608653
Distributed localization using noisy distance and angle information
Localization is an important and extensively studied problem in ad-hoc wireless sensor networks. Given the connectivity graph of the sensor nodes,along with additional local information (e.g. distances, angles, orientations etc.), the goal is to reconstruct the global geometry of the network. In this paper, we study the problem of localization with noisy distance and angle information. With no noise at all, the localization problem with both angle (with orientation) and distance information is trivial. However, in the presence of even a small amount of noise, we prove that the localization problem is NP hard.Localization with accurate distance information and relative angle information is also hard. These hardness results motivate our study of approximation schemes. We relax the non-convex constraints to approximating convex constraints and propose linear programs (LP) for two formulations of the resulting localization problem, which we call the weak deployment and strong deployment problems.These two formulations give upper and lower bounds on the location uncertainty respectively: No sensor is located outside its weak deployment region, and each sensor can be anywhere in its strong deployment region without violating the approximate distance and angle constraints. Though LP-based algorithms are usually solved by centralized methods, we propose distributed, iterative methods, which are provably convergent to the centralized algorithm solutions. We give simulation results for the distributed algorithms, evaluating the convergence rate, dependence on measurement noises,and robustness to link dynamics.
11557966
An algorithmic study of manufacturing paperclips and other folded structures
We study algorithmic aspects of bending wires and sheet metal into a specified structure. Problems of this type are closely related to the question of deciding whether a simple non-self-intersecting wire structure (a carpenter's ruler) can be straightened, a problem that was open for several years and has only recently been solved in the affirmative. If we impose some of the constraints that are imposed by the manufacturing process, we obtain quite different results. In particular, we study the variant of the carpenter's ruler problem in which there is a restriction that only one joint can be modified at a time. For a linkage that does not self-intersect or self-touch, the recent results of Connelly et al. and Streinu imply that it can always be straightened, modifying one joint at a time. However, we show that for a linkage with even a single vertex degeneracy, it becomes NP-hard to decide if it can be straightened while altering only one joint at a time. If we add the restriction that each joint can be altered at most once, we show that the problem is NP-complete even without vertex degeneracies. In the special case, arising in wire forming manufacturing, that each joint can be altered at most once, and must be done sequentially from one or both ends of the linkage, we give an efficient algorithm to determine if a linkage can be straightened.
16007270
Point Probe Decision Trees for Geometric Concept Classes
A fundamental problem in model-based computer vision is that of identifying to which of a given set of concept classes of geometric models an observed model belongs. Considering a “probe” to be an oracle that tells whether or not the observed model is present at a given point in an image, we study the problem of computing efficient strategies (“decision trees”) for probing an image, with the goal to minimize the number of probes necessary (in the worst case) to determine in which class the observed model belongs. We prove a hardness result and give strategies that obtain decision trees whose height is within a log factor of optimal.
207132060
Terrain Decomposition and Layered Manufacturing
We consider a problem that arises in generating three-dimensional models by methods of layered manufacturing: How does one decompose a given model P into a small number of sub-models each of which is a terrain polyhedron? Terrain polyhedra have a base facet such that, for each point of the polyhedron, the line segment joining the point to its orthogonal projection on the base facet lies within the polyhedron. Terrain polyhedra are exactly the class of polyhedral models for which it is possible to construct the model using layered manufacturing (with layers parallel to the base facet), without the need for constructing "supports" (which must later be removed). In order to maximize the integrity of a prototype, one wants to minimize the number of individual sub-models that are manufactured and then glued together. We show that it is NP-hard to decide if a three-dimensional model P of genus 0 can be decomposed into k terrain polyhedra. We also prove a two-dimensional version of this theorem, for the case in which P is a polygonal region with holes. Both results still hold if we are restricted to isothetic objects and/or axis-parallel layering directions.
57900050
Computational Geometry Column 42
A compendium of thirty previously published open problems in computational geometry is presented.
