arXiv subjects

Marc van Kreveld

Publications and source records attributed to Marc van Kreveld.

At least 19 recordsLinked to original sources

arXiv2025

Minimizing Vertical Length in Linked Bar Charts

A linked bar chart is the augmentation of a traditional bar chart where each bar is partitioned into blocks and pairs of blocks are linked using orthogonal lines that pass over intermediate bars. The order of the blocks readily influences the legibility of the links. We study the algorithmic problem of minimizing the vertical length of these links, for a fixed bar order. The main challenge lies with ``dependent'' links, whose vertical link length cannot be optimized independently per bar. We show that, if the dependent links form a forest, the problem can be solved in $O(nm)$ time, for n bars and m links. If the dependent links between non-adjacent bars form a forest, the problem admits an $O(n^4m)$-time algorithm. Finally, we show that the general case is fixed-parameter tractable in the maximum number of links that are connected to one bar.

cs.CG↗

arXiv2025

Computing Largest Subsets of Points Whose Convex Hulls have Bounded Area and Diameter

We study the problem of computing a convex region with bounded area and diameter that contains the maximum number of points from a given point set $P$. We show that this problem can be solved in $O(n^6k)$ time and $O(n^3k)$ space, where $n$ is the size of $P$ and $k$ is the maximum number of points in the found region. We experimentally compare this new algorithm with an existing algorithm that does the same but without the diameter constraint, which runs in $O(n^3k)$ time. For the new algorithm, we use different diameters. We use both synthetic data and data from an application in cancer detection, which motivated our research.

cs.CG↗

arXiv2025

Faster Fréchet Distance Approximation through Truncated Smoothing

The Fréchet distance is a commonly used distance measure for curves. Computing the Fréchet distance between two polygonal curves of $n$ vertices takes roughly quadratic time, and conditional lower bounds suggest that approximating to within a factor $3$ cannot be done in strongly-subquadratic time, even in one dimension. Currently, the best approximation algorithms present trade-offs between approximation quality and running time. At SoCG 2021, Colombe and Fox presented an $O((n^3 / α^2) \log n)$-time $α$-approximate algorithm for curves in arbitrary dimensions, for any $α\in [\sqrt{n}, n]$. In this work, we give an $α$-approximate algorithm with a significantly faster running time of $O((n^2 / α) \log n)$, for any $α\in [1, n]$. In particular, we give the first strongly-subquadratic $n^\varepsilon$-approximation algorithm, for any constant $\varepsilon \in (0, 1/2]$. For curves in one dimension we further improve the running time to $O((n^2 / α^3) \log^2 n)$, for $α\in [1, n^{1/3}]$. Both of our algorithms rely on a linear-time simplification procedure that in one dimension reduces the complexity of the reachable free space to $O(n^2 / α)$ without making sacrifices in the asymptotic approximation factor.

cs.CG↗

arXiv2025

The Geodesic Fréchet Distance Between Two Curves Bounding a Simple Polygon

The Fréchet distance is a popular similarity measure that is well-understood for polygonal curves in $\mathbb{R}^d$: near-quadratic time algorithms exist, and conditional lower bounds suggest that these results cannot be improved significantly, even in one dimension and when approximating with a factor less than three. We consider the special case where the curves bound a simple polygon and distances are measured via geodesics inside this simple polygon. Here the conditional lower bounds do not apply; Efrat $et$ $al.$ (2002) were able to give a near-linear time $2$-approximation algorithm. In this paper, we significantly improve upon their result: we present a $(1+\varepsilon)$-approximation algorithm, for any $\varepsilon > 0$, that runs in $\mathcal{O}(\frac{1}{\varepsilon} (n+m \log n) \log nm \log \frac{1}{\varepsilon})$ time for a simple polygon bounded by two curves with $n$ and $m$ vertices, respectively. To do so, we show how to compute the reachability of specific groups of points in the free space at once, by interpreting the free space as one between separated one-dimensional curves. We solve this one-dimensional problem in near-linear time, generalizing a result by Bringmann and Künnemann (2015). Finally, we give a linear time exact algorithm if the two curves bound a convex polygon.

