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Tim Ophelders

Publications and source records attributed to Tim Ophelders.

At least 19 recordsLinked to original sources
arXiv2026

A Strongly-Subquadratic $(3+\varepsilon)$-Approximation for the Fr\'echet Distance for Paths in Metric Spaces

The Fr\'echet distance is a well-studied distance measure for paths in a metric space. It is mostly studied for paths in $d$-dimensional Euclidean space. Here, computing the Fr\'echet distance between two polylines takes time roughly quadratic in the number of vertices. Assuming the strong exponential time hypothesis (SETH), it cannot be approximated to within a factor less than $3$ in strongly-subquadratic time. Recently, it was shown that for any $\varepsilon>0$, there exists a randomized algorithm that can compute a $(7+\varepsilon)$-approximation in strongly-subquadratic expected time [Cheng, Huang, and Zhang; STOC'25]. For polylines with $n$ and $m$ vertices in a Euclidean space of constant dimension, where $n \geq m$, their algorithm takes $O(nm^{0.99} \log(n/\varepsilon))$ time in expectation. We present a deterministic approximation algorithm that significantly improves upon the approximation factor and running time. Specifically, our algorithm computes a $(3+\varepsilon)$-approximation in $O(nm^{2/3} \log n \cdot \log (\frac{1}{\varepsilon} \log n))$ time. Our algorithm nearly matches the conditional lower bound on the approximation factor implied by SETH. For polylines in $\mathbb{R}$, we present a $3$-approximation algorithm that runs in $O(nm^{2/3} \log^{5/3} n)$ time, and exactly matches the conditional lower bound. For our results, we introduce a general strongly-subquadratic time $3$-approximate decision algorithm. This algorithm makes no assumptions on the ambient metric space, and relies only on standard assumptions on the so-called free space of the input paths. Under some mild assumptions, our decision algorithm leads to a $(3+\varepsilon)$-approximation algorithm in general metric spaces. These assumptions hold automatically for polylines in any metric space $(\mathbb{R}^d, L_p)$ with $p \geq 1$.

cs.CG↗
arXiv2026

Which Vertical Graphs are Non VPHT Reconstructible?

The verbose persistent homology transform (VPHT) is a topological summary of shapes in Euclidean space. Assuming general position, the VPHT is injective, meaning shapes can be reconstructed using only the VPHT. In this work, we investigate cases in which the VPHT is not injective, focusing on a simple setting of degeneracy; graphs whose vertices are all collinear. We identify both necessary properties and sufficient properties for non-reconstructibility of such graphs, bringing us closer to a complete classification.

cs.CG↗
arXiv2026

Reconfiguration of Squares Using a Constant Number of Moves Each

Multi-robot motion planning is a hard problem. We investigate restricted variants of the problem where square robots are allowed to slide over an arbitrary curve to a new position only a constant number of times each. We show that the problem remains NP-hard in most cases, except when the squares have unit size and when the problem is unlabeled, i.e., the location of each square in the target configuration is left unspecified.

cs.CG↗
arXiv2025

Locally Correct Interleavings between Merge Trees

Temporal sequences of terrains arise in various application areas. To analyze them efficiently, one generally needs a suitable abstraction of the data as well as a method to compare and match them over time. In this paper we consider merge trees as a topological descriptor for terrains and the interleaving distance as a method to match and compare them. An interleaving between two merge trees consists of two maps, one in each direction. These maps must satisfy ancestor relations and hence introduce a ''shift'' between points and their image. An optimal interleaving minimizes the maximum shift; the interleaving distance is the value of this shift. However, to study the evolution of merge trees over time, we need not only a number but also a meaningful matching between the two trees. The two maps of an optimal interleaving induce a matching, but due to the bottleneck nature of the interleaving distance, this matching fails to capture local similarities between the trees. In this paper we hence propose a notion of local optimality for interleavings. To do so, we define the residual interleaving distance, a generalization of the interleaving distance that allows additional constraints on the maps. This allows us to define locally correct interleavings, which use a range of shifts across the two merge trees that reflect the local similarity well. We give a constructive proof that a locally correct interleaving always exists.

cs.CG↗
arXiv2025

Counting Triangulations of Fixed Cardinal Degrees

A fixed set of vertices in the plane may have multiple planar straight-line triangulations in which the degree of each vertex is the same. As such, the degree information does not completely determine the triangulation. We show that even if we know, for each vertex, the number of neighbors in each of the four cardinal directions, the triangulation is not completely determined. In fact, we show that counting such triangulations is #P-hard via a reduction from #3-regular bipartite planar vertex cover.

