natural generalization of
Dijkstra's algorithm. Our algorithm runs in time, where~ and are the number of vertices and
edges of the graph, respectively, and is the number of intersections
between edges and the boundary of the risk zone. We present simulations on
robotic platforms demonstrating both the natural paths produced by our cost
function and the computational efficiency of our algorithm
Intriguingly Simple and Efficient Time-Dependent Routing in Road Networks
We study the earliest arrival problem in road networks with static
time-dependent functions as arc weights. We propose and evaluate the following
simple algorithm: (1) average the travel time in k time windows, (2) compute a
shortest time-independent path within each window and mark the edges in these
paths, and (3) compute a shortest time-dependent path in the original graph
restricted to the marked edges. Our experimental evaluation shows that this
simple algorithm yields near optimal results on well-established benchmark
instances. We additionally demonstrate that the error can be further reduced by
additionally considering alternative routes at the expense of more marked
edges. Finally, we show that the achieved subgraphs are small enough to be able
to efficiently implement profile queries using a simple sampling-based
approach. A highlight of our introduced algorithms is that they do not rely on
linking and merging profile functions
Fast Shortest Path Distance Estimation in Large Networks
We study the problem of preprocessing a large graph so that point-to-point shortest-path queries can be answered very fast. Computing shortest paths is a well studied problem, but exact algorithms do not scale to huge graphs encountered on the web, social networks, and other applications.
In this paper we focus on approximate methods for distance estimation, in particular using landmark-based distance indexing. This approach involves selecting a subset of nodes as landmarks and computing (offline) the distances from each node in the graph to those landmarks. At runtime, when the distance between a pair of nodes is needed, we can estimate it quickly by combining the precomputed distances of the two nodes to the landmarks.
We prove that selecting the optimal set of landmarks is an NP-hard problem, and thus heuristic solutions need to be employed. Given a budget of memory for the index, which translates directly into a budget of landmarks, different landmark selection strategies can yield dramatically different results in terms of accuracy. A number of simple methods that scale well to large graphs are therefore developed and experimentally compared. The simplest methods choose central nodes of the graph, while the more elaborate ones select central nodes that are also far away from one another. The efficiency of the suggested techniques is tested experimentally using five different real world graphs with millions of edges; for a given accuracy, they require as much as 250 times less space than the current approach in the literature which considers selecting landmarks at random.
Finally, we study applications of our method in two problems arising naturally in large-scale networks, namely, social search and community detection.Yahoo! Research (internship
Shortest Paths and Distances with Differential Privacy
We introduce a model for differentially private analysis of weighted graphs
in which the graph topology is assumed to be public and the private
information consists only of the edge weights . This can
express hiding congestion patterns in a known system of roads. Differential
privacy requires that the output of an algorithm provides little advantage,
measured by privacy parameters and , for distinguishing
between neighboring inputs, which are thought of as inputs that differ on the
contribution of one individual. In our model, two weight functions are
considered to be neighboring if they have distance at most one.
We study the problems of privately releasing a short path between a pair of
vertices and of privately releasing approximate distances between all pairs of
vertices. We are concerned with the approximation error, the difference between
the length of the released path or released distance and the length of the
shortest path or actual distance.
For privately releasing a short path between a pair of vertices, we prove a
lower bound of on the additive approximation error for fixed
. We provide a differentially private algorithm that matches
this error bound up to a logarithmic factor and releases paths between all
pairs of vertices. The approximation error of our algorithm can be bounded by
the number of edges on the shortest path, so we achieve better accuracy than
the worst-case bound for vertex pairs that are connected by a low-weight path
with vertices.