cs.CG↗

arXiv2025

A near-linear time exact algorithm for the $L_1$-geodesic Fréchet distance between two curves on the boundary of a simple polygon

Let $P$ be a polygon with $k$ vertices. Let $R$ and $B$ be two simple, interior disjoint curves on the boundary of $P$, with $n$ and $m$ vertices. We show how to compute the Fréchet distance between $R$ and $B$ using the geodesic $L_1$-distance in $P$ in $\mathcal{O}(k \log nm + (n+m) (\log^2 nm \log k + \log^4 nm))$ time.

cs.CG↗

arXiv2025

Minimum spanning blob-trees

We investigate blob-trees, a new way of connecting a set of points, by a mixture of enclosing them by cycles (as in the convex hull) and connecting them by edges (as in a spanning tree). We show that a minimum-cost blob-tree for $n$ points can be computed in $O(n^3)$ time.

cs.CG↗

arXiv2024

Robust Classification of Dynamic Bichromatic point Sets in R2

Let $R \cup B$ be a set of $n$ points in $\mathbb{R}^2$, and let $k \in 1..n$. Our goal is to compute a line that "best" separates the "red" points $R$ from the "blue" points $B$ with at most $k$ outliers. We present an efficient semi-online dynamic data structure that can maintain whether such a separator exists. Furthermore, we present efficient exact and approximation algorithms that compute a linear separator that is guaranteed to misclassify at most $k$, points and minimizes the distance to the farthest outlier. Our exact algorithm runs in $O(nk + n \log n)$ time, and our $(1+\varepsilon)$-approximation algorithm runs in $O(\varepsilon^{-1/2}((n + k^2) \log n))$ time. Based on our $(1+\varepsilon)$-approximation algorithm we then also obtain a semi-online data structure to maintain such a separator efficiently.

cs.CG↗

arXiv2024

Capturing the Shape of a Point Set with a Line Segment

Detecting location-correlated groups in point sets is an important task in a wide variety of applications areas. In addition to merely detecting such groups, the group's shape carries meaning as well. In this paper, we represent a group's shape using a simple geometric object, a line segment. Specifically, given a radius $r$, we say a line segment is representative of a point set $P$ if it is within distance $r$ of each point $p \in P$. We aim to find the shortest such line segment. This problem is equivalent to stabbing a set of circles of radius $r$ using the shortest line segment. We describe an algorithm to find the shortest representative segment in $O(n \log h + h \log^3 h)$ time. Additionally, we show how to maintain a stable approximation of the shortest representative segment when the points in $P$ move.

cs.CG↗

arXiv2024

The Complexity of Geodesic Spanners

A geometric $t$-spanner for a set $S$ of $n$ point sites is an edge-weighted graph for which the (weighted) distance between any two sites $p,q \in S$ is at most $t$ times the original distance between $p$ and~$q$. We study geometric $t$-spanners for point sets in a constrained two-dimensional environment $P$. In such cases, the edges of the spanner may have non-constant complexity. Hence, we introduce a novel spanner property: the spanner complexity, that is, the total complexity of all edges in the spanner. Let $S$ be a set of $n$ point sites in a simple polygon $P$ with $m$ vertices. We present an algorithm to construct, for any fixed integer $k \geq 1$, a $2\sqrt{2}k$-spanner with complexity $O(mn^{1/k} + n\log^2 n)$ in $O(n\log^2n + m\log n + K)$ time, where $K$ denotes the output complexity. When we relax the restriction that the edges in the spanner are shortest paths, such that an edge in the spanner can be any path between two sites, we obtain for any constant $\varepsilon \in (0,2k)$ a relaxed geodesic $(2k + \varepsilon)$-spanner of the same complexity, where the constant is dependent on $\varepsilon$. When we consider sites in a polygonal domain $P$ with holes, we can construct a relaxed geodesic $6k$-spanner of complexity $O(mn^{1/k} + n\log^2 n)$ in $O((n+m)\log^2n\log m+ K)$ time. Additionally, for any constant $\varepsilon \in (0,1)$ and integer constant $t \geq 2$, we show a lower bound for the complexity of any $(t-\varepsilon)$-spanner of $Ω(mn^{1/(t-1)} + n)$.

