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Joachim Gudmundsson

Publications and source records attributed to Joachim Gudmundsson.

At least 19 recordsLinked to original sources

Linear time single-source shortest path algorithms in Euclidean graph classes

In the celebrated paper of Henzinger, Klein, Rao and Subramanian (1997), it was shown that planar graphs admit a linear time single-source shortest path algorithm. Their algorithm unfortunately does not extend to Euclidean graph classes. We give criteria and prove that any Euclidean graph class satisfying the criteria admits a linear time single-source shortest path algorithm. As a main ingredient, we show that the contracted graphs of these Euclidean graph classes admit sublinear separators.

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Minimum Exposure Motion Planning

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].

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The Mutual Visibility Problem for Fat Robots with Lights

Given a set of $n\geq 1$ unit disk robots in the Euclidean plane, we consider the fundamental problem of providing mutual visibility to them: the robots must reposition themselves to reach a configuration where they all see each other. This problem arises under obstructed visibility, where a robot cannot see another robot if there is a third robot on the straight line segment between them. This problem was solved by Sharma et al. [ICDCN, 2018] in the luminous robots model, where each robot is equipped with an externally visible light that can assume colors from a fixed set of colors, using 9 colors and $O(n)$ rounds. In this work, we present an algorithm that requires only 2 colors and $O(n)$ rounds. The number of colors is optimal since at least two colors are required even for point robots [Di Luna et al., Information and Computation, 2017].

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Theoretical analysis of beaconless geocast protocols in 1D

Beaconless geocast protocols are routing protocols used to send messages in mobile ad-hoc wireless networks, in which the only information available to each node is its own location. Messages get routed in a distributed manner: each node uses local decision rules based on the message source and destination, and its own location. In this paper we analyze six different beaconless geocast protocols, focusing on two relevant 1D scenarios. The selection of protocols reflects the most relevant types of protocols proposed in the literature, including those evaluated in previous computer simulations. We present a formal and structured analysis of the maximum number of messages that a node can receive, for each protocol, in each of the two scenarios. This is a measure of the network load incurred by each protocol. Our analysis, that for some of the protocols requires an involved probabilistic analysis, confirms behaviors that had been observed only through simulations before.

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Oriented Spanners

Given a point set $P$ in the Euclidean plane and a parameter $t$, we define an \emph{oriented $t$-spanner} $G$ as an oriented subgraph of the complete bi-directed graph such that for every pair of points, the shortest closed walk in $G$ through those points is at most a factor $t$ longer than the shortest cycle in the complete graph on $P$. We investigate the problem of computing sparse graphs with small oriented dilation. As we can show that minimising oriented dilation for a given number of edges is NP-hard in the plane, we first consider one-dimensional point sets. While obtaining a $1$-spanner in this setting is straightforward, already for five points such a spanner has no plane embedding with the leftmost and rightmost point on the outer face. This leads to restricting to oriented graphs with a one-page book embedding on the one-dimensional point set. For this case we present a dynamic program to compute the graph of minimum oriented dilation that runs in $\mathcal{O}(n^7)$ time for $n$ points, and a greedy algorithm that computes a $5$-spanner in $\mathcal{O}(n\log n)$ time. Expanding these results finally gives us a result for two-dimensional point sets: we prove that for convex point sets the greedy triangulation results in a plane oriented $t$-spanner with $t=7.2 \cdot t_g$, where $t_g$ is an upper bound on the dilation of the greedy triangulation.

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A well-separated pair decomposition for low density graphs

Low density graphs are considered to be a realistic graph class for modelling road networks. It has advantages over other popular graph classes for road networks, such as planar graphs, bounded highway dimension graphs, and spanners. We believe that low density graphs have the potential to be a useful graph class for road networks, but until now, its usefulness is limited by a lack of available tools. In this paper, we develop two fundamental tools for low density graphs, that is, a well-separated pair decomposition and an approximate distance oracle. We believe that by expanding the algorithmic toolbox for low density graphs, we can help provide a useful and realistic graph class for road networks, which in turn, may help explain the many efficient and practical heuristics available for road networks.

