Jacobus Conradi

Sarita de Berg, Jacobus Conradi, Ivor van der Hoog et al.

Recently, a natural variant of the Art Gallery problem, known as the \emph{Contiguous Art Gallery problem} was proposed. Given a simple polygon $P$, the goal is to partition its boundary $\partial P$ into the smallest number of contiguous segments such that each segment is completely visible from some point in $P$. Unlike the classical Art Gallery problem, which is NP-hard, this variant is polynomial-time solvable. At SoCG~2025, three independent works presented algorithms for this problem, each achieving a running time of $O(k n^5 \log n)$ (or $O(n^6\log n)$), where $k$ is the size of an optimal solution. Interestingly, these results were obtained using entirely different approaches, yet all led to roughly the same asymptotic complexity, suggesting that such a running time might be inherent to the problem. We show that this is not the case. In the real RAM-model, the prevalent model in computational geometry, we present an $O(n \log n)$-time algorithm, achieving an $O(k n^4)$ factor speed-up over the previous state-of-the-art. We also give a straightforward sorting-based lower bound by reducing from the set intersection problem. We thus show that the Contiguous Art Gallery problem is in $Θ(n \log n)$.

Jacobus Conradi, Ivor van der Hoog, Eva Rotenberg

The continuous Frechet distance between two polygonal curves is classically computed by exploring their free space diagram. Recently, Har-Peled, Raichel, and Robson [SoCG'25] proposed a radically different approach: instead of directly traversing the continuous free space, they approximate the distance by computing paths in a discrete graph derived from the discrete free space, recursively bisecting edges until the discrete distance converges to the continuous Frechet distance. They implement this so-called frog-based technique and report substantial practical speedups over the state of the art. We revisit the frog-based approach and address three of its limitations. First, the method does not compute the Frechet distance exactly. Second, the recursive bisection procedure only introduces the monotonicity events required to realise the Frechet distance asymptotically, that is, only in the limit. Third, the applied simplification technique is heuristic. Motivated by theoretical considerations, we develop new techniques that guarantee exactness, polynomial-time convergence, and near-optimal lossless simplifications. We provide an open-source C++ implementation of our variant. Our primary contribution is an extensive empirical evaluation. As expected, exact computation introduces overhead and increases the median running time. Yet, our method is often faster in the worst case, the slowest ten percent of instances, or even on average due to its convergence guarantees. More surprisingly, in our experiments, the implementation of Bringmann, Kuennemann, and Nusser [SoCG'19] consistently outperforms all frog-based approaches in practice. This appears to contrast published claims of the efficiency of the frog-based techniques. These results thereby provide nuanced perspective on frogs: highlighting both the theoretical appeal, but also the practical limitations.

Sepideh Aghamolaei, Kevin Buchin, Timothy M. Chan et al.

Computing the convex hull of a planar $n$-point set $P$ is one of the most fundamental problems in computational geometry. It has an $Ω(n \log n)$ lower bound in the algebraic computation tree model, and many convex hull algorithms match this bound. Classical results show that, under special input assumptions, sub-$O(n \log n)$ algorithms are possible. For instance, when the points are given in lexicographic or angular order, the convex hull can be computed in linear time. Even under the weaker assumption that the sequence of points corresponds to the ordered vertices of a simple polygonal chain, linear-time algorithms exist. This naturally raises the question: can the convex hull of a point set be computed in sub-$O(n \log n)$ time under weaker input assumptions? We answer this positively. Under the promise that the input sequence contains the convex hull as a subsequence, we give a deterministic $O(n \sqrt{\log n})$-time algorithm to compute the convex hull of $P$. With randomisation, we achieve expected running time $O(n \log^{\varepsilon} n)$ for any constant $\varepsilon > 0$. We find this surprising, as points not on the convex hull may behave adversarially toward our convex hull construction algorithm. Yet the promise that \emph{only} the hull points are sorted suffices for $o(n \log n)$-time algorithms. Finally, we show that this promise is tight: if it is even slightly broken, i.e., allowing just one hull point to appear out of order, we prove an adversarial $Ω(n \log n)$-time lower bound. Consequently, the promise cannot be verified with fewer than $Ω(n \log n)$ comparisons. This also negatively resolves an open problem of Löffler and Raichel, who conjectured sub-$O(n \log n)$-time algorithms for computing the convex hull of a supersequence containing the hull as a subsequence.

Jacobus Conradi, Ivor van der Hoog, Frederikke Uldahl et al.

The Fréchet distance is a popular distance measure between trajectories or curves in space, or between walks in graphs. We study computing the Fréchet distance between walks in the $d$-dimensional grid graphs, i.e. $\mathbb{Z}^d$ where points share an edge if they differ by one in one coordinate. We give an algorithm, that for two simple paths on $n$ vertices, $(1+\varepsilon)$-approximates the Fréchet distance in time $\widetilde{O}((\frac{n}{\varepsilon})^{2-2/d} +n)$. We complement this by a near-matching fine-grained lower bound: for constant dimensions $d \geq 3$, there is no $O((\varepsilon^{2/d}(\frac{n}{\varepsilon})^{2-2/d})^{1-δ})$ algorithm for any $δ>0$ unless the Orthogonal Vector Hypothesis fails. Thus, our results are tight up to a factor $\varepsilon^{2/d}$ and $\log(n)$-factors. We extend our results to imbalanced lower and upper bounds, where the curves have $n$ and $m$ vertices respectively, and also obtain near-tight bounds. Driemel, Har-Peled and Wenk [DCG'12] studied \emph{realistic assumptions} for curves to speed up Fréchet distance computation. One of these assumptions is $λ$-low density and they can compute a $(1+\varepsilon)$-approximation between $λ$-low dense curves in time $\widetilde{O}( \varepsilon^{-2} λ^2 n^{2(1-1/d)})$. By adapting our lower bound, we show that their algorithm has a tight dependency on $n$ and a tight dependency on $\varepsilon$ as $d$ goes to infinity. A gap remains in terms of $λ$.