A Unified Approach to Tail Estimates for Randomized Incremental Construction

By combining several interesting applications of random sampling in geometric algorithms like point location, linear programming, segment intersections, binary space partitioning, Clarkson and Shor [Kenneth L. Clarkson and Peter W. Shor, 1989] developed a general framework of randomized incremental construction (RIC ). The basic idea is to add objects in a random order and show that this approach yields efficient/optimal bounds on expected running time. Even quicksort can be viewed as a special case of this paradigm. However, unlike quicksort, for most of these problems, sharper tail estimates on their running times are not known. Barring some promising attempts in [Kurt Mehlhorn et al., 1993; Kenneth L. Clarkson et al., 1992; Raimund Seidel, 1991], the general question remains unresolved. In this paper we present a general technique to obtain tail estimates for RIC and and provide applications to some fundamental problems like Delaunay triangulations and construction of Visibility maps of intersecting line segments. The main result of the paper is derived from a new and careful application of Freedman\u27s [David Freedman, 1975] inequality for Martingale concentration that overcomes the bottleneck of the better known Azuma-Hoeffding inequality. Further, we explore instances, where an RIC based algorithm may not have inverse polynomial tail estimates. In particular, we show that the RIC time bounds for trapezoidal map can encounter a running time of Omega (n log n log log n) with probability exceeding 1/(sqrt{n)}. This rules out inverse polynomial concentration bounds within a constant factor of the O(n log n) expected running time

From Proximity to Utility: A Voronoi Partition of Pareto Optima

We present an extension of Voronoi diagrams where when considering which site a client is going to use, in addition to the site distances, other site attributes are also considered (for example, prices or weights). A cell in this diagram is then the locus of all clients that consider the same set of sites to be relevant. In particular, the precise site a client might use from this candidate set depends on parameters that might change between usages, and the candidate set lists all of the relevant sites. The resulting diagram is significantly more expressive than Voronoi diagrams, but naturally has the drawback that its complexity, even in the plane, might be quite high. Nevertheless, we show that if the attributes of the sites are drawn from the same distribution (note that the locations are fixed), then the expected complexity of the candidate diagram is near linear. To this end, we derive several new technical results, which are of independent interest. In particular, we provide a high-probability, asymptotically optimal bound on the number of Pareto optima points in a point set uniformly sampled from the $d$-dimensional hypercube. To do so we revisit the classical backward analysis technique, both simplifying and improving relevant results in order to achieve the high-probability bounds

Sampling-based Algorithms for Optimal Motion Planning

During the last decade, sampling-based path planning algorithms, such as Probabilistic RoadMaps (PRM) and Rapidly-exploring Random Trees (RRT), have been shown to work well in practice and possess theoretical guarantees such as probabilistic completeness. However, little effort has been devoted to the formal analysis of the quality of the solution returned by such algorithms, e.g., as a function of the number of samples. The purpose of this paper is to fill this gap, by rigorously analyzing the asymptotic behavior of the cost of the solution returned by stochastic sampling-based algorithms as the number of samples increases. A number of negative results are provided, characterizing existing algorithms, e.g., showing that, under mild technical conditions, the cost of the solution returned by broadly used sampling-based algorithms converges almost surely to a non-optimal value. The main contribution of the paper is the introduction of new algorithms, namely, PRM* and RRT*, which are provably asymptotically optimal, i.e., such that the cost of the returned solution converges almost surely to the optimum. Moreover, it is shown that the computational complexity of the new algorithms is within a constant factor of that of their probabilistically complete (but not asymptotically optimal) counterparts. The analysis in this paper hinges on novel connections between stochastic sampling-based path planning algorithms and the theory of random geometric graphs.Comment: 76 pages, 26 figures, to appear in International Journal of Robotics Researc