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Shortest Paths of Mutually Visible Robots

Given a set of $n$ point robots inside a simple polygon $P$, the task is to move the robots from their starting positions to their target positions along their shortest paths, while the mutual visibility of these robots is preserved. Previous work only considered two robots. In this paper, we present an $O(mn)$ time algorithm, where $m$ is the complexity of the polygon, when all the starting positions lie on a line segment $S$, all the target positions lie on a line segment $T$, and $S$ and $T$ do not intersect. We also argue that there is no polynomial-time algorithm, whose running time depends only on $n$ and $m$, that uses a single strategy for the case where $S$ and $T$ intersect.

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Player-Team Heterogeneous Interaction Graph Transformer for Soccer Outcome Prediction

Predicting soccer match outcomes is a challenging task due to the inherently unpredictable nature of the game and the numerous dynamic factors influencing results. While it conventionally relies on meticulous feature engineering, deep learning techniques have recently shown a great promise in learning effective player and team representations directly for soccer outcome prediction. However, existing methods often overlook the heterogeneous nature of interactions among players and teams, which is crucial for accurately modeling match dynamics. To address this gap, we propose HIGFormer (Heterogeneous Interaction Graph Transformer), a novel graph-augmented transformer-based deep learning model for soccer outcome prediction. HIGFormer introduces a multi-level interaction framework that captures both fine-grained player dynamics and high-level team interactions. Specifically, it comprises (1) a Player Interaction Network, which encodes player performance through heterogeneous interaction graphs, combining local graph convolutions with a global graph-augmented transformer; (2) a Team Interaction Network, which constructs interaction graphs from a team-to-team perspective to model historical match relationships; and (3) a Match Comparison Transformer, which jointly analyzes both team and player-level information to predict match outcomes. Extensive experiments on the WyScout Open Access Dataset, a large-scale real-world soccer dataset, demonstrate that HIGFormer significantly outperforms existing methods in prediction accuracy. Furthermore, we provide valuable insights into leveraging our model for player performance evaluation, offering a new perspective on talent scouting and team strategy analysis.

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A WSPD, Separator and Small Tree Cover for c-packed Graphs

The $c$-packedness property, proposed in 2010, is a geometric property that captures the spatial distribution of a set of edges. Despite the recent interest in $c$-packedness, its utility has so far been limited to Fréchet distance problems. An open problem is whether a wider variety of algorithmic and data structure problems can be solved efficiently under the $c$-packedness assumption, and more specifically, on $c$-packed graphs. In this paper, we prove two fundamental properties of $c$-packed graphs: that there exists a linear-size well-separated pair decomposition under the graph metric, and there exists a constant size balanced separator. We then apply these fundamental properties to obtain a small tree cover for the metric space and distance oracles under the shortest path metric. In particular, we obtain a tree cover of constant size, an exact distance oracle of near-linear size and an approximate distance oracle of linear size.

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A Tail Estimate with Exponential Decay for the Randomized Incremental Construction of Search Structures

The Randomized Incremental Construction (RIC) of search DAGs for point location in planar subdivisions, nearest-neighbor search in 2D points, and extreme point search in 3D convex hulls, are well known to take ${\cal O}(n \log n)$ expected time for structures of ${\cal O}(n)$ expected size. Moreover, searching takes w.h.p. ${\cal O}(\log n)$ comparisons in the first and w.h.p. ${\cal O}(\log^2 n)$ comparisons in the latter two DAGs. However, the expected depth of the DAGs and high probability bounds for their size are unknown. Using a novel analysis technique, we show that the three DAGs have w.h.p. i) a size of ${\cal O}(n)$, ii) a depth of ${\cal O}(\log n)$, and iii) a construction time of ${\cal O}(n \log n)$. One application of these new and improved results are \emph{remarkably simple} Las Vegas verifiers to obtain search DAGs with optimal worst-case bounds. This positively answers the conjectured logarithmic search cost in the DAG of Delaunay triangulations [Guibas et al.; ICALP 1990] and a conjecture on the depth of the DAG of Trapezoidal subdivisions [Hemmer et al.; ESA 2012].