For privately releasing all-pairs distances, we show that for trees we can
release all distances with approximation error for fixed
privacy parameters. For arbitrary bounded-weight graphs with edge weights in
we can release all distances with approximation error
A comprehensive theory of cascading via-paths and the reciprocal pointer chain method
In this paper, we consolidate and expand upon the current theory and
potential applications of the set of best \emph{cascading via-paths} (CVPs)
and the \emph{reciprocal pointer chain} (RPC) method for identifying them. CVPs
are a collection of up to paths between a source and a target node in a
graph , computed using two shortest path trees, that have
distinctive properties relative to other path sets. They have been shown to be
particularly useful in geospatial applications, where they are an intuitive and
efficient means for identifying a set of spatially diverse alternatives to the
single shortest path between the source and target. However, spatial diversity
is not intrinsic to paths in a graph, and little theory has been developed
outside of application to describe the nature of these paths and the RPC method
in general. Here we divorce the RPC method from its typical geospatial
applications and develop a comprehensive theory of CVPs from an abstract
graph-theoretic perspective. Restricting ourselves to properties of the CVPs
and of the entire set of -best CVPs that can be computed in , we are able to then propose, among other things, new and efficient
approaches to problems such as generating a diverse set of paths and to
computing the shortest loopless paths between two nodes in a graph. We
conclude by demonstrating the new theory in practice, first for a typical
application of finding alternative routes in road networks and then for a novel
application of identifying layer-boundaries in ground-penetrating radar (GPR)
data. It is our hope that by generalizing the RPC method, providing a sound
theoretical foundation, and demonstrating novel uses, we are able to broaden
its perceived applicability and stimulate new research in this area, both
applied and theoretical
On the Complexity of Hub Labeling
Hub Labeling (HL) is a data structure for distance oracles. Hierarchical HL
(HHL) is a special type of HL, that received a lot of attention from a
practical point of view. However, theoretical questions such as NP-hardness and
approximation guarantee for HHL algorithms have been left aside. In this paper
we study HL and HHL from the complexity theory point of view. We prove that
both HL and HHL are NP-hard, and present upper and lower bounds for the
approximation ratios of greedy HHL algorithms used in practice. We also
introduce a new variant of the greedy HHL algorithm and a proof that it
produces small labels for graphs with small highway dimension
Customizable Contraction Hierarchies
We consider the problem of quickly computing shortest paths in weighted
graphs given auxiliary data derived in an expensive preprocessing phase. By
adding a fast weight-customization phase, we extend Contraction Hierarchies by
Geisberger et al to support the three-phase workflow introduced by Delling et
al. Our Customizable Contraction Hierarchies use nested dissection orders as
suggested by Bauer et al. We provide an in-depth experimental analysis on large
road and game maps that clearly shows that Customizable Contraction Hierarchies
are a very practicable solution in scenarios where edge weights often change
Towards Knowledge-Enriched Path Computation
Directions and paths, as commonly provided by navigation systems, are usually
derived considering absolute metrics, e.g., finding the shortest path within an
underlying road network. With the aid of crowdsourced geospatial data we aim at
obtaining paths that do not only minimize distance but also lead through more
popular areas using knowledge generated by users. We extract spatial relations
such as "nearby" or "next to" from travel blogs, that define closeness between
pairs of points of interest (PoIs) and quantify each of these relations using a
probabilistic model. Subsequently, we create a relationship graph where each
node corresponds to a PoI and each edge describes the spatial connection
between the respective PoIs. Using Bayesian inference we obtain a probabilistic
measure of spatial closeness according to the crowd. Applying this measure to
the corresponding road network, we obtain an altered cost function which does
not exclusively rely on distance, and enriches an actual road networks taking
crowdsourced spatial relations into account. Finally, we propose two routing
algorithms on the enriched road networks. To evaluate our approach, we use
Flickr photo data as a ground truth for popularity. Our experimental results --
based on real world datasets -- show that the paths computed w.r.t.\ our
alternative cost function yield competitive solutions in terms of path length
while also providing more "popular" paths, making routing easier and more
informative for the user.Comment: Accepted as a short paper at ACM SIGSPATIAL GIS 201
Path-Fault-Tolerant Approximate Shortest-Path Trees
Let be an -nodes non-negatively real-weighted undirected graph.
In this paper we show how to enrich a {\em single-source shortest-path tree}
(SPT) of with a \emph{sparse} set of \emph{auxiliary} edges selected from
, in order to create a structure which tolerates effectively a \emph{path
failure} in the SPT. This consists of a simultaneous fault of a set of at
most adjacent edges along a shortest path emanating from the source, and it
is recognized as one of the most frequent disruption in an SPT. We show that,
for any integer parameter , it is possible to provide a very sparse
(i.e., of size ) auxiliary structure that carefully
approximates (i.e., within a stretch factor of ) the true
shortest paths from the source during the lifetime of the failure. Moreover, we
show that our construction can be further refined to get a stretch factor of
and a size of for the special case , and that it can be
converted into a very efficient \emph{approximate-distance sensitivity oracle},
that allows to quickly (even in optimal time, if ) reconstruct the
shortest paths (w.r.t. our structure) from the source after a path failure,
thus permitting to perform promptly the needed rerouting operations. Our
structure compares favorably with previous known solutions, as we discuss in
the paper, and moreover it is also very effective in practice, as we assess
through a large set of experiments.Comment: 21 pages, 3 figures, SIROCCO 201
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