Motion Planning For Micro Aerial Vehicles

A Micro Aerial Vehicle (MAV) is capable of agile motion in 3D making it an ideal platform for developments of planning and control algorithms. For fully autonomous MAV systems, it is essential to plan motions that are both dynamically feasible and collision-free in cluttered environments. Recent work demonstrates precise control of MAVs using time-parameterized trajectories that satisfy feasibility and safety requirements. However, planning such trajectories is non-trivial, especially when considering constraints, such as optimality and completeness. For navigating in unknown environments, the capability for fast re-planning is also critical. Considering all of these requirements, motion planning for MAVs is a challenging problem. In this thesis, we examine trajectory planning algorithms for MAVs and present methodologies that solve a wide range of planning problems. We first introduce path planning and geometric control methods, which produce spatial paths that are inadequate for high speed flight, but can be used to guide trajectory optimization. We then describe optimization-based trajectory planning and demonstrate this method for solving navigation problems in complex 3D environments. When the initial state is not fixed, an optimization-based method is prone to generate sub-optimal trajectories. To address this challenge, we propose a search-based approach using motion primitives to plan resolution complete and sub-optimal trajectories. This algorithm can also be used to solve planning problems with constraints such as motion uncertainty, limited field-of-view and moving obstacles. The proposed methods can run in real time and are applicable for real-world autonomous navigation, even with limited on-board computational resources. This thesis includes a carefully analysis of the strengths and weaknesses of our planning paradigm and algorithms, and demonstration of their performance through simulation and experiments

Nonrigid reconstruction of 3D breast surfaces with a low-cost RGBD camera for surgical planning and aesthetic evaluation

Accounting for 26% of all new cancer cases worldwide, breast cancer remains the most common form of cancer in women. Although early breast cancer has a favourable long-term prognosis, roughly a third of patients suffer from a suboptimal aesthetic outcome despite breast conserving cancer treatment. Clinical-quality 3D modelling of the breast surface therefore assumes an increasingly important role in advancing treatment planning, prediction and evaluation of breast cosmesis. Yet, existing 3D torso scanners are expensive and either infrastructure-heavy or subject to motion artefacts. In this paper we employ a single consumer-grade RGBD camera with an ICP-based registration approach to jointly align all points from a sequence of depth images non-rigidly. Subtle body deformation due to postural sway and respiration is successfully mitigated leading to a higher geometric accuracy through regularised locally affine transformations. We present results from 6 clinical cases where our method compares well with the gold standard and outperforms a previous approach. We show that our method produces better reconstructions qualitatively by visual assessment and quantitatively by consistently obtaining lower landmark error scores and yielding more accurate breast volume estimates

Near-Optimal Decremental SSSP in Dense Weighted Digraphs

In the decremental Single-Source Shortest Path problem (SSSP), we are given a weighted directed graph $G=(V,E,w)$ undergoing edge deletions and a source vertex $r \in V$; let $n = |V|, m = |E|$ and $W$ be the aspect ratio of the graph. The goal is to obtain a data structure that maintains shortest paths from $r$ to all vertices in $V$ and can answer distance queries in $O(1)$ time, as well as return the corresponding path $P$ in $O(|P|)$ time. This problem was first considered by Even and Shiloach [JACM'81], who provided an algorithm with total update time $O(mn)$ for unweighted undirected graphs; this was later extended to directed weighted graphs [FOCS'95, STOC'99]. There are conditional lower bounds showing that $O(mn)$ is in fact near-optimal [ESA'04, FOCS'14, STOC'15, STOC'20]. In a breakthrough result, Forster et al. showed that it is possible to achieve total update time $mn^{0.9+o(1)}\log W$ if the algorithm is allowed to return $(1+{\epsilon})$-approximate paths, instead of exact ones [STOC'14, ICALP'15]. No further progress was made until Probst Gutenberg and Wulff-Nilsen [SODA'20] provided a new approach for the problem, which yields total time $\tilde{O}(\min{m^{2/3}n^{4/3}\log W, (mn)^{7/8} \log W})$. Our result builds on this recent approach, but overcomes its limitations by introducing a significantly more powerful abstraction, as well as a different core subroutine. Our new framework yields a decremental $(1+{\epsilon})$-approximate SSSP data structure with total update time $\tilde{O}(n^2 \log^4 W)$. Our algorithm is thus near-optimal for dense graphs with polynomial edge-weights. Our framework can also be applied to sparse graphs to obtain total update time $\tilde{O}(mn^{2/3} \log^3 W)$. Our main technique allows us to convert SSSP algorithms for DAGs to ones for general graphs, which we believe has significant potential to influence future work.Comment: Accepted to FOCS'2

Unstructured Grid Generation Techniques and Software

The Workshop on Unstructured Grid Generation Techniques and Software was conducted for NASA to assess its unstructured grid activities, improve the coordination among NASA centers, and promote technology transfer to industry. The proceedings represent contributions from Ames, Langley, and Lewis Research Centers, and the Johnson and Marshall Space Flight Centers. This report is a compilation of the presentations made at the workshop