cs.CG↗

arXiv2024

Towards Mobility Data Science (Vision Paper)

Mobility data captures the locations of moving objects such as humans, animals, and cars. With the availability of GPS-equipped mobile devices and other inexpensive location-tracking technologies, mobility data is collected ubiquitously. In recent years, the use of mobility data has demonstrated significant impact in various domains including traffic management, urban planning, and health sciences. In this paper, we present the emerging domain of mobility data science. Towards a unified approach to mobility data science, we envision a pipeline having the following components: mobility data collection, cleaning, analysis, management, and privacy. For each of these components, we explain how mobility data science differs from general data science, we survey the current state of the art and describe open challenges for the research community in the coming years.

cs.OH↗

arXiv2023

Collision Detection for Modular Robots -- it is easy to cause collisions and hard to avoid them

We consider geometric collision-detection problems for modular reconfigurable robots. Assuming the nodes (modules) are connected squares on a grid, we investigate the complexity of deciding whether collisions may occur, or can be avoided, if a set of expansion and contraction operations is executed. We study both discrete- and continuous-time models, and allow operations to be coupled into a single parallel group. Our algorithms to decide if a collision may occur run in $O(n^2\log^2 n)$ time, $O(n^2)$ time, or $O(n\log^2 n)$ time, depending on the presence and type of coupled operations, in a continuous-time model for a modular robot with $n$ nodes. To decide if collisions can be avoided, we show that a very restricted version is already NP-complete in the discrete-time model, while the same problem is polynomial in the continuous-time model. A less restricted version is NP-hard in the continuous-time model.

cs.CG↗

arXiv2023

Reconstructing Graphs from Connected Triples

We introduce a new model of indeterminacy in graphs: instead of specifying all the edges of the graph, the input contains all triples of vertices that form a connected subgraph. In general, different (labelled) graphs may have the same set of connected triples, making unique reconstruction of the original graph from the triples impossible. We identify some families of graphs (including triangle-free graphs) for which all graphs have a different set of connected triples. We also give algorithms that reconstruct a graph from a set of triples, and for testing if this reconstruction is unique. Finally, we study a possible extension of the model in which the subsets of size $k$ that induce a connected graph are given for larger (fixed) values of $k$.

cs.DM↗

arXiv2023

Approximating the Earth Mover's Distance between sets of geometric objects

Given two distributions $P$ and $S$ of equal total mass, the Earth Mover's Distance measures the cost of transforming one distribution into the other, where the cost of moving a unit of mass is equal to the distance over which it is moved. We give approximation algorithms for the Earth Mover's Distance between various sets of geometric objects. We give a $(1 + \varepsilon)$-approximation when $P$ is a set of weighted points and $S$ is a set of line segments, triangles or $d$-dimensional simplices. When $P$ and $S$ are both sets of line segments, sets of triangles or sets of simplices, we give a $(1 + \varepsilon)$-approximation with a small additive term. All algorithms run in time polynomial in the size of $P$ and $S$, and actually calculate the transport plan (that is, a specification of how to move the mass), rather than just the cost. To our knowledge, these are the first combinatorial algorithms with a provable approximation ratio for the Earth Mover's Distance when the objects are continuous rather than discrete points.

cs.CG↗

arXiv2022

A Subquadratic $n^ε$-approximation for the Continuous Fréchet Distance

The Fréchet distance is a commonly used similarity measure between curves. It is known how to compute the continuous Fréchet distance between two polylines with $m$ and $n$ vertices in $\mathbb{R}^d$ in $O(mn (\log \log n)^2)$ time; doing so in strongly subquadratic time is a longstanding open problem. Recent conditional lower bounds suggest that it is unlikely that a strongly subquadratic algorithm exists. Moreover, it is unlikely that we can approximate the Fréchet distance to within a factor $3$ in strongly subquadratic time, even if $d=1$. The best current results establish a tradeoff between approximation quality and running time. Specifically, Colombe and Fox (SoCG, 2021) give an $O(α)$-approximate algorithm that runs in $O((n^3 / α^2) \log n)$ time for any $α\in [\sqrt{n}, n]$, assuming $m = n$. In this paper, we improve this result with an $O(α)$-approximate algorithm that runs in $O((n + mn / α) \log^3 n)$ time for any $α\in [1, n]$, assuming $m \leq n$ and constant dimension $d$.

