Publications (203)54.44 Total impact
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ABSTRACT: We introduce a new model of algorithmic tile selfassembly called sizedependent assembly. In previous models, supertiles are stable when the total strength of the bonds between any two halves exceeds some constant temperature. In this model, this constant temperature requirement is replaced by an nondecreasing temperature function $\tau : \mathbb{N} \rightarrow \mathbb{N}$ that depends on the size of the smaller of the two halves. This generalization allows supertiles to become unstable and break apart, and captures the increased forces that large structures may place on the bonds holding them together. We demonstrate the power of this model in two ways. First, we give fixed tile sets that assemble constantheight rectangles and squares of arbitrary input size given an appropriate temperature function. Second, we prove that deciding whether a supertile is stable is coNPcomplete. Both results contrast with known results for fixed temperature.  [Show abstract] [Hide abstract]
ABSTRACT: We consider staged selfassembly systems, in which squareshaped tiles can be added to bins in several stages. Within these bins, the tiles may connect to each other, depending on the glue types of their edges. Previous work by Demaine et al. showed that a relatively small number of tile types suffices to produce arbitrary shapes in this model. However, these constructions were only based on a spanning tree of the geometric shape, so they did not produce full connectivity of the underlying grid graph in the case of shapes with holes; designing fully connected assemblies with a polylogarithmic number of stages was left as a major open problem. We resolve this challenge by presenting new systems for staged assembly that produce fully connected polyominoes in O(log^2 n) stages, for various scale factors and temperature {\tau} = 2 as well as {\tau} = 1. Our constructions work even for shapes with holes and uses only a constant number of glues and tiles. Moreover, the underlying approach is more geometric in nature, implying that it promised to be more feasible for shapes with compact geometric description.  [Show abstract] [Hide abstract]
ABSTRACT: 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.  [Show abstract] [Hide abstract]
ABSTRACT: We consider the problem of organizing a scattered group of $n$ robots in twodimensional 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 equallyspaced 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 lowcost robots. 
Conference Paper: Computing MaxMin Edge Length Triangulations
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ABSTRACT: In 1991, Edelsbrunner and Tan gave an O(n^2) algorithm for finding the MinMax Length triangulation of a set of points in the plane, but stated the complexity of finding a MaxMin Edge Length Triangulation (MELT) as a natural open problem. We resolve this longstanding problem by showing that computing a MELT is NPcomplete. Moreover, we prove that (unless P=NP), there is no polynomialtime approximation algorithm that can approximate MELT within any polynomial factor. While this may be taken as conclusive evidence from a theoretical point of view that the problem is hopelessly intractable, it still makes sense to consider powerful optimization methods, such as integer programming (IP), in order to obtain provably optimal solutions for intances of nontrivial size. A straightforward IP based on pairwise disjointness of the Θ(n^2) segments between the n points has Θ(n^4) constraints, making this IP hopelessly intractable from a practical point of view, even for relatively small n. The main algorithm engineering twist of this paper is to demonstrate how the combination of geometric insights with refined methods of combinatorial optimization can still help to put together an exact method capable of computing optimal MELT solutions for planar point sets up to n = 200. Our key idea is to exploit specific geometric properties in combination with more compact IP formulations, such that we are able to drastically reduce the IPs. On the practical side, we combine two of the most powerful software packages for the individual components: CGAL for carrying out the geometric computations, and CPLEX for solving the IPs. In addition, we discuss specific analytic aspects of the speedup for random point sets.  [Show abstract] [Hide abstract]
ABSTRACT: 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.  [Show abstract] [Hide abstract]
ABSTRACT: In this paper we explore the power of geometry to overcome the limitations of noncooperative selfassembly. We define a generalization of the abstract Tile Assembly Model (aTAM), such that a tile system consists of a collection of polyomino tiles, the Polyomino Tile Assembly Model (polyTAM), and investigate the computational powers of polyTAM systems at temperature 1, where attachment among tiles occurs without glue cooperation. Systems composed of the unitsquare tiles of the aTAM at temperature 1 are believed to be incapable of Turing universal computation (while cooperative systems, with temperature > 1, are able). As our main result, we prove that for any polyomino $P$ of size 3 or greater, there exists a temperature1 polyTAM system containing only shape$P$ tiles that is computationally universal. Our proof leverages the geometric properties of these larger (relative to the aTAM) tiles and their abilities to effectively utilize geometric blocking of particular growth paths of assemblies, while allowing others to complete. To round out our main result, we provide strong evidence that size1 (i.e. aTAM tiles) and size2 polyomino systems are unlikely to be computationally universal by showing that such systems are incapable of geometric bitreading, which is a technique common to all currently known temperature1 computationally universal systems. We further show that larger polyominoes with a limited number of binding positions are unlikely to be computationally universal, as they are only as powerful as temperature1 aTAM systems. Finally, we connect our work with other work on domino selfassembly to show that temperature1 assembly with at least 2 distinct shapes, regardless of the shapes or their sizes, allows for universal computation. 
Conference Paper: On the Chromatic Art Gallery Problem,

