Sarita de Berg

Sarita de Berg, Guillermo Esteban, Rodrigo I. Silveira et al.

In this paper, we consider the Weighted Region Problem. In the Weighted Region Problem, the length of a path is defined as the sum of the weights of the subpaths within each region, where the weight of a subpath is its Euclidean length multiplied by a weight $ α\geq 0 $ depending on the region. We study a restricted version of the problem of determining shortest paths through a single weighted rectangular region. We prove that even this very restricted version of the problem is unsolvable within the Algebraic Computation Model over the Rational Numbers (ACMQ). On the positive side, we provide the equations for the shortest paths that are computable within the ACMQ. Additionally, we provide equations for the bisectors between regions of the Shortest Path Map for a source point on the boundary of (or inside) the rectangular region.

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 $Θ(n \log n)$.

Sarita de Berg, Ivor van der Hoog, Eva Rotenberg et al.

We maintain a $(1+\varepsilon)$-spanner over the disk intersection graph of a dynamic set of disks. We restrict all disks to have their diameter in $[4,Ψ]$ for some fixed and known $Ψ$. The resulting $(1+\varepsilon)$-spanner has size $O(n \varepsilon^{-2} \log Ψ\log (\varepsilon^{-1}))$, where $n$ is the present number of disks. We develop a novel use of persistent data structures to dynamically maintain our $(1+\varepsilon)$-spanner. Our approach requires $O(\varepsilon^{-2} n \log^4 n \log Ψ)$ space and has an $O( \left( \fracΨ{\varepsilon} \right)^2 \log^4 n \log^2 Ψ\log^2 (\varepsilon^{-1}))$ expected amortised update time. For constant $\varepsilon$ and $Ψ$, this spanner has near-linear size, uses near-linear space and has polylogarithmic update time. Furthermore, we observe that for any $\varepsilon < 1$, our spanner also serves as a connectivity data structure. With a slight adaptation of our techniques, this leads to better bounds for dynamically supporting connectivity queries in a disk intersection graph. In particular, we improve the space usage when compared to the dynamic data structure of (Baumann et al., DCG'24), replacing the linear dependency on $Ψ$ by a polylogarithmic dependency. Finally, we generalise our results to $d$-dimensional hypercubes.

Sarita de Berg, Joachim Gudmundsson, Peter Kramer et al.

We investigate multiple fundamental variants of the classic coordinated motion planning (CMP) problem for unit square robots in the plane under the $L_1$ metric. In coordinated motion planning, we are given two arrangements of $k$ robots and are tasked with finding a movement schedule that minimizes a certain objective function. The two most prominent objective functions are the sum of distances traveled (Min-Sum) and the latest time of arrival (Min-Makespan). Both objectives have previously been studied extensively. We introduce a new objective function for CMP in the plane. The proposed Min-Exposure objective function defines a set of polygonal regions in the plane that provide cover and asks for a schedule with minimal elapsed time during which at least one robot is partially or fully outside of these regions. We give an $\mathcal{O}(n^4\log n)$ time algorithm that computes exposure-minimal schedules for $k=2$ robots, and an XP algorithm for arbitrary $k$. As a result of independent interest, we leverage new insights to prove that both the Min-Makespan and Min-Sum objectives are fixed-parameter tractable (FPT) parameterized by the number of robots. Our parameterized complexity results generalize known FPT results for rectangular grid graphs [Eiben, Ganian, and Kanj, SoCG'23].

Sarita de Berg, Nynne Maria Foldager Bække, Frida Astrup Eriksen et al.

In the imprecise geometry model, the input is an imprecise point set, which is a family of regions $F = (R_1, \ldots,R_n)$, where for each $R_i$ one may retrieve the true point $p_i \in R_i$. By preprocessing $F$, we can construct the output, in our case the Pareto front, on $P$ faster. We efficiently construct the Pareto front of an imprecise point set in the plane. Efficiency is interpreted in two ways: minimizing (i) the number of retrievals, and (ii) the computation time used to determine the set of regions that must be retrieved and to construct the Pareto front. We present an algorithm to construct the Pareto front for possibly overlapping rectangles that is \emph{instance-optimal} with respect to the number of retrievals, meaning that for every fixed input $(F, P)$, there is no algorithm that retrieves asymptotically fewer regions to compute the output. This is a strong algorithmic quality, as it means that our algorithm is competitive even to clairvoyant algorithms which know a correct guess of the output and only have to verify its correctness. In terms of algorithmic running time, instance-optimality is provably unobtainable. We instead present an algorithm which is within a $\log n$-factor of instance-optimality. This generalizes earlier results to overlapping input regions, at only a minor cost in running time. For unit squares, we present an algorithm that is not only instance-optimal in the number of retrievals, but also \emph{universally} optimal in terms of running time, meaning that for any fixed set of regions $F$, no algorithm has a better worst-case running time for all possible point sets $P$. This is the first universally optimal algorithm for overlapping planar input. Compared to previous work, this result improves the degree of overlap, the preprocessing time, the number of retrievals, and the running time.