geographic information system developers

Abstract: We study the fully-dynamic all-pairs minimum cut problem on planar graphs. Given an $n$-vertex undirected planar graph $G=(V,E)$ undergoing edge insertions and deletions, the goal is to efficiently process these updates while supporting $s$-$t$ minimum cut queries for any pair of vertices $s$ and $t$. We give a $(1+\varepsilon)$-approximate dynamic algorithm that supports edge updates and minimum cut queries in $n^{o(1)}$ time, for any $1/\text{poly}(\log n) < \varepsilon < 1$. For general graphs, previous results either are limited to partially dynamic settings or achieve only logarithmic-factor approximations. The main conceptual and technical contribution is to show that a suitable adaptation of the planar emulator of Chang, Krauthgamer, and Tan~[STOC'22] satisfies the stronger pattern emulator introduced by Chen and Tan~[ICALP'25]. This leads almost immediately to a nearly linear-time construction of pattern emulators, which, when combined with the recent fully dynamic $(1+\varepsilon)$-approximate all-pairs shortest paths algorithm of Filtser, Goranci, Patel, and Probst Gutenberg~[FOCS'24], yields our result.

Abstract: We study the geometric set cover problems of covering and partitioning a polygon into a minimum number of triangles. While the decision version of the covering problem is known to be APX-hard and $\exists\mathbb R$-complete (even for polygons without holes and vertices in general position), the computational complexity of partitioning a simple polygon into a minimum number of triangles is open. Our main result refutes a 40 year old conjecture by Asano, Asano, and Pinter stating that the triangle vertices of a minimum partition lie on intersection points of special lines, which are iteratively generated by the polygon vertices and the intersection points. In particular, we show that there exist polygons where all minimum triangle partitions need Steiner points with irrational coordinates. This indicates that the problem might be more complex than previously expected. Complementarily, we provide a simple constant factor approximation to compute a minimum triangle partition of a given polygon (possibly with holes). In the context of coverings, we devise the first constant factor approximation for computing a minimum triangle cover of a simple polygon. For general polygons, an $O(\log n)$-approximation follows by close inspection of previous work.

Abstract: Given a set $P$ of $n$ point sites in the plane, the $k$ nearest neighbors problem is to build a data structure that, for a query point $v$ and a query integer $k$, reports the $k$ nearest point sites of $v$ in $P$ efficiently. For the Euclidean metric, after four decades of efforts, an optimal data structure has been developed with $O(n)$ space, $O(\log n+k)$ query time and $O(n\log n)$ preprocessing time [Mat92b, Cha00, AC09, CT16]. In contrast, for a family of distance functions, the state-of-the-art data structure [AES99, KMR+20, Liu22] attains $O(n\log\log n)$ space, $O(\log n+k)$ query time and expected $O(n\log n)$ preprocessing time. An open question is the existence of an optimal data structure with $O(n)$ space and $O(\log n + k)$ query time for distance functions more general than the Euclidean metric. Such an optimal data structure also remains unknown for non-point sites in the Euclidean metric, even for disk sites or, more generally, additively weighted Euclidean distances. We develop an optimal $k$-nearest-neighbors data structure for additively weighted Euclidean distances with $O(n)$ space, $O(\log n + k)$ query time and expected $O(n\log n)$ preprocessing time. Since this data structure can be adjusted to disk sites, it serves as an initial result toward optimality for non-point sites. We also adapt our data structure to point sites under a family of convex metrics, including commonly used $L_p$ metrics and uniform orientation metrics. The data structure can be further extended to distance functions that satisfy certain axioms. Thus, we demonstrate the existence of optimal $k$-nearest-neighbors data structures for distance functions more general than point sites in the Euclidean metric. Our main technical advance is to generalize Matoušek's shallow partition theorem for hyperplanes [Mat92b] to these settings, while achieving a logarithmic crossing number.

Abstract: In this paper, we explore the $(1,2)$-center problem for polygonal curves under continuous Fréchet distance. The $(k,\ell)$-center problem, in general, is known to be NP-hard. Aronov, Filtser, Horton, Katz, and Sheikhan (WADS'19) gave a polynomial-time algorithm for the $(1,2)$-center of curves in the plane under the discrete Fréchet distance. To the best of our knowledge, the $(1,2)$-center under continuous Fréchet distance has not been studied yet. We present a polynomial time algorithm to solve the problem exactly under both $\mathbb{L}_2$ and $\mathbb{L}_\infty$ norm, running in $O\bigr((n^2r+nr^2)^{2+ε}\bigl)$ time for curves in the plane where $r$ is the number of input curves and $n$ is the maximum complexity of any curve. Further, for curves in any dimension $d$, the expected time to compute the center using the algorithm is $O\bigr((n^2r+nr^2)^{2(d-1)+ε}\bigl)$. We have also shown that an $(1+\widetildeε)-$factor approximation of $(1,2)$-center can be computed in $O(n^2r+nr^2+1/ε^s)$ time for any $ε>\widetildeε>0$ and some constant $s$ for curves in the plane. For curves in the plane, we have shown that, with the center restricted to be horizontal, we can compute the exact center in $O(n^2r+nr^2)$ time. An algorithm has been introduced to find a $3$-factor approximation of the $(1,2)$-center in time linear in the number of curves. A formulation was introduced by de Berg, Mehrabi, and Ophelders (CCCG'17) to measure Fréchet distance between a curve and a query segment under $\mathbb{L}_2$ norm for curves in the plane. We have shown the formulation is valid under both $\mathbb{L}_2$ and $\mathbb{L}_\infty$ norm for curves in $\mathbb{R}^d$.

