A t-spanner of a point set X in a metric space (𝒳, δ) is a graph G with vertex set P such that, for any pair of points u,v ∈ X, the distance between u and v in G is at most t times δ(u,v). We study the problem of maintaining a spanner for a dynamic point set X - that is, when X undergoes a sequence of insertions and deletions - in a metric space of constant doubling dimension. For any constant ε > 0, we maintain a (1+ε)-spanner of P whose total weight remains within a constant factor of the weight of the minimum spanning tree of X. Each update (insertion or deletion) can be performed in poly(log Φ) time, where Φ denotes the aspect ratio of X. Prior to our work, no efficient dynamic algorithm for maintaining a light-weight spanner was known even for point sets in low-dimensional Euclidean space.

Given a weighted digraph G, a (t,g,μ)-DAG cover is a collection of g dominating DAGs D_1,… ,D_g such that all distances are approximately preserved: for every pair (u,v) of vertices, min_id_{D_i}(u,v) ≤ t⋅ d_G(u,v), and the total number of non-G edges is bounded by |(∪_i D_i)⧵ G| ≤ μ. Assadi, Hoppenworth, and Wein [STOC 25] and Filtser [SODA 26] studied DAG covers for general digraphs. This paper initiates the study of Steiner DAG cover, where the DAGs are allowed to contain Steiner points. We obtain Steiner DAG covers on the important classes of planar digraphs and low-treewidth digraphs. Specifically, we show that any digraph with treewidth tw admits a (1,2,Õ(n⋅tw))-Steiner DAG cover. For planar digraphs we provide a (1+ε,2,Õ_ε(n))-Steiner DAG cover. We also demonstrate a stark difference between Steiner and non-Steiner DAG covers. As a lower bound, we show that any non-Steiner DAG cover for graphs with treewidth 1 with stretch t < 2 and sub-quadratic number of extra edges requires Ω(log n) DAGs.

Given an edge-weighted graph G and a subset of vertices T called terminals, an $\alpha$-distance-approximating minor ($\alpha$-DAM) of G is a graph minor H of G that contains all terminals, such that the distance between every pair of terminals is preserved up to a factor of $\alpha$. Distance-approximating minor would be an effective distance-sketching structure on minor-closed family of graphs; in the constant-stretch regime it generalizes the well-known Steiner Point Removal problem by allowing the existence of (a small number of) non-terminal vertices. Unfortunately, in the ($1+\varepsilon$) regime the only known DAM construction for planar graphs relies on overlaying $\tilde{O}_{\varepsilon}(|T|)$ shortest paths in G, which naturally leads to a quadratic bound in the number of terminals [Cheung, Goranci, and Henzinger, ICALP 2016]. We break the quadratic barrier and build the first ($1+\varepsilon$)-distance-approximating minor for k-terminal planar graphs and minor-free graphs of near-linear size $\tilde{O}_{\varepsilon}(k)$. In addition to the near-optimality in size, the construction relies only on the existence of shortest-path separators [Abraham and Gavoille, PODC 2006] and $\varepsilon$-covers [Thorup, J. ACM 2004]. Consequently, this provides an alternative and simpler construction to the near-linear-size emulator for planar graphs [Chang, Krauthgamer, and Tan, STOC 2022], as well as the first near-linear-size emulator for minor-free graphs. Our DAM can be constructed in near-linear time.

We introduce the aggregated clustering problem, where one is given T instances of a center-based clustering task over the same n points, but under different metrics. The goal is to open k centers to minimize an aggregate of the clustering costs - e.g., the average or maximum - where the cost is measured via k-center/median/means objectives. More generally, we minimize a norm Ψ over the T cost values. We show that for T ≥ 3, the problem is inapproximable to any finite factor in polynomial time. For T = 2, we give constant-factor approximations. We also show W[2]-hardness when parameterized by k, but obtain f(k,T)poly(n)-time 3-approximations when parameterized by both k and T. When the metrics have structure, we obtain efficient parameterized approximation schemes (EPAS). If all T metrics have bounded ε-scatter dimension, we achieve a (1+ε)-approximation in f(k,T,ε)poly(n) time. If the metrics are induced by edge weights on a common graph G of bounded treewidth tw, and Ψ is the sum function, we get an EPAS in f(T,ε,tw)poly(n,k) time. Conversely, unless (randomized) ETH is false, any finite factor approximation is impossible if parametrized by only T, even when the treewidth is tw = Ω(polylog n).

Cohen-Addad, Le, Pilipczuk, and Pilipczuk [CLPP23] recently constructed a stochastic embedding with expected 1 + ε distortion of n-vertex planar graphs (with polynomial aspect ratio) into graphs of treewidth O (ε-1 log13n ). Their embedding is the first to achieve polylogarithmic treewidth. However, there remains a large gap between the treewidth of their embedding and the treewidth lower bound of Ω(log n ) shown by Carroll and Goel [CG04]. In this work, we substantially narrow the gap by constructing a stochastic embedding with treewidth O (ε-1 log3 n ).

