4,059 research outputs found
Group Marching Tree: Sampling-Based Approximately Optimal Motion Planning on GPUs
This paper presents a novel approach, named the Group Marching Tree (GMT*)
algorithm, to planning on GPUs at rates amenable to application within control
loops, allowing planning in real-world settings via repeated computation of
near-optimal plans. GMT*, like the Fast Marching Tree (FMT) algorithm, explores
the state space with a "lazy" dynamic programming recursion on a set of samples
to grow a tree of near-optimal paths. GMT*, however, alters the approach of FMT
with approximate dynamic programming by expanding, in parallel, the group of
all active samples with cost below an increasing threshold, rather than only
the minimum cost sample. This group approximation enables low-level parallelism
over the sample set and removes the need for sequential data structures, while
the "lazy" collision checking limits thread divergence---all contributing to a
very efficient GPU implementation. While this approach incurs some
suboptimality, we prove that GMT* remains asymptotically optimal up to a
constant multiplicative factor. We show solutions for complex planning problems
under differential constraints can be found in ~10 ms on a desktop GPU and ~30
ms on an embedded GPU, representing a significant speed up over the state of
the art, with only small losses in performance. Finally, we present a scenario
demonstrating the efficacy of planning within the control loop (~100 Hz)
towards operating in dynamic, uncertain settings
A Combinatorial Algorithm for All-Pairs Shortest Paths in Directed Vertex-Weighted Graphs with Applications to Disc Graphs
We consider the problem of computing all-pairs shortest paths in a directed
graph with real weights assigned to vertices.
For an 0-1 matrix let be the complete weighted graph
on the rows of where the weight of an edge between two rows is equal to
their Hamming distance. Let be the weight of a minimum weight spanning
tree of
We show that the all-pairs shortest path problem for a directed graph on
vertices with nonnegative real weights and adjacency matrix can be
solved by a combinatorial randomized algorithm in time
As a corollary, we conclude that the transitive closure of a directed graph
can be computed by a combinatorial randomized algorithm in the
aforementioned time.
We also conclude that the all-pairs shortest path problem for uniform disk
graphs, with nonnegative real vertex weights, induced by point sets of bounded
density within a unit square can be solved in time
A Nonlinear Dimensionality Reduction Framework Using Smooth Geodesics
Existing dimensionality reduction methods are adept at revealing hidden
underlying manifolds arising from high-dimensional data and thereby producing a
low-dimensional representation. However, the smoothness of the manifolds
produced by classic techniques over sparse and noisy data is not guaranteed. In
fact, the embedding generated using such data may distort the geometry of the
manifold and thereby produce an unfaithful embedding. Herein, we propose a
framework for nonlinear dimensionality reduction that generates a manifold in
terms of smooth geodesics that is designed to treat problems in which manifold
measurements are either sparse or corrupted by noise. Our method generates a
network structure for given high-dimensional data using a nearest neighbors
search and then produces piecewise linear shortest paths that are defined as
geodesics. Then, we fit points in each geodesic by a smoothing spline to
emphasize the smoothness. The robustness of this approach for sparse and noisy
datasets is demonstrated by the implementation of the method on synthetic and
real-world datasets.Comment: 13 pages, 7 figures, submitted to Pattern Recognitio
A Survey of Shortest-Path Algorithms
A shortest-path algorithm finds a path containing the minimal cost between
two vertices in a graph. A plethora of shortest-path algorithms is studied in
the literature that span across multiple disciplines. This paper presents a
survey of shortest-path algorithms based on a taxonomy that is introduced in
the paper. One dimension of this taxonomy is the various flavors of the
shortest-path problem. There is no one general algorithm that is capable of
solving all variants of the shortest-path problem due to the space and time
complexities associated with each algorithm. Other important dimensions of the
taxonomy include whether the shortest-path algorithm operates over a static or
a dynamic graph, whether the shortest-path algorithm produces exact or
approximate answers, and whether the objective of the shortest-path algorithm
is to achieve time-dependence or is to only be goal directed. This survey
studies and classifies shortest-path algorithms according to the proposed
taxonomy. The survey also presents the challenges and proposed solutions
associated with each category in the taxonomy
Integrating asymptotically-optimal path planning with local optimization
Many robots operating in unpredictable environments require an online path
planning algorithm that can quickly compute high quality paths. Asymptotically
optimal planners are capable of finding the optimal path, but can be slow to
converge. Local optimisation algorithms are capable of quickly improving a
solution, but are not guaranteed to converge to the optimal solution. In this
paper we develop a new way to integrate an asymptotically optimal planners with
a local optimiser. We test our approach using RRTConnect* with a short-cutting
local optimiser. Our approach results in a significant performance improvement
when compared with the state-of-the-art RRTConnect* asymptotically optimal
planner and computes paths that are 31\% faster to execute when both are given
3 seconds of planning time
Towards Fully Environment-Aware UAVs: Real-Time Path Planning with Online 3D Wind Field Prediction in Complex Terrain
Today, low-altitude fixed-wing Unmanned Aerial Vehicles (UAVs) are largely
limited to primitively follow user-defined waypoints. To allow fully-autonomous
remote missions in complex environments, real-time environment-aware navigation
is required both with respect to terrain and strong wind drafts. This paper
presents two relevant initial contributions: First, the literature's first-ever
3D wind field prediction method which can run in real time onboard a UAV is
presented. The approach retrieves low-resolution global weather data, and uses
potential flow theory to adjust the wind field such that terrain boundaries,
mass conservation, and the atmospheric stratification are observed. A
comparison with 1D LIDAR data shows an overall wind error reduction of 23% with
respect to the zero-wind assumption that is mostly used for UAV path planning
today. However, given that the vertical winds are not resolved accurately
enough further research is required and identified. Second, a sampling-based
path planner that considers the aircraft dynamics in non-uniform wind
iteratively via Dubins airplane paths is presented. Performance optimizations,
e.g. obstacle-aware sampling and fast 2.5D-map collision checks, render the
planner 50% faster than the Open Motion Planning Library (OMPL) implementation.
