arXiv subjects

Mitchell Jones

Publications and source records attributed to Mitchell Jones.

11 recordsLinked to original sources

arXiv2022

Lane-Level Route Planning for Autonomous Vehicles

We present an algorithm that, given a representation of a road network in lane-level detail, computes a route that minimizes the expected cost to reach a given destination. In doing so, our algorithm allows us to solve for the complex trade-offs encountered when trying to decide not just which roads to follow, but also when to change between the lanes making up these roads, in order to -- for example -- reduce the likelihood of missing a left exit while not unnecessarily driving in the leftmost lane. This routing problem can naturally be formulated as a Markov Decision Process (MDP), in which lane change actions have stochastic outcomes. However, MDPs are known to be time-consuming to solve in general. In this paper, we show that -- under reasonable assumptions -- we can use a Dijkstra-like approach to solve this stochastic problem, and benefit from its efficient $O(n \log n)$ running time. This enables an autonomous vehicle to exhibit lane-selection behavior as it efficiently plans an optimal route to its destination.

cs.RO↗

arXiv2020

A Note on Stabbing Convex Bodies with Points, Lines, and Flats

$\newcommand{\eps}{\varepsilon}\newcommand{\tldO}{\widetilde{O}}$Consider the problem of constructing weak $\eps$-nets where the stabbing elements are lines or $k$-flats instead of points. We study this problem in the simplest setting where it is still interesting -- namely, the uniform measure of volume over the hypercube $[0,1]^d\bigr.$. Specifically, a $(k,\eps)$-net is a set of $k$-flats, such that any convex body in $[0,1]^d$ of volume larger than $\eps$ is stabbed by one of these $k$-flats. We show that for $k \geq 1$, one can construct $(k,\eps)$-nets of size $O(1/\eps^{1-k/d})$. We also prove that any such net must have size at least $\Omega(1/\eps^{1-k/d})$. As a concrete example, in three dimensions all $\eps$-heavy bodies in $[0,1]^3$ can be stabbed by $\Theta(1/\eps^{2/3})$ lines. Note, that these bounds are \emph{sublinear} in $1/\eps$, and are thus somewhat surprising. The new construction also works for points providing a weak $\eps$-net of size $O(\tfrac{1}{\eps}\log^{d-1} \tfrac{1}{\eps} )$.

cs.CG↗

arXiv2020

Cutting cycles of rods in space is FPT

In this short note, we show that cutting cycles of rods is fixed-parameter tractable by reducing the problem to computing a feedback vertex set in a mixed graph.

cs.CG↗

arXiv2019

Optimal Algorithms for Geometric Centers and Depth

$\renewcommand{\Re}{\mathbb{R}}$ We develop a general randomized technique for solving "implic it" linear programming problems, where the collection of constraints are defined implicitly by an underlying ground set of elements. In many cases, the structure of the implicitly defined constraints can be exploited in order to obtain efficient linear program solvers. We apply this technique to obtain near-optimal algorithms for a variety of fundamental problems in geometry. For a given point set $P$ of size $n$ in $\Re^d$, we develop algorithms for computing geometric centers of a point set, including the centerpoint and the Tukey median, and several other more involved measures of centrality. For $d=2$, the new algorithms run in $O(n\log n)$ expected time, which is optimal, and for higher constant $d>2$, the expected time bound is within one logarithmic factor of $O(n^{d-1})$, which is also likely near optimal for some of the problems.

cs.CG↗

arXiv2019

Proof of Dudley's Convex Approximation

We provide a self contained proof of a result of Dudley [Dud64]} which shows that a bounded convex-body in $\Re^d$ can be $\varepsilon$-approximated, by the intersection of $O_d\bigl(\varepsilon^{-(d-1)/2} \bigr)$ halfspaces, where $O_d$ hides constants that depends on $d$.