cs.CG↗

arXiv2022

Abstract morphing using the Hausdorff distance and Voronoi diagrams

This paper introduces two new abstract morphs for two $2$-dimensional shapes. The intermediate shapes gradually reduce the Hausdorff distance to the goal shape and increase the Hausdorff distance to the initial shape. The morphs are conceptually simple and apply to shapes with multiple components and/or holes. We prove some basic properties relating to continuity, containment, and area. Then we give an experimental analysis that includes the two new morphs and a recently introduced abstract morph that is also based on the Hausdorff distance (Van Kreveld et al. Between shapes, using the Hausdorff distance. Computational Geometry 100:101817, 2022). We show results on the area and perimeter development throughout the morph, and also the number of components and holes. A visual comparison shows that one of the new morphs appears most attractive.

cs.CG↗

arXiv2021

Topological Stability of Kinetic $k$-Centers

We study the $k$-center problem in a kinetic setting: given a set of continuously moving points $P$ in the plane, determine a set of $k$ (moving) disks that cover $P$ at every time step, such that the disks are as small as possible at any point in time. Whereas the optimal solution over time may exhibit discontinuous changes, many practical applications require the solution to be stable: the disks must move smoothly over time. Existing results on this problem require the disks to move with a bounded speed, but this model allows positive results only for $k<3$. Hence, the results are limited and offer little theoretical insight. Instead, we study the topological stability of $k$-centers. Topological stability was recently introduced and simply requires the solution to change continuously, but may do so arbitrarily fast. We prove upper and lower bounds on the ratio between the radii of an optimal but unstable solution and the radii of a topologically stable solution -- the topological stability ratio -- considering various metrics and various optimization criteria. For $k = 2$ we provide tight bounds, and for small $k > 2$ we can obtain nontrivial lower and upper bounds. Finally, we provide an algorithm to compute the topological stability ratio in polynomial time for constant $k$.

cs.CG↗

arXiv2021

Between Shapes, Using the Hausdorff Distance

Given two shapes $A$ and $B$ in the plane with Hausdorff distance $1$, is there a shape $S$ with Hausdorff distance $1/2$ to and from $A$ and $B$? The answer is always yes, and depending on convexity of $A$ and/or $B$, $S$ may be convex, connected, or disconnected. We show that our result can be generalised to give an interpolated shape between $A$ and $B$ for any interpolation variable $α$ between $0$ and $1$, and prove that the resulting morph has a bounded rate of change with respect to $α$. Finally, we explore a generalization of the concept of a Hausdorff middle to more than two input sets. We show how to approximate or compute this middle shape, and that the properties relating to the connectedness of the Hausdorff middle extend from the case with two input sets. We also give bounds on the Hausdorff distance between the middle set and the input.

cs.CG↗

arXiv2020

Gourds: a sliding-block puzzle with turning

We propose a new kind of sliding-block puzzle, called Gourds, where the objective is to rearrange 1 x 2 pieces on a hexagonal grid board of 2n + 1 cells with n pieces, using sliding, turning and pivoting moves. This puzzle has a single empty cell on a board and forms a natural extension of the 15-puzzle to include rotational moves. We analyze the puzzle and completely characterize the cases when the puzzle can always be solved. We also study the complexity of determining whether a given set of colored pieces can be placed on a colored hexagonal grid board with matching colors. We show this problem is NP-complete for arbitrarily many colors, but solvable in randomized polynomial time if the number of colors is a fixed constant.

cs.DS↗