cs.CC↗
arXiv2025

Computing the Fr\'echet Distance When Just One Curve is $c$-Packed: A Simple Almost-Tight Algorithm

We study approximating the continuous Fr\'echet distance of two curves with complexity $n$ and $m$, under the assumption that only one of the two curves is $c$-packed. Driemel, Har{-}Peled and Wenk DCG'12 studied Fr\'echet distance approximations under the assumption that both curves are $c$-packed. In $\mathbb{R}^d$, they prove a $(1+\varepsilon)$-approximation in $\tilde{O}(d \, c\,\frac{n+m}{\varepsilon})$ time. Bringmann and K\"unnemann IJCGA'17 improved this to $\tilde{O}(c\,\frac{n + m }{\sqrt{\varepsilon}})$ time, which they showed is near-tight under SETH. Recently, Gudmundsson, Mai, and Wong ISAAC'24 studied our setting where only one of the curves is $c$-packed. They provide an involved $\tilde{O}( d \cdot (c+\varepsilon^{-1})(cn\varepsilon^{-2} + c^2m\varepsilon^{-7} + \varepsilon^{-2d-1}))$-time algorithm when the $c$-packed curve has $n$ vertices and the arbitrary curve has $m$, where $d$ is the dimension in Euclidean space. In this paper, we show a simple technique to compute a $(1+\varepsilon)$-approximation in $\mathbb{R}^d$ in time $O(d \cdot c\,\frac{n+m}{\varepsilon}\log\frac{n+m}{\varepsilon})$ when one of the curves is $c$-packed. Our approach is not only simpler than previous work, but also significantly improves the dependencies on $c$, $\varepsilon$, and $d$. Moreover, it almost matches the asymptotically tight bound for when both curves are $c$-packed. Our algorithm is robust in the sense that it does not require knowledge of $c$, nor information about which of the two input curves is $c$-packed.

cs.CG↗
arXiv2025

ParkView: Visualizing Monotone Interleavings

Merge trees are a powerful tool from topological data analysis that is frequently used to analyze scalar fields. The similarity between two merge trees can be captured by an interleaving: a pair of maps between the trees that jointly preserve ancestor relations in the trees. Interleavings can have a complex structure; visualizing them requires a sense of (drawing) order which is not inherent in this purely topological concept. However, in practice it is often desirable to introduce additional geometric constraints, which leads to variants such as labeled or monotone interleavings. Monotone interleavings respect a given order on the leaves of the merge trees and hence have the potential to be visualized in a clear and comprehensive manner. In this paper, we introduce ParkView: a schematic, scalable encoding for monotone interleavings. ParkView captures both maps of the interleaving using an optimal decomposition of both trees into paths and corresponding branches. We prove several structural properties of monotone interleavings, which support a sparse visual encoding using active paths and hedges that can be linked using a maximum of 6 colors for merge trees of arbitrary size. We show how to compute an optimal path-branch decomposition in linear time and illustrate ParkView on a number of real-world datasets.

cs.CG↗
arXiv2025

The Geodesic Fr\'echet Distance Between Two Curves Bounding a Simple Polygon

The Fr\'echet 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 K\"unnemann (2015). Finally, we give a linear time exact algorithm if the two curves bound a convex polygon.

cs.CG↗
arXiv2025

Sweeping Orders for Simplicial Complex Reconstruction

Standard sweep algorithms require an order of discrete points in Euclidean space, and rely on the property that, at a given point, all points in the halfspace below come earlier in this order. We are motivated by the problem of reconstructing a graph in $\mathbb{R}^d$ from vertex locations and degree information, which was addressed using standard sweep algorithms by Fasy et al. We generalize this to the reconstruction of general simplicial complexes. As our main ingredient, we introduce a generalized \emph{sweeping order} on $i$-simplices, maintaining the property that, at a given $i$-simplex $\sigma$, all $(i+1)$-dimensional cofaces of $\sigma$ in the halfspace below $\sigma$ have an $i$-dimensional face that appeared earlier in the order ("below" with respect to some direction perpendicular to $\sigma$). We then go on to incorporate computing such sweeping orders to reconstruct an unknown simplicial complex $K$, starting with only its vertex locations, i.e., its $0$-skeleton. Specifically, once we have found the $i$-skeleton of $K$, we compute a sweeping order for the $i$-simplices, and use it to reconstruct the $(i+1)$-skeleton of $K$ by querying the \emph{indegree}, or the number of $(i+1)$-simplices incident to and below a given $i$-simplex. In addition to generalizing the algorithm by Fasy et al. to simplicial complexes, we improve upon the running time of their central subroutine of radially finding edges above a vertex.