Abstract: Global video geo-localization aims to infer the geographic location of a video worldwide, evaluating performance across four geographic hierarchies: city, state/province, country, and continent. Existing methods typically employ one-way uniform sampling to process video frames and train independent classifiers for each hierarchy, which leads to the loss of key geographic cues and prediction conflicts between hierarchies, especially for complex multi-shot edited videos. To address these limitations, we propose GeoGAT, which integrates bidirectional temporal sampling with graph attention networks (GATs). Specifically, GeoGAT extracts forward and offset-reversed frame sequences to construct complementary spatiotemporal features. These fused features are then fed into a predefined geographical hierarchy graph, where GATs perform structure-aware message passing, while a dual-constraint mechanism prunes predictions to eliminate cross-hierarchy conflicts. We construct GeoGAT10k, comprising 9,720 multi-shot edited videos from 166 cities worldwide, specifically to benchmark generalization ability on complex video structures. Experimental results on CityGuessr68k and GeoGAT10k demonstrate that GeoGAT eliminates hierarchical conflicts entirely and achieves state-of-the-art performance across all four geographic hierarchies. On CityGuessr68k, GeoGAT outperforms the strongest baseline, evaluated under both classification and retrieval protocols, by 2.6 percentage points at the city level. On the more challenging GeoGAT10k with multi-shot edited videos, the accuracy improvement exceeds 24 percentage points, validating strong generalization to complex real-world scenarios.

Abstract: When designing algorithms for geometric graphs, exploiting structural parameters can lead to significantly improved bounds. Two prominent parameters in this context are $c$-packedness and $λ$-low density, both of which locally restrict graph complexity. Parameterized algorithms based on these parameters have been developed for computing well-separated pair decompositions, balanced separators, as well as distance oracles. Nevertheless the practical applicability of algorithms parameterized by $c$ or $λ$ remains unclear. While $c$-packed and $λ$-low-density graphs have been proposed as realistic models for road networks, the actual parameter values of large real-world instances have so far remained unknown, and existing theoretical guarantees are partially too loose for practical usage. In this paper we first devise scalable implementations for the approximate computation of $c$ and the exact computation of $λ$. Our experiments on road networks with millions of edges reveals a significant gap between the two parameters. On the theoretical side we prove that $c\in O(λ\sqrt n)$ which complements the known result that $λ\in O(c)$. Furthermore we present improved parameterized algorithms for balanced separator computation that reduce the separator size in theory and practice. We also show how to compute a tree decomposition with a width linear in the respective parameterized balanced separator size in polynomial time. This structural result yields a variety of new algorithmic consequences. Among them is an exact distance oracle with query time $O(c)$ for $c$-packed graphs after polynomial-time preprocessing, which improves upon the previous $O(c\log n)$ bound. Our experiments show that the proposed techniques efficiently produce small balanced separators and enable the construction of concise exact distance oracles on large road networks.

Abstract: Geographic applications need every object inside a radius that satisfies a semantic threshold, yet embedding indexes return approximate top-ranked lists and may omit qualifying records silently. We present FRESH-GEORANGE, a semantic- spatial range design that separates source-watermark freshness from optional record age. Geographic cells and semantic mi- croblocks provide admissible pruning bounds; a graph proposes verification order but supplies no correctness evidence. Exact mode scans every nonprunable block and the delta overlay. Certified mode may stop early and reports a deterministic query- specific recall lower bound from verified answers and unresolved records. A reproducible CPU pilot uses 2,500 real OpenFlights airport records, a 2,000-record base, and 740 simulated insert, delete, and text-revision events; it evaluates 180 unique queries over five seeds. Exact mode achieved 100.00% set recall on every query. The 95-percent mode achieved 99.91% empirical mean recall with a 99.41% reported mean certificate and no observed bound violation. However, its 7.24 ms median latency was 5.85 times the 1.24 ms spatial-first exact baseline, and full-history delta replay became slower than rebuilding at larger batches. The prototype therefore validates the completeness mechanism, not performance superiority or production freshness. Submission- scale evaluation requires real map diffs, official recent baselines, and truly incremental versioned maintenance.

Abstract: Delaunay triangulations are a fundamental class of plane spanners, and determining their worst-case stretch factor has been a longstanding problem in computational geometry. We prove an upper bound of 1.65, improving the previous bound of 1.998 and reducing the gap to the known lower bound of 1.5932 by a factor of more than seven. The result holds for every planar Delaunay triangulation, including configurations with collinear or cocircular sites. Our main contribution is a Bellman formulation of the disk-chain bound underlying the proof. By comparing shortest-path length with additive progress along the query segment, we obtain an exact recursion whose state records only the current disk, the incoming chord, and the difference between two prefix distances. We show that a bound for this chain class holds if and only if a potential satisfies three local inequalities for initialization, transitions, and termination. The associated Bellman value function is the pointwise smallest feasible potential, giving a precise target for constructing an upper bound. We construct such a potential using a function of one variable. Geometric monotonicity reduces its feasibility to inequalities that are affine in this function and its derivative. A spline construction, certified by exact arithmetic and rigorous interval bounds, yields the stretch bound of 1.65. We also give a dual certificate showing that every feasible quadratic profile under the same conditions requires a certified constant greater than 1.67.