cs.CG↗

arXiv2019

Some Geometric Applications of Anti-Chains

We present an algorithmic framework for computing anti-chains of maximum size in geometric posets. Specifically, posets in which the entities are geometric objects, where comparability of two entities is implicitly defined but can be efficiently tested. Computing the largest anti-chain in a poset can be done in polynomial time via maximum-matching in a bipartite graph, and this leads to several efficient algorithms for the following problems, each running in (roughly) $O(n^{3/2})$ time: (A) Computing the largest Pareto-optimal subset of a set of $n$ points in $\mathbb{R}^d$. (B) Given a set of disks in the plane, computing the largest subset of disks such that no disk contains another. This is quite surprising, as the independent version of this problem is computationally hard. (C) Given a set of axis-aligned rectangles, computing the largest subset of non-crossing rectangles.

cs.CG↗

arXiv2019

Active Learning a Convex Body in Low Dimensions

Consider a set $P \subseteq \Re^d$ of $n$ points, and a convex body $C$ provided via a separation oracle. The task at hand is to decide for each point of $P$ if it is in $C$ using the fewest number of oracle queries. We show that one can solve this problem in two and three dimensions using $O( h(P) \log n)$ queries, where $h(P)$ is the largest subset of points of $P$ in convex position. Furthermore, we show that in two dimensions one can solve this problem using $O( v(P,C) \log^2 n )$ oracle queries, where $v(P, C)$ is a lower bound on the minimum number of queries that any algorithm for this specific instance requires.

cs.CG↗

arXiv2018

On Locality-Sensitive Orderings and their Applications

For any constant $d$ and parameter $\varepsilon > 0$, we show the existence of (roughly) $1/\varepsilon^d$ orderings on the unit cube $[0,1)^d$, such that any two points $p,q\in [0,1)^d$ that are close together under the Euclidean metric are "close together" in one of these linear orderings in the following sense: the only points that could lie between $p$ and $q$ in the ordering are points with Euclidean distance at most $\varepsilon\| p - q \|$ from $p$ or $q$. These orderings are extensions of the $\mathcal{Z}$-order, and they can be efficiently computed. Functionally, the orderings can be thought of as a replacement to quadtrees and related structures (like well-separated pair decompositions). We use such orderings to obtain surprisingly simple algorithms for a number of basic problems in low-dimensional computational geometry, including (i) dynamic approximate bichromatic closest pair, (ii) dynamic spanners, (iii) dynamic approximate minimum spanning trees, (iv) static and dynamic fault-tolerant spanners, and (v) approximate nearest neighbor search.

cs.CG↗

arXiv2018

Few Cuts Meet Many Point Sets

We study the problem of how to breakup many point sets in $\mathbb{R}^d$ into smaller parts using a few splitting (shared) hyperplanes. This problem is related to the classical Ham-Sandwich Theorem. We provide a logarithmic approximation to the optimal solution using the greedy algorithm for submodular optimization.

cs.CG↗

arXiv2017

Journey to the Center of the Point Set

$\renewcommand{\Re}{\mathbb{R}} \newcommand{\eps}{\varepsilon} \newcommand{\Net}{S} \newcommand{\tldO}{{\widetilde{O}}} \newcommand{\body}{C} $ We revisit an algorithm of Clarkson etal [CEMST96], that computes (roughly) a $1/(4d^2)$-centerpoint in $\tldO(d^9)$ time, for a point set in $\Re^d$, where $\tldO$ hides polylogarithmic terms. We present an improved algorithm that computes (roughly) a $1/d^2$-centerpoint with running time $\tldO(d^7)$. While the improvements are (arguably) mild, it is the first progress on this well known problem in over twenty years. The new algorithm is simpler, and the running time bound follows by a simple random walk argument, which we believe to be of independent interest. We also present several new applications of the improved centerpoint algorithm.

cs.CG↗

arXiv2017

On Separating Points by Lines

Given a set $P$ of $n$ points in the plane, its separability is the minimum number of lines needed to separate all its pairs of points from each other. We show that the minimum number of lines needed to separate $n$ points, picked randomly (and uniformly) in the unit square, is $\Theta( n^{2/3})$, where $\Theta$ hides polylogarithmic factors. In addition, we provide a fast approximation algorithm for computing the separability of a given point set in the plane. Finally, we point out the connection between separability and partitions.

cs.CG↗