Sándor P. Fekete

Oswin Aichholzer, Joseph Dorfer, Sándor P. Fekete et al.

We give an overview of the 2026 Computational Geometry Challenge targeting the problem of finding a Central Triangulation under Parallel Flip Operations in triangulations of point sets. A flip is the parallel exchange of a set of edges in a triangulation with opposing diagonals of the convex quadrilaterals containing them. The challenge objective was, given a set of triangulations of a fixed point set, to determine a central triangulation with respect to parallel flip distances. More precisely, this asks for a triangulation that minimizes the sum of flip distances to all elements of the input

Sándor P. Fekete, Phillip Keldenich, Dominik Krupke et al.

We give an overview of the 2021 Computational Geometry Challenge, which targeted the problem of optimally coordinating a set of robots by computing a family of collision-free trajectories for a set set S of n pixel-shaped objects from a given start configuration into a desired target configuration.

Erik D. Demaine, Sándor P. Fekete, Phillip Keldenich et al.

We present a number of breakthroughs for coordinated motion planning, in which the objective is to reconfigure a swarm of labeled convex objects by a combination of parallel, continuous, collision-free translations into a given target arrangement. Problems of this type can be traced back to the classic work of Schwartz and Sharir (1983), who gave a method for deciding the existence of a coordinated motion for a set of disks between obstacles; their approach is polynomial in the complexity of the obstacles, but exponential in the number of disks. Other previous work has largely focused on {\em sequential} schedules, in which one robot moves at a time. We provide constant-factor approximation algorithms for minimizing the execution time of a coordinated, {\em parallel} motion plan for a swarm of robots in the absence of obstacles, provided some amount of separability. Our algorithm achieves {\em constant stretch factor}: If all robots are at most $d$ units from their respective starting positions, the total duration of the overall schedule is $O(d)$. Extensions include unlabeled robots and different classes of robots. We also prove that finding a plan with minimal execution time is NP-hard, even for a grid arrangement without any stationary obstacles. On the other hand, we show that for densely packed disks that cannot be well separated, a stretch factor $Ω(N^{1/4})$ may be required. On the positive side, we establish a stretch factor of $O(N^{1/2})$ even in this case.

Aaron T. Becker, Erik D. Demaine, Sándor P. Fekete et al.

We investigate algorithmic control of a large swarm of mobile particles (such as robots, sensors, or building material) that move in a 2D workspace using a global input signal (such as gravity or a magnetic field). We show that a maze of obstacles to the environment can be used to create complex systems. We provide a wide range of results for a wide range of questions. These can be subdivided into external algorithmic problems, in which particle configurations serve as input for computations that are performed elsewhere, and internal logic problems, in which the particle configurations themselves are used for carrying out computations. For external algorithms, we give both negative and positive results. If we are given a set of stationary obstacles, we prove that it is NP-hard to decide whether a given initial configuration of unit-sized particles can be transformed into a desired target configuration. Moreover, we show that finding a control sequence of minimum length is PSPACE-complete. We also work on the inverse problem, providing constructive algorithms to design workspaces that efficiently implement arbitrary permutations between different configurations. For internal logic, we investigate how arbitrary computations can be implemented. We demonstrate how to encode dual-rail logic to build a universal logic gate that concurrently evaluates and, nand, nor, and or operations. Using many of these gates and appropriate interconnects, we can evaluate any logical expression. However, we establish that simulating the full range of complex interactions present in arbitrary digital circuits encounters a fundamental difficulty: a fan-out gate cannot be generated. We resolve this missing component with the help of 2x1 particles, which can create fan-out gates that produce multiple copies of the inputs. Using these gates we provide rules for replicating arbitrary digital circuits.

Arun V. Mahadev, Dominik Krupke, Jan-Marc Reinhardt et al.

This paper investigates efficient techniques to collect and concentrate an under-actuated particle swarm despite obstacles. Concentrating a swarm of particles is of critical importance in health-care for targeted drug delivery, where micro-scale particles must be steered to a goal location. Individual particles must be small in order to navigate through micro-vasculature, but decreasing size brings new challenges. Individual particles are too small to contain on-board power or computation and are instead controlled by a global input, such as an applied fluidic flow or electric field. To make progress, this paper considers a swarm of robots initialized in a grid world in which each position is either free-space or obstacle. This paper provides algorithms that collect all the robots to one position and compares these algorithms on the basis of efficiency and implementation time.

Dominik Krupke, Michael Hemmer, James McLurkin et al.