Article: One Tile to Rule Them All: Simulating Any Tile Assembly System with a Single Universal Tile
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ABSTRACT: In the classical model of tile selfassembly, unit square tiles translate in the plane and attach edgewise to form large crystalline structures. This model of selfassembly has been shown to be capable of asymptotically optimal assembly of arbitrary shapes and, via informationtheoretic arguments, increasingly complex shapes necessarily require increasing numbers of distinct types of tiles. We explore the possibility of complex and efficient assembly using systems consisting of a single tile. Our main result shows that any system of square tiles can be simulated using a system with a single tile that is permitted to flip and rotate. We also show that systems of single tiles restricted to translation only can simulate cellular automata for a limited number of steps given an appropriate seed assembly, and that any longerrunning simulation must induce infinite assembly.  [Show abstract] [Hide abstract]
ABSTRACT: We consider optimization techniques for a problem that requires a valid scheduling and allocation of tasks on Field Programmable Gate Arrays (FPGAs). A concrete application on a scientific space instrument arises in the context of ESA's Solar Orbiter mission; making use of dynamic reconfiguration allows a good and flexible use of resources, but the resulting packing and scheduling problems in the presence of inhomogeneous allocation resources are quite challenging. In our scenario, modules are described by three parameters: their resource demands for different types of resources, their priority, and their clock frequency. These are to be allocated on an FPGA that provides a number of different resources that are available at specific locations. We first present an Integer Program that partitions the tasks in such a way that all constraints can be met and the reconfiguration overhead is minimized, and then give methods for allocating the processing modules of the partitioned tasks on the FPGA. We evaluate our methods on a real application of the Solar Orbiter PHI instrument. The results obtained indicate computational efficiency and a remarkable solution quality. 
Article: Covering Folded Shapes
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ABSTRACT: Can folding a piece of paper flat make it larger? We explore whether a shape $S$ must be scaled to cover a flatfolded copy of itself. We consider both single folds and arbitrary folds (continuous piecewise isometries $S\rightarrow R^2$). The underlying problem is motivated by computational origami, and is related to other covering and fixturing problems, such as Lebesgue's universal cover problem and force closure grasps. In addition to considering special shapes (squares, equilateral triangles, polygons and disks), we give upper and lower bounds on scale factors for single folds of convex objects and arbitrary folds of simply connected objects. 
Article: Covering Folded Shapes
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ABSTRACT: Can folding a piece of paper flat make it larger? We explore whether a shape $S$ must be scaled to cover a flatfolded copy of itself. We consider both single folds and arbitrary folds (continuous piecewise isometries $S\rightarrow R^2$). The underlying problem is motivated by computational origami, and is related to other covering and fixturing problems, such as Lebesgue's universal cover problem and force closure grasps. In addition to considering special shapes (squares, equilateral triangles, polygons and disks), we give upper and lower bounds on scale factors for single folds of convex objects and arbitrary folds of simply connected objects. 
Article: Costoblivious storage reallocation
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ABSTRACT: Databases need to allocate and free blocks of storage on disk. Freed blocks introduce holes where no data is stored. Allocation systems attempt to reuse such deallocated regions in order to minimize the footprint on disk. If previously allocated blocks cannot be moved, the problem is called the memory allocation problem, which is known to have a logarithmic overhead in the footprint. This paper defines the storage reallocation problem, where previously allocated blocks can be moved, or reallocated, but at some cost. The algorithms presented here are cost oblivious, in that they work for a broad and reasonable class of cost functions, even when they do not know what the cost function is. The objective is to minimize the storage footprint, that is, the largest memory address containing an allocated object, while simultaneously minimizing the reallocation costs. This paper gives asymptotically optimal algorithms for storage reallocation, in which the storage footprint is at most (1+epsilon) times optimal, and the reallocation cost is at most (1/epsilon) times the original allocation cost, which is also optimal. The algorithms are cost oblivious as long as the allocation/reallocation cost function is subadditive. 
Article: Online Square Packing with Gravity
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ABSTRACT: We analyze the problem of packing squares in an online fashion: Given a semiinfinite strip of width 1 and an unknown sequence of squares of side length in [0,1] that arrive from above, one at a time. The objective is to pack these items as they arrive, minimizing the resulting height. Just like in the classical game of Tetris, each square must be moved along a collisionfree path to its final destination. In addition, we account for gravity in both motion (squares must never move up) and position (any final destination must be supported from below). A similar problem has been considered before; the best previous result is by Azar and Epstein, who gave a 4competitive algorithm in a setting without gravity (i.e., with the possibility of letting squares “hang in the air”) based on ideas of shelf packing: Squares are assigned to different horizontal levels, allowing an analysis that is reminiscent of some binpacking arguments. We apply a geometric analysis to establish a competitive factor of 3.5 for the bottomleft heuristic and present a $\frac{34}{13} \approx 2.6154$ competitive algorithm. 
Article: Online SquareintoSquare Packing