cs.CG↗
arXiv2024

Faster, Deterministic and Space Efficient Subtrajectory Clustering

Given a trajectory $T$ and a distance $\Delta$, we wish to find a set $C$ of curves of complexity at most $\ell$, such that we can cover $T$ with subcurves that each are within Fr\'echet distance $\Delta$ to at least one curve in $C$. We call $C$ an $(\ell,\Delta)$-clustering and aim to find an $(\ell,\Delta)$-clustering of minimum cardinality. This problem variant was introduced by Akitaya $et$ $al.$ (2021) and shown to be NP-complete. The main focus has therefore been on bicriteria approximation algorithms, allowing for the clustering to be an $(\ell, \Theta(\Delta))$-clustering of roughly optimal size. We present algorithms that construct $(\ell,4\Delta)$-clusterings of $\mathcal{O}(k \log n)$ size, where $k$ is the size of the optimal $(\ell, \Delta)$-clustering. We use $\mathcal{O}(n^3)$ space and $\mathcal{O}(k n^3 \log^4 n)$ time. Our algorithms significantly improve upon the clustering quality (improving the approximation factor in $\Delta$) and size (whenever $\ell \in \Omega(\log n / \log k)$). We offer deterministic running times improving known expected bounds by a factor near-linear in $\ell$. Additionally, we match the space usage of prior work, and improve it substantially, by a factor super-linear in $n\ell$, when compared to deterministic results.

cs.CG↗
arXiv2024

The Complexity of Geodesic Spanners using Steiner Points

A geometric $t$-spanner $\mathcal{G}$ on a set $S$ of $n$ point sites in a metric space $P$ is a subgraph of the complete graph on $S$ such that for every pair of sites $p,q$ the distance in $\mathcal{G}$ is a most $t$ times the distance $d(p,q)$ in $P$. We call a connection between two sites a \emph{link}. In some settings, such as when $P$ is a simple polygon with $m$ vertices and a link is a shortest path in $P$, links can consist of $\Theta (m)$ segments and thus have non-constant complexity. The spanner complexity is a measure of how compact a spanner is, which is equal to the sum of the complexities of all links in the spanner. In this paper, we study what happens if we are allowed to introduce $k$ Steiner points to reduce the spanner complexity. We study such Steiner spanners in simple polygons, polygonal domains, and edge-weighted trees. We show that Steiner points have only limited utility. For a spanner that uses $k$ Steiner points, we provide an $\Omega(mn^{1/(t+1)}/k^{1/(t+1)})$ lower bound on the worst-case complexity of any $(t-\varepsilon)$-spanner, for any constant $\varepsilon \in (0,1)$ and integer constant $t \geq 2$. Additionally, we show NP-hardness for the problem of deciding whether a set of sites in a polygonal domain admits a $3$-spanner with a given maximum complexity using $k$ Steiner points. On the positive side, for trees we show how to build a $2t$-spanner that uses $k$ Steiner points of complexity $O(mn^{1/t}/k^{1/t} + n \log (n/k))$, for any integer $t \geq 1$. We generalize this to forests, and use it to obtain a $2\sqrt{2}t$-spanner in a simple polygon with complexity $O(mn^{1/t}(\log k)^{1+1/t}/k^{1/t} + n\log^2 n)$. When a link can be any path between two sites, we show how to improve the spanning ratio to $(2k+\varepsilon)$, for any constant $\varepsilon \in (0,2k)$, and how to build a $6t$-spanner in a polygonal domain with the same complexity.

cs.CG↗
arXiv2024

Faster Fr\'echet Distance Approximation through Truncated Smoothing

The Fr\'echet distance is a commonly used distance measure for curves. Computing the Fr\'echet 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 $O((n^3 / \alpha^2) \log n)$-time $\alpha$-approximate algorithm for curves in arbitrary dimensions, for any $\alpha \in [\sqrt{n}, n]$. In this work, we give an $\alpha$-approximate algorithm with a significantly faster running time of $O((n^2 / \alpha) \log n)$, for any $\alpha \in [1, n]$. 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 $O((n^2 / \alpha^3) \log^2 n)$, for $\alpha \in [1, n^{1/3}]$. Both of our algorithms rely on a linear-time simplification procedure that in one dimension reduces the complexity of the reachable free space to $O(n^2 / \alpha)$ without making sacrifices in the asymptotic approximation factor.