We consider the problem of organizing a scattered group of $n$ robots in two-dimensional space, with geometric maximum distance $D$ between robots. The communication graph of the swarm is connected, but there is no central authority for organizing it. We want to arrange them into a sorted and equally-spaced array between the robots with lowest and highest label, while maintaining a connected communication network. In this paper, we describe a distributed method to accomplish these goals, without using central control, while also keeping time, travel distance and communication cost at a minimum. We proceed in a number of stages (leader election, initial path construction, subtree contraction, geometric straightening, and distributed sorting), none of which requires a central authority, but still accomplishes best possible parallelization. The overall arraying is performed in $O(n)$ time, $O(n^2)$ individual messages, and $O(nD)$ travel distance. Implementation of the sorting and navigation use communication messages of fixed size, and are a practical solution for large populations of low-cost robots.

Dominik Krupke, Maximilian Ernestus, Michael Hemmer et al.

We present a number of powerful local mechanisms for maintaining a dynamic swarm of robots with limited capabilities and information, in the presence of external forces and permanent node failures. We propose a set of local continuous algorithms that together produce a generalization of a Euclidean Steiner tree. At any stage, the resulting overall shape achieves a good compromise between local thickness, global connectivity, and flexibility to further continuous motion of the terminals. The resulting swarm behavior scales well, is robust against node failures, and performs close to the best known approximation bound for a corresponding centralized static optimization problem.

Daniela Maftuleac, Seoung Kyou Lee, Sandor P. Fekete et al.

We present and analyze methods for patrolling an environment with a distributed swarm of robots. Our approach uses a physical data structure - a distributed triangulation of the workspace. A large number of stationary "mapping" robots cover and triangulate the environment and a smaller number of mobile "patrolling" robots move amongst them. The focus of this work is to develop, analyze, implement and compare local patrolling policies. We desire strategies that achieve full coverage, but also produce good coverage frequency and visitation times. Policies that provide theoretical guarantees for these quantities have received some attention, but gaps have remained. We present: 1) A summary of how to achieve coverage by building a triangulation of the workspace, and the ensuing properties. 2) A description of simple local policies (LRV, for Least Recently Visited and LFV, for Least Frequently Visited) for achieving coverage by the patrolling robots. 3) New analytical arguments why different versions of LRV may require worst case exponential time between visits of triangles. 4) Analytical evidence that a local implementation of LFV on the edges of the dual graph is possible in our scenario, and immensely better in the worst case. 5) Experimental and simulation validation for the practical usefulness of these policies, showing that even a small number of weak robots with weak local information can greatly outperform a single, powerful robots with full information and computational capabilities.

Aaron Becker, Erik D. Demaine, Sándor P. Fekete et al.

Micro- and nanorobots are often controlled by global input signals, such as an electromagnetic or gravitational field. These fields move each robot maximally until it hits a stationary obstacle or another stationary robot. This paper investigates 2D motion-planning complexity for large swarms of simple mobile robots (such as bacteria, sensors, or smart building material). In previous work we proved it is NP-hard to decide whether a given initial configuration can be transformed into a desired target configuration; in this paper we prove a stronger result: the problem of finding an optimal control sequence is PSPACE-complete. On the positive side, we show we can build useful systems by designing obstacles. We present a reconfigurable hardware platform and demonstrate how to form arbitrary permutations and build a compact absolute encoder. We then take the same platform and use dual-rail logic to build a universal logic gate that concurrently evaluates AND, NAND, NOR and OR operations. Using many of these gates and appropriate interconnects we can evaluate any logical expression.

SeoungKyou Lee, Aaron Becker, Sándor P. Fekete et al.

This paper presents a distributed approach for exploring and triangulating an unknown region using a multi- robot system. The objective is to produce a covering of an unknown workspace by a fixed number of robots such that the covered region is maximized, solving the Maximum Area Triangulation Problem (MATP). The resulting triangulation is a physical data structure that is a compact representation of the workspace; it contains distributed knowledge of each triangle, adjacent triangles, and the dual graph of the workspace. Algorithms can store information in this physical data structure, such as a routing table for robot navigation Our algorithm builds a triangulation in a closed environment, starting from a single location. It provides coverage with a breadth-first search pattern and completeness guarantees. We show the computational and communication requirements to build and maintain the triangulation and its dual graph are small. Finally, we present a physical navigation algorithm that uses the dual graph, and show that the resulting path lengths are within a constant factor of the shortest-path Euclidean distance. We validate our theoretical results with experiments on triangulating a region with a system of low-cost robots. Analysis of the resulting quality of the triangulation shows that most of the triangles are of high quality, and cover a large area. Implementation of the triangulation, dual graph, and navigation all use communication messages of fixed size, and are a practical solution for large populations of low-cost robots.