A $(1+\varepsilon)\textit{-stretch tree cover}$ of a metric space is a collection of trees, where every pair of points has a $(1+\varepsilon)$-stretch path in one of the trees. The celebrated $\textit{Dumbbell Theorem}$ [Arya et~al. STOC'95] states that any set of $n$ points in $d$-dimensional Euclidean space admits a $(1+\varepsilon)$-stretch tree cover with $O_d(\varepsilon^{-d} \cdot \log(1/\varepsilon))$ trees, where the $O_d$ notation suppresses terms that depend solely on the dimension~$d$. The running time of their construction is $O_d(n \log n \cdot \frac{\log(1/\varepsilon)}{\varepsilon^{d}} + n \cdot \varepsilon^{-2d})$. Since the same point may occur in multiple levels of the tree, the $\textit{maximum degree}$ of a point in the tree cover may be as large as $Ω(\log Φ)$, where $Φ$ is the aspect ratio of the input point set. In this work we present a $(1+\varepsilon)$-stretch tree cover with $O_d(\varepsilon^{-d+1} \cdot \log(1/\varepsilon))$ trees, which is optimal (up to the $\log(1/\varepsilon)$ factor). Moreover, the maximum degree of points in any tree is an $\textit{absolute constant}$ for any $d$. As a direct corollary, we obtain an optimal {routing scheme} in low-dimensional Euclidean spaces. We also present a $(1+\varepsilon)$-stretch $\textit{Steiner}$ tree cover (that may use Steiner points) with $O_d(\varepsilon^{(-d+1)/{2}} \cdot \log(1/\varepsilon))$ trees, which too is optimal. The running time of our two constructions is linear in the number of edges in the respective tree covers, ignoring an additive $O_d(n \log n)$ term; this improves over the running time underlying the Dumbbell Theorem.

The notion of shortcut partition, introduced recently by Chang, Conroy, Le, Milenković, Solomon, and Than [CCL+23], is a new type of graph partition into low-diameter clusters. Roughly speaking, the shortcut partition guarantees that for every two vertices u and v in the graph, there exists a path between u and v that intersects only a few clusters. They proved that any planar graph admits a shortcut partition and gave several applications, including a construction of tree cover for arbitrary planar graphs with stretch 1 + ɛ and O(1) many trees for any fixed ɛ ∈ (0,1). However, the construction heavily exploits planarity in multiple steps, and is thus inherently limited to planar graphs.

While research on the geometry of planar graphs has been active in the past decades, many properties of planar metrics remain mysterious. This paper studies a fundamental aspect of the planar graph geometry: covering planar metrics by a small collection of simpler metrics. Specifically, a tree cover of a metric space $(X, \delta)$ is a collection of trees, so that every pair of points u and v in X has a low-distortion path in at least one of the trees.The celebrated “Dumbbell Theorem” [ADM+95] states that any low-dimensional Euclidean space admits a tree cover with $O(1)$ trees and distortion $1+\varepsilon$, for any fixed $\varepsilon \in(0,1)$. This result has found numerous algorithmic applications, and has been generalized to the wider family of doubling metrics [BFN19]. Does the same result hold for planar metrics? A positive answer would add another evidence to the well-observed connection between Euclidean/doubling metrics and planar metrics.In this work, we answer this fundamental question affirmatively. Specifically, we show that for any given fixed $\varepsilon \in(0,1)$, any planar metric can be covered by $O(1)$ trees with distortion $1+\varepsilon$. Our result for planar metrics follows from a rather general framework: First we reduce the problem to constructing tree covers with additive distortion. Then we introduce the notion of shortcut partition, and draw connection between shortcut partition and additive tree cover. Finally we prove the existence of shortcut partition for any planar metric, using new insights regarding the grid-like structure of planar graphs. To demonstrate the power of our framework:•We establish additional tree cover results beyond planar metrics; in particular, we present an $O(1)$-size tree cover with distortion $1+\varepsilon$ for bounded treewidth metrics;•We obtain several algorithmic applications in planar graphs from our tree cover. The grid-like structure is a technical contribution that we believe is of independent interest. We showcase its applicability beyond tree cover by constructing a simpler and better embedding of planar graphs into $O(1)$-treewidth graphs with small additive distortion, resolving an open problem in this line of research.

Regular irradiation of indoor environments with ultraviolet C (UVC) light has become a regular task for many in-door settings as a result of COVID-19, but current robotic systems attempting to automate it suffer from high costs and inefficient irradiation. In this paper, we propose a purpose-made inexpensive robotic platform with off-the-shelf components and standard navigation software that, with a novel algorithm for finding optimal irradiation locations, addresses both shortcomings to offer affordable and efficient solutions for UVC irradiation. We demonstrate in simulations the efficacy of the algorithm and show a prototypical run of the autonomous integrated robotic system in an indoor environment. In our sample instances, our proposed algorithm reduces the time needed by roughly 30% while it increases the coverage by a factor of 35% (when compared to the best possible placement of a static light).