How can a set of identical mobile agents coordinate their motions to transform their arrangement from a given starting to a desired goal configuration? We consider this question in the context of actual physical devices called Catoms, which can perform reconfiguration, but need to maintain connectivity at all times to ensure communication and energy supply. We demonstrate and animate algorithmic results, in particular a proof of hardness, as well as an algorithm that guarantees constant stretch for certain classes of arrangements: If mapping the start configuration to the target configuration requires a maximum Manhattan distance of d, then the total duration of our overall schedule is in 𝒪(d), which is optimal up to constant factors.

Motivated by map labeling, Funke, Krumpe, and Storandt [IWOCA 2016] introduced the following problem: we are given a sequence of n disks in the plane. Initially, all disks have radius 0, and they grow at constant, but possibly different, speeds. Whenever two disks touch, the one with the higher index disappears. The goal is to determine the elimination order, i.e., the order in which the disks disappear. We provide the first general subquadratic algorithm for this problem. Our solution extends to other shapes (e.g., rectangles), and it works in any fixed dimension. We also describe an alternative algorithm that is based on quadtrees. Its running time is O ( n ( log ⁡ n + min ⁡ { log ⁡ Δ , log ⁡ Φ } ) ) , where Δ is the ratio of the fastest and the slowest growth rate and Φ is the ratio of the largest and the smallest distance between two disk centers. This improves the running times of previous algorithms by Funke, Krumpe, and Storandt [IWOCA 2016], Bahrdt et al. [ALENEX 2017], and Funke and Storandt [EuroCG 2017]. Finally, we give an Ω ( n log ⁡ n ) lower bound, showing that our quadtree algorithms are almost tight.

Given a planar subdivision with n vertices, each face having a cone of possible directions of travel, our goal is to decide which vertices of the subdivision can be reached from a given starting point s. We give an O(n log n)-time algorithm for this problem, as well as an Omega(n log n) lower bound in the algebraic computation tree model. We prove that the generalization where two cones of directions per face are allowed is NP-hard.

Motivated by map labeling, we study the problem in which we are given a collection of n disks in the plane that grow at possibly different speeds. Whenever two disks meet, the one with the higher index disappears. This problem was introduced by Funke, Krumpe, and Storandt[IWOCA 2016]. We provide the first general subquadratic algorithm for computing the times and the order of disappearance. Our algorithm also works for other shapes (such as rectangles) and in any fixed dimension. Using quadtrees, we provide an alternative algorithm that runs in near linear time, although this second algorithm has a logarithmic dependence on either the ratio of the fastest speed to the slowest speed of disks or the spread of the disk centers (the ratio of the maximum to the minimum distance between them). Our result improves the running times of previous algorithms by Funke, Krumpe, and Storandt [IWOCA 2016], Bahrdt et al. [ALENEX 2017], and Funke and Storandt [EWCG 2017]. Finally, we give an \Omega(n\log n) lower bound on the problem, showing that our quadtree algorithms are almost tight.

We present a new algorithm for computing the straight skeleton of a polygon. For a polygon with n vertices, among which r are reflex vertices, we give a deterministic algorithm that reduces the straight skeleton computation to a motorcycle graph computation in O ( n (log n )log r ) time. It improves on the previously best known algorithm for this reduction, which is randomized, and runs in expected O ( n sqrt h + 1 log 2 n ) time for a polygon with h holes. Using known motorcycle graph algorithms, our result yields improved time bounds for computing straight skeletons. In particular, we can compute the straight skeleton of a nondegenerate polygon in O ( n (log n )log r + r 4/3 + ε ) time for any ε > 0. On degenerate input, our time bound increases to O ( n (log n )log r + r 17/11 + ε ).

Let be a planar subdivision with n vertices. Each face of has a weight from [1, ρ] ∪ {∞}. A path inside a face has cost equal to the product of its length and the face weight. In general, the cost of a path is the sum of the subpath costs in the faces intersected by the path. For any ε ∊ (0, 1), we present a fully polynomial-time approximation scheme that finds a (1 + ε)-approximate shortest path between two given points in in time, where k is the smallest integer such that the sum of the k smallest angles in is at least π. Therefore, our running time can be as small as if there are O(1) small angles and it is in the worst case. Our algorithm relies on a new triangulation refinement method, which produces a triangulation of size O(n + k2) such that no triangle has two angles less than min{π/(2k), π/12}.

We present a new optimization technique that yields the first FPTAS for several geometric problems. These problems reduce to optimizing a sum of nonnegative, constant description complexity algebraic functions. We first give an FPTAS for optimizing such a sum of algebraic functions, and then we apply it to several geometric optimization problems. We obtain the first FPTAS for two fundamental geometric shape-matching problems in fixed dimension: maximizing the volume of overlap of two polyhedra under rigid motions and minimizing their symmetric difference. We obtain the first FPTAS for other problems in fixed dimension, such as computing an optimal ray in a weighted subdivision, finding the largest axially symmetric subset of a polyhedron, and computing minimum-area hulls.