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Pattern Formation for Fat Robots with Memory

Given a set of $n\geq 1$ autonomous, anonymous, indistinguishable, silent, and possibly disoriented mobile unit disk (i.e., fat) robots operating following Look-Compute-Move cycles in the Euclidean plane, we consider the Pattern Formation problem: from arbitrary starting positions, the robots must reposition themselves to form a given target pattern. This problem arises under obstructed visibility, where a robot cannot see another robot if there is a third robot on the straight line segment between the two robots. We assume that a robot's movement cannot be interrupted by an adversary and that robots have a small $O(1)$-sized memory that they can use to store information, but that cannot be communicated to the other robots. To solve this problem, we present an algorithm that works in three steps. First it establishes mutual visibility, then it elects one robot to be the leader, and finally it forms the required pattern. The whole algorithm runs in $O(n) + O(q \log n)$ rounds with probability at least $1 - n^{-q}$. The algorithms are collision-free and do not require the knowledge of the number of robots.

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Pattern Formation for Fat Robots with Lights

Given a set of $n\geq 1$ unit disk robots in the Euclidean plane, we consider the Pattern Formation problem, i.e., the robots must reposition themselves to form a given target pattern. This problem arises under obstructed visibility, where a robot cannot see another robot if there is a third robot on the straight line segment between the two robots. Recently, this problem was solved in the asynchonous model for fat robots that agree on at least one axis in the robots with lights model where each robot is equipped with an externally visible persistent light that can assume colors from a fixed set of colors [K. Bose, R. Adhikary, M. K. Kundu, and B. Sau. Arbitrary pattern formation by opaque fat robots with lights. CALDAM, pages 347-359, 2020]. In this work, we reduce the number of colors needed and remove the axis-agreement requirement in the fully synchronous model. In particular, we present an algorithm requiring 7 colors when scaling the target pattern is allowed and an 8-color algorithm if scaling is not allowed. Our algorithms run in $O(n) + O(q \log n)$ rounds with probability at least $1 - n^{-q}$.

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Approximating the Fréchet distance when only one curve is $c$-packed

One approach to studying the Fréchet distance is to consider curves that satisfy realistic assumptions. By now, the most popular realistic assumption for curves is $c$-packedness. Existing algorithms for computing the Fréchet distance between $c$-packed curves require both curves to be $c$-packed. In this paper, we only require one of the two curves to be $c$-packed. Our result is a nearly-linear time algorithm that $(1+\varepsilon)$-approximates the Fréchet distance between a $c$-packed curve and a general curve in $\mathbb R^d$, for constant values of $\varepsilon$, $d$ and $c$.

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Map-Matching Queries under Fréchet Distance on Low-Density Spanners

Map matching is a common task when analysing GPS tracks, such as vehicle trajectories. The goal is to match a recorded noisy polygonal curve to a path on the map, usually represented as a geometric graph. The Fréchet distance is a commonly used metric for curves, making it a natural fit. The map-matching problem is well-studied, yet until recently no-one tackled the data structure question: preprocess a given graph so that one can query the minimum Fréchet distance between all graph paths and a polygonal curve. Recently, Gudmundsson, Seybold, and Wong [SODA 2023, arXiv:2211.02951] studied this problem for arbitrary query polygonal curves and $c$-packed graphs. In this paper, we instead require the graphs to be $λ$-low-density $t$-spanners, which is significantly more representative of real-world networks. We also show how to report a path that minimises the distance efficiently rather than only returning the minimal distance, which was stated as an open problem in their paper.

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Bicriteria approximation for minimum dilation graph augmentation

Spanner constructions focus on the initial design of the network. However, networks tend to improve over time. In this paper, we focus on the improvement step. Given a graph and a budget $k$, which $k$ edges do we add to the graph to minimise its dilation? Gudmundsson and Wong [TALG'22] provided the first positive result for this problem, but their approximation factor is linear in $k$. Our main result is a $(2 \sqrt[r]{2} \ k^{1/r},2r)$-bicriteria approximation that runs in $O(n^3 \log n)$ time, for all $r \geq 1$. In other words, if $t^*$ is the minimum dilation after adding any $k$ edges to a graph, then our algorithm adds $O(k^{1+1/r})$ edges to the graph to obtain a dilation of $2rt^*$. Moreover, our analysis of the algorithm is tight under the Erdős girth conjecture.