cs.CG↗
arXiv2023

Optimal In-Place Compaction of Sliding Cubes

The sliding cubes model is a well-established theoretical framework that supports the analysis of reconfiguration algorithms for modular robots consisting of face-connected cubes. The best algorithm currently known for the reconfiguration problem, by Abel and Kominers [arXiv, 2011], uses O(n3) moves to transform any n-cube configuration into any other n-cube configuration. As is common in the literature, this algorithm reconfigures the input into an intermediate canonical shape. In this paper we present an in-place algorithm that reconfigures any n-cube configuration into a compact canonical shape using a number of moves proportional to the sum of coordinates of the input cubes. This result is asymptotically optimal. Furthermore, our algorithm directly extends to dimensions higher than three.

cs.CG↗
arXiv2023

Relating Interleaving and Fr\'echet Distances via Ordered Merge Trees

Merge trees are a common topological descriptor for data with a hierarchical component, such as terrains and scalar fields. The interleaving distance, in turn, is a common distance for comparing merge trees. However, the interleaving distance for merge trees is solely based on the hierarchical structure, and disregards any other geometrical or topological properties that might be present in the underlying data. Furthermore, the interleaving distance is NP-hard to compute. In this paper, we introduce a form of ordered merge trees that can capture intrinsic order present in the data. We further define a natural variant of the interleaving distance, the monotone interleaving distance, which is an order-preserving distance for ordered merge trees. Analogously to the regular interleaving distance for merge trees, we show that the monotone variant has three equivalent definitions in terms of two maps, a single map, or a labelling. Furthermore, we establish a connection between the monotone interleaving distance of ordered merge trees and the Fr\'echet distance of 1D curves. As a result, the monotone interleaving distance between two ordered merge trees can be computed exactly in near-quadratic time in their complexity. The connection between the monotone interleaving distance and the Fr\'echet distance builds a new bridge between the fields of topological data analysis, where interleaving distances are a common tool, and computational geometry, where Fr\'echet distances are studied extensively.

cs.CG↗
arXiv2023

Simply Realising an Imprecise Polyline is NP-hard

We consider the problem of deciding, given a sequence of regions, if there is a choice of points, one for each region, such that the induced polyline is simple or weakly simple, meaning that it can touch but not cross itself. Specifically, we consider the case where each region is a translate of the same shape. We show that the problem is NP-hard when the shape is a unit-disk or unit-square. We argue that the problem is NP-complete when the shape is a vertical unit-segment.

cs.CG↗
arXiv2023

Shortest Paths in Portalgons

Any surface that is intrinsically polyhedral can be represented by a collection of simple polygons (fragments), glued along pairs of equally long oriented edges, where each fragment is endowed with the geodesic metric arising from its Euclidean metric. We refer to such a representation as a portalgon, and we call two portalgons equivalent if the surfaces they represent are isometric. We analyze the complexity of shortest paths in portalgons. We call a fragment happy if any shortest path on the portalgon visits it at most a constant number of times. A portalgon is happy if all of its fragments are happy. We present an efficient algorithm to compute shortest paths on happy portalgons. The number of times that a shortest path visits a fragment is unbounded in general. We contrast this by showing that the intrinsic Delaunay triangulation of any polyhedral surface corresponds to a happy portalgon. Since computing the intrinsic Delaunay triangulation may be inefficient, we provide an efficient algorithm to compute happy portalgons for a restricted class of portalgons.

cs.CG↗
arXiv2022

Morphing Planar Graph Drawings Through 3D

In this paper, we investigate crossing-free 3D morphs between planar straight-line drawings. We show that, for any two (not necessarily topologically equivalent) planar straight-line drawings of an $n$-vertex planar graph, there exists a piecewise-linear crossing-free 3D morph with $O(n^2)$ steps that transforms one drawing into the other. We also give some evidence why it is difficult to obtain a linear lower bound (which exists in 2D) for the number of steps of a crossing-free 3D morph.

cs.CG↗