207159283
Minimum-cost coverage of point sets by disks
We consider a class of geometric facility location problems in which the