[Show abstract][Hide abstract] ABSTRACT: We introduce a new model of algorithmic tile self-assembly called
size-dependent 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 constant-height rectangles and squares of arbitrary input
size given an appropriate temperature function. Second, we prove that deciding
whether a supertile is stable is coNP-complete. Both results contrast with
known results for fixed temperature.
[Show abstract][Hide abstract] ABSTRACT: We consider staged self-assembly systems, in which square-shaped 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
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.
[Show abstract][Hide abstract] 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 long-standing problem by showing that computing a MELT is NP-complete. Moreover, we prove that (unless P=NP), there is no polynomial-time 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 non-trivial 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.
SIAM Meeting on Algorithm Engineering & Experiments (ALENEX); 01/2015
[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
non-cooperative self-assembly. 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
unit-square 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 temperature-1 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 size-1 (i.e.
aTAM tiles) and size-2 polyomino systems are unlikely to be computationally
universal by showing that such systems are incapable of geometric bit-reading,
which is a technique common to all currently known temperature-1
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 temperature-1 aTAM systems. Finally,
we connect our work with other work on domino self-assembly to show that
temperature-1 assembly with at least 2 distinct shapes, regardless of the
shapes or their sizes, allows for universal computation.
[Show abstract][Hide abstract] ABSTRACT: In the classical model of tile self-assembly, unit square tiles translate in the plane and attach edgewise to form large crystalline structures. This model of self-assembly has been shown to be capable of asymptotically optimal assembly of arbitrary shapes and, via information-theoretic 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 longer-running 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.
2014 NASA/ESA Conference on Adaptive Hardware and Systems (AHS); 07/2014
[Show abstract][Hide abstract] ABSTRACT: Can folding a piece of paper flat make it larger? We explore whether a shape
$S$ must be scaled to cover a flat-folded 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.
[Show abstract][Hide abstract] ABSTRACT: Can folding a piece of paper flat make it larger? We explore whether a shape $S$ must be scaled to cover a flat-folded 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.
[Show abstract][Hide abstract] 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.
[Show abstract][Hide abstract] ABSTRACT: We analyze the problem of packing squares in an online fashion: Given a semi-infinite 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 collision-free 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 4-competitive 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 bin-packing arguments. We apply a geometric analysis to establish a competitive factor of 3.5 for the bottom-left heuristic and present a
$\frac{34}{13} \approx 2.6154$
-competitive algorithm.
[Show abstract][Hide abstract] 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 landmark-based 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 \NP-hard 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 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.
Proceedings - IEEE International Conference on Robotics and Automation 02/2014; DOI:10.1109/ICRA.2014.6907856
[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 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.
[Show abstract][Hide abstract] 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 freedom---all 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 NP-hard 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.
International Symposium on Algorithms and Experiments for Sensor Systems, Wireless Networks and Distributed Robotics (ALGOSENSORS); 09/2013
[Show abstract][Hide abstract] ABSTRACT: 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
problem is known to be NP-hard, and its inherent continuous structure (with
both the set of points that need to be guarded and the set of points that can
be used for guarding being uncountably infinite) makes it difficult to apply a
straightforward formulation as an Integer Linear Program. We use an iterative
primal-dual 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.
Particularly useful are 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 NP-complete, but
exploiting the underlying geometric structure, 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. We also characterize all
facets for finite AGP relaxations with coefficients in {0, 1, 2}.
Finally, we demonstrate the practical usefulness of our approach with
improved solution quality and speed for a wide array of large benchmark
instances; as it turns out, our results yield a significant improvement.