Test cases in Alpine terrain show that the wind-aware planning performs up to
50x less iterations than shortest-path planning and is thus slower in low
winds, but that it tends to deliver lower-cost paths in stronger winds. More
importantly, in contrast to the shortest-path planner, it always delivers
collision-free paths. Overall, our initial research demonstrates the
feasibility of 3D wind field prediction from a UAV and the advantages of
wind-aware planning. This paves the way for follow-up research on
fully-autonomous environment-aware navigation of UAVs in real-life missions and
complex terrain
Computing Geodesic Distances in Tree Space
We present two algorithms for computing the geodesic distance between
phylogenetic trees in tree space, as introduced by Billera, Holmes, and
Vogtmann (2001). We show that the possible combinatorial types of shortest
paths between two trees can be compactly represented by a partially ordered
set. We calculate the shortest distance along each candidate path by converting
the problem into one of finding the shortest path through a certain region of
Euclidean space. In particular, we show there is a linear time algorithm for
finding the shortest path between a point in the all positive orthant and a
point in the all negative orthant of R^k contained in the subspace of R^k
consisting of all orthants with the first i coordinates non-positive and the
remaining coordinates non-negative for 0 <= i <= k.Comment: 24 pages, 7 figures; v2: substantially revised for clarit
Network Alignment by Discrete Ollivier-Ricci Flow
In this paper, we consider the problem of approximately aligning/matching two
graphs. Given two graphs and , the
objective is to map nodes to nodes such that when
have an edge in , very likely their corresponding nodes in
are connected as well. This problem with subgraph isomorphism as a
special case has extra challenges when we consider matching complex networks
exhibiting the small world phenomena. In this work, we propose to use `Ricci
flow metric', to define the distance between two nodes in a network. This is
then used to define similarity of a pair of nodes in two networks respectively,
which is the crucial step of network alignment. %computed by discrete graph
curvatures and graph Ricci flows. Specifically, the Ricci curvature of an edge
describes intuitively how well the local neighborhood is connected. The graph
Ricci flow uniformizes discrete Ricci curvature and induces a Ricci flow metric
that is insensitive to node/edge insertions and deletions. With the new metric,
we can map a node in to a node in whose distance vector to only a
few preselected landmarks is the most similar. The robustness of the graph
metric makes it outperform other methods when tested on various complex graph
models and real world network data sets (Emails, Internet, and protein
interaction networks)\footnote{The source code of computing Ricci curvature and
Ricci flow metric are available:
https://github.com/saibalmars/GraphRicciCurvature}.Comment: Appears in the Proceedings of the 26th International Symposium on
Graph Drawing and Network Visualization (GD 2018
Network Topology Mapping from Partial Virtual Coordinates and Graph Geodesics
For many important network types (e.g., sensor networks in complex harsh
environments and social networks) physical coordinate systems (e.g.,
Cartesian), and physical distances (e.g., Euclidean), are either difficult to
discern or inapplicable. Accordingly, coordinate systems and characterizations
based on hop-distance measurements, such as Topology Preserving Maps (TPMs) and
Virtual-Coordinate (VC) systems are attractive alternatives to Cartesian
coordinates for many network algorithms. Herein, we present an approach to
recover geometric and topological properties of a network with a small set of
distance measurements. In particular, our approach is a combination of shortest
path (often called geodesic) recovery concepts and low-rank matrix completion,
generalized to the case of hop-distances in graphs. Results for sensor networks
embedded in 2-D and 3-D spaces, as well as a social networks, indicates that
the method can accurately capture the network connectivity with a small set of
measurements. TPM generation can now also be based on various context
appropriate measurements or VC systems, as long as they characterize different
nodes by distances to small sets of random nodes (instead of a set of global
anchors). The proposed method is a significant generalization that allows the
topology to be extracted from a random set of graph shortest paths, making it
applicable in contexts such as social networks where VC generation may not be
possible.Comment: 17 pages, 9 figures. arXiv admin note: substantial text overlap with
arXiv:1712.1006
Geometric Algorithms with Limited Workspace: A Survey
In the limited workspace model, we consider algorithms whose input resides in
read-only memory and that use only a constant or sublinear amount of writable
memory to accomplish their task. We survey recent results in computational
geometry that fall into this model and that strive to achieve the lowest
possible running time. In addition to discussing the state of the art, we give
some illustrative examples and mention open problems for further research.Comment: 18 pages, 3 figure
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