We present a new algorithm for computing motorcycle graphs that runs in O(n4/3+ε) time for any ε>0, improving on all previously known algorithms. The main application of this result is to computing the straight skeleton of a polygon. It allows us to compute the straight skeleton of a non-degenerate polygon with h holes in O(n √(h+1) log2 n + n4/3+ε) expected time. If all input coordinates are O(log n)-bit rational numbers, we can compute the straight skeleton of a (possibly degenerate) polygon with h holes in expected time O(n √{h+1}log3 n). In particular, it means that we can compute the straight skeleton of a simple polygon in O(n log3n) expected time if all input coordinates are O(log n)-bit rationals, while all previously known algorithms have worst-case running time ω(n3/2).

We present a new optimization technique that yields the first FPTAS for several geometric problems. These problems reduce to optimizing a sum of non-negative, constant description-complexity algebraic functions. We first give a FPTAS for optimizing such a sum of algebraic functions, and then we apply it to several geometric optimization problems. We obtain the first FPTAS for two fundamental geometric shape matching problems in fixed dimension: maximizing the volume of overlap of two polyhedra under rigid motions, and minimizing their symmetric difference. We obtain the first FPTAS for other problems in fixed dimension, such as computing an optimal ray in a weighted subdivision, finding the largest axially symmetric subset of a polyhedron, and computing minimum area hulls.

We present a data structure for answering approximate shortest path queries in a planar subdivision from a fixed source. Let $\rho\geqslant1$ be a real number. Distances in each face of this subdivision are measured by a possibly asymmetric convex distance function whose unit disk is contained in a concentric unit Euclidean disk and contains a concentric Euclidean disk with radius $1/\rho$. Different convex distance functions may be used for different faces, and obstacles are allowed. Let $\varepsilon$ be any number strictly between 0 and 1. Our data structure returns a $(1+\varepsilon)$ approximation of the shortest path cost from the fixed source to a query destination in $O(\log\frac{\rho n}{\varepsilon})$ time. Afterwards, a $(1+\varepsilon)$-approximate shortest path can be reported in $O(\log n)$ time plus the complexity of the path. The data structure uses $O(\frac{\rho^2n^3}{\varepsilon^2}\log\frac{\rho n}{\varepsilon})$ space and can be built in $O(\frac{\rho^2n^3}{\varepsilon^2}(\log\frac{\rho n}{\varepsilon})^2)$ time. Our time and space bounds do not depend on any other parameter; in particular, they do not depend on any geometric parameter of the subdivision such as the minimum angle.

Our goal is to find an approximate shortest path for a point robot moving in a planar subdivision with n vertices. Let $\rho\geq 1$ be a real number. Distances in each face of this subdivision are measured by a convex distance function whose unit disk is contained in a concentric unit Euclidean disk and contains a concentric Euclidean disk with radius $1/\rho$. Different convex distance functions may be used for different faces, and obstacles are allowed. These convex distance functions may be asymmetric. For any $\varepsilon\in(0,1)$ and for any two points $v_s$ and $v_d$, we give an algorithm that finds a path from $v_s$ to $v_d$ whose cost is at most $(1+\varepsilon)$ times the optimal. Our algorithm runs in $O(\frac{\rho^2\log \rho}{\varepsilon^2}n^3 \log(\frac{\rho n}\varepsilon))$ time. This bound does not depend on any other parameters; in particular it does not depend on the minimum angle in the subdivision. We give applications to two special cases that have been considered before: the weighted region problem and motion planning in the presence of uniform flows. For the weighted region problem with weights in $[1,\rho]\cup \{\infty\}$, the time bound of our algorithm improves to $O(\frac{\rho\log \rho}{\varepsilon}n^3 \log(\frac{\rho n}\varepsilon))$.

We present a data structure for answering approximate shortest path queries ina planar subdivision from a fixed source. Let ρ ≥ 1 be a real number.Distances in each face of this subdivision are measured by a possiblyasymmetric convex distance function whose unit disk is contained in aconcentric unit Euclidean disk, and contains a concentric Euclidean disk withradius 1/ρ. Different convex distance functions may be used for differentfaces, and obstacles are allowed. Let ε be any number strictly between 0and 1. Our data structure returns a (1+ε)approximation of the shortest path cost from the fixed source to a querydestination in O(logρn/ε) time. Afterwards, a(1+ε)-approximate shortest path can be reported in time linear in itscomplexity. The data structure uses O(ρ2 n4/ε2 log ρn/ε) space and can be built in O((ρ2 n4)/(ε2)(log ρn/ε)2) time. Our time and space bounds do not depend onany geometric parameter of the subdivision such as the minimum angle.

Given two compact convex sets P and Q in the plane, we compute an image of P under a rigid motion that approximately maximizes the overlap with Q. More precisely, for any ε > 0, we compute a rigid motion such that the area of overlap is at least 1 - ε times the maximum possible overlap. Our algorithm uses O(1/ε) extreme point and line intersection queries on P and Q, plus O((1/ε2) log(1/ε)) running time. If only translations are allowed, the extra running time reduces to O((1/ε) log(1/ε)). If P and Q are convex polygons with n vertices in total, the total running time is O((1/ε) log n + (1/ε2) log(1/ε)) for rigid motions and O((1/ε) log n + (1/ε) log(1/ε)) for translations.

Let B be a point robot moving in the plane, whose path is constrained to forward motions with a curvature at most 1, and let P be a convex polygon with n vertices.Given a starting configuration (a location and a direction of travel) for B inside P, we characterize the region of all points of P that can be reached by B, and show that it has linear complexity.