7753820
Approximate Geometric Pattern Matching Under Rigid Motions
We present techniques for matching point-sets in two and three dimensions under rigid-body transformations. We prove bounds on the worst-case performance of these algorithms to be within a small constant factor of optimal and conduct experiments to show that the average performance of these matching algorithms is often better than that predicted by the worst-case bounds.
10533033
On the Sequential Nature of Unification
Abstract The problem of unification of terms is log-space complete for P. In deriving this lower bound no use is made of the potentially concise representation of terms by directed acyclic graphs. In addition, the problem remains complete even if infinite substitutions are allowed. A consequence of this result is that parallelism cannot significantly improve on the best sequential solutions for unification. However, we show that for the problem of term matching, an important subcase of unification, there is a good parallel algorithm using O ( log 2 n ) time and n O(1) processors on a PRAM. For the O ( log 2 n ) parallel time upper bound we assume that the terms are represented by directed acyclic graphs; if the longer string representation is used we obtain an O ( log n ) parallel time bound.
2523975
Multiset rewriting and the complexity of bounded security protocols
We formalize the Dolev-Yao model of security protocols, using a notation based on multiset rewriting with existentials. The goals are to provide a simple formal notation for describing security protocols, to formalize the assumptions of the Dolev-Yao model using this notation, and to analyze the complexity of the secrecy problem under various restrictions. We prove that, even for the case where we restrict the size of messages and the depth of message encryption, the secrecy problem is undecidable for the case of an unrestricted number of protocol roles and an unbounded number of new nonces. We also identify several decidable classes, including a DEXP-complete class when the number of nonces is restricted, and an NP-complete class when both the number of nonces and the number of roles is restricted. We point out a remaining open complexity problem, and discuss the implications these results have on the general topic of protocol analysis.
800932
A probabilistic poly-time framework for protocol analysis
We develop a framework for analyzing security protocols in which protocol adversaries may be arbitrary probabilistic polynomial-time processes. In this framework, protocols are written in a restricted form of pi-calculus and security may expressed as a form or observational equivalence, a standard relation from programming language theory that involves quantifying over possible environments that might interact with the protocol. Using an asymptotic notion of probabilistic equivalence, we relate observational equivalence to polynomial-time statistical tests and discuss some example protocols to illustrate the potential strengths of our approach.
10944537
The Complexity of Local Proof Search in Linear Logic
Abstract Proof search in linear logic is known to be difficult: the provability of propositional linear logic formulas is undecidable. Even without the modalities, multiplicative-additive fragment of propositional linear logic, mall , is known to be PSPACE-complete, and the pure multiplicative fragment, mll , is known to be np -complete. However, this still leaves open the possibility that there might be proof search heuristics (perhaps involving randomization) that often lead to a proof if there is one, or always lead to something close to a proof. One approach to these problems is to study strategies for proof games. A class of linear logic proof games is developed, each with a numeric score that depends on the number of certain preferred axioms used in a complete or partial proof tree. Using recent techniques for proving lower bounds on optimization problems, the complexity of these games is analyzed for the fragment mll extended with additive constants and for the fragment MALL. It is shown that no efficient heuristics exist unless there is an unexpected collapse in the complexity hierarchy.
452980
On Abstraction and the Expressive Power of Programming Languages
We present a tentative theory of programming language expressiveness based on reductions (language translations) that preserve observational equivalence. These are called “abstraction-preserving” because of a connection with a definition of “abstraction” or “information-hiding” mechanism. If there is an abstraction-preserving reduction from one language to another, then essentially every function on natural numbers that is definable in the first is also definable in the second. Moreover, regardless of the set of first-order functions definable in either language, no programming language with an abstraction mechanism can be reduced to a language without. Since Lisp with user-defined special forms does not have an abstraction mechanism, it is therefore not “universal” in this theory, in spite of the ability to define every partial recursive function on the natural numbers. Several examples and counter-examples to abstraction-preserving reductions are given. We do not know whether there is a natural universal language with respect to abstraction-preserving reduction.