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Spanner for the $0/1/\infty$ weighted region problem

We consider the problem of computing an approximate weighted shortest path in a weighted subdivision, with weights assigned from the set $\{0, 1, \infty\}$. We present a data structure $B$, which stores a set of convex, non-overlapping regions. These include zero-cost regions (0-regions) with a weight of $0$ and obstacles with a weight of $\infty$, all embedded in a plane with a weight of $1$. The data structure $B$ can be constructed in expected time $O(N + (n/\varepsilon^3)(\log(n/\varepsilon) + \log N))$, where $n$ is the total number of regions, $N$ represents the total complexity of the regions, and $1 + \varepsilon$ is the approximation factor, for any $0 < \varepsilon < 1$. Using $B$, one can compute an approximate weighted shortest path from any point $s$ to any point $t$ in $O(N + n/\varepsilon^3 + (n/\varepsilon^2) \log(n/\varepsilon) + (\log N)/\varepsilon)$ time. In the special case where the 0-regions and obstacles are polygons (not necessarily convex), $B$ contains a $(1 + \varepsilon)$-spanner of the input vertices.

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Approximating Multiplicatively Weighted Voronoi Diagrams: Efficient Construction with Linear Size

Given a set of $n$ sites from $\mathbb{R}^d$, each having some positive weight factor, the Multiplicatively Weighted Voronoi Diagram is a subdivision of space that associates each cell to the site whose weighted Euclidean distance is minimal for all points in the cell. We give novel approximation algorithms that output a cube-based subdivision such that the weighted distance of a point with respect to the associated site is at most $(1+\varepsilon)$ times the minimum weighted distance, for any fixed parameter $\varepsilon \in (0,1)$. The diagram size is $O_d(n \log(1/\varepsilon)/\varepsilon^{d-1})$ and the construction time is within an $O_D(\log(n)/\varepsilon^{(d+5)/2})$-factor of the size bound. We also prove a matching lower bound for the size, showing that the proposed method is the first to achieve \emph{optimal size}, up to $Θ(1)^d$-factors. In particular, the obscure $\log(1/\varepsilon)$ factor is unavoidable. As a by-product, we obtain a factor $d^{O(d)}$ improvement in size for the unweighted case and $O(d \log(n) + d^2 \log(1/\varepsilon))$ point-location time in the subdivision, improving the known query bound by one $d$-factor. The key ingredients of our approximation algorithms are the study of convex regions that we call cores, an adaptive refinement algorithm to obtain optimal size, and a novel notion of \emph{bisector coresets}, which may be of independent interest. In particular, we show that coresets with $O_d(1/\varepsilon^{(d+3)/2})$ worst-case size can be computed in near-linear time.

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Map matching queries on realistic input graphs under the Fréchet distance

Map matching is a common preprocessing step for analysing vehicle trajectories. In the theory community, the most popular approach for map matching is to compute a path on the road network that is the most spatially similar to the trajectory, where spatial similarity is measured using the Fréchet distance. A shortcoming of existing map matching algorithms under the Fréchet distance is that every time a trajectory is matched, the entire road network needs to be reprocessed from scratch. An open problem is whether one can preprocess the road network into a data structure, so that map matching queries can be answered in sublinear time. In this paper, we investigate map matching queries under the Fréchet distance. We provide a negative result for geometric planar graphs. We show that, unless SETH fails, there is no data structure that can be constructed in polynomial time that answers map matching queries in $O((pq)^{1-δ})$ query time for any $δ> 0$, where $p$ and $q$ are the complexities of the geometric planar graph and the query trajectory, respectively. We provide a positive result for realistic input graphs, which we regard as the main result of this paper. We show that for $c$-packed graphs, one can construct a data structure of $\tilde O(cp)$ size that can answer $(1+\varepsilon)$-approximate map matching queries in $\tilde O(c^4 q \log^4 p)$ time, where $\tilde O(\cdot)$ hides lower-order factors and dependence on $\varepsilon$.

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