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ABSTRACT: In 1967, Moon and Moser proved a tight bound on the critical density of squares in squares: any set of squares with a total area of at most 1/2 can be packed into a unit square, which is tight. The proof requires full knowledge of the set, as the algorithmic solution consists in sorting the objects by decreasing size, and packing them greedily into shelves. Since then, the online version of the problem has remained open; the best upper bound is still 1/2, while the currently best lower bound is 1/3, due to Han et al. (2008). In this paper, we present a new lower bound of 11/32, based on a dynamic shelf allocation scheme, which may be interesting in itself. We also give results for the closely related problem in which the size of the square container is not fixed, but must be dynamically increased in order to ac commodate online sequences of objects. For this variant, we establish an upper bound of 3/7 for the critical density, and a lower bound of 1/8. When aiming for accommodating an online sequence of squares, this corresponds to a 2.82... competitive method for minimizing the required container size, and a lower bound of 1.33 . . . for the achievable factor.  [Show abstract] [Hide abstract]
ABSTRACT: In the original Art Gallery Problem (AGP), one seeks the minimum number of guards required to cover a polygon $P$. We consider the Chromatic AGP (CAGP), where the guards are colored. As long as $P$ is completely covered, the number of guards does not matter, but guards with overlapping visibility regions must have different colors. This problem has applications in landmarkbased mobile robot navigation: Guards are landmarks, which have to be distinguishable (hence the colors), and are used to encode motion primitives, \eg, "move towards the red landmark". Let $\chi_G(P)$, the chromatic number of $P$, denote the minimum number of colors required to color any guard cover of $P$. We show that determining, whether $\chi_G(P) \leq k$ is \NPhard for all $k \geq 2$. Keeping the number of colors minimal is of great interest for robot navigation, because less types of landmarks lead to cheaper and more reliable recognition.  [Show abstract] [Hide abstract]
ABSTRACT: 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 motionplanning complexity for large swarms of simple mobile robots (such as bacteria, sensors, or smart building material). In previous work we proved it is NPhard 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 PSPACEcomplete. 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 dualrail 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.  [Show abstract] [Hide abstract]
ABSTRACT: 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 breadthfirst 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 shortestpath Euclidean distance. We validate our theoretical results with experiments on triangulating a region with a system of lowcost 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 lowcost robots. 
Conference Paper: Reconfiguring Massive Particle Swarms with Limited, Global Control
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ABSTRACT: In this paper we investigate 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. Upon activation of the field, each robot moves maximally in the same direction, until it hits a stationary obstacle or another stationary robot. In an open workspace this system model is of limited use because it has only two controllable degrees of freedomall robots receive the same inputs and move uniformly. We show that adding a maze of obstacles to the environment can make the system drastically more complex but also more useful. If we are given a fixed set of stationary obstacles, we prove that it is NPhard to decide whether a given initial configuration can be transformed into a desired target configuration. On the positive side, we provide constructive algorithms to design workspaces that efficiently implement arbitrary permutations between different configurations. 
Article: Facets for Art Gallery Problems
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ABSTRACT: We demonstrate how polyhedral methods of mathematical programming can be developed for and applied to computing optimal solutions for large instances of a classical geometric optimization problem with an uncountable number of constraints and variables. The Art Gallery Problem (AGP) asks for placing a minimum number of stationary guards in a polygonal region P, such that all points in P are guarded. The AGP is NPhard, even to approximate. Due to the infinite number of points to be guarded as well as possible guard positions, applying mathematical programming methods for computing provably optimal solutions is far from straightforward. In this paper, we use an iterative primaldual relaxation approach for solving AGP instances to optimality. At each stage, a pair of LP relaxations for a finite candidate subset of primal covering and dual packing constraints and variables is considered; these correspond to possible guard positions and points that are to be guarded. Of particular interest are additional cutting planes for eliminating fractional solutions. We identify two classes of facets, based on Edge Cover and Set Cover (SC) inequalities. Solving the separation problem for the latter is NPcomplete, but exploiting the underlying geometric structure of the AGP, we show that large subclasses of fractional SC solutions cannot occur for the AGP. This allows us to separate the relevant subset of facets in polynomial time. Finally, we characterize all facets for finite AGP relaxations with coefficients in {0, 1, 2}. We demonstrate the practical usefulness of our approach with improved solution quality and speed for a wide array of large benchmark instances.
Publication Stats
3k  Citations  
54.44  Total Impact Points  
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Institutions

19702014

Technische Universität Braunschweig
 • Department of Computer Science
 • Institut für Mathematische Optimierung
Brunswyck, Lower Saxony, Germany


2010

New York State Institute for Basic Research in Developmental Disabilities
New York, New York, United States


19702005

University of Cologne
 Mathematical Institute
Köln, North RhineWestphalia, Germany


19702004

Technische Universität Berlin
 Department of Mathematics
Berlín, Berlin, Germany


1993

Stony Brook University
스토니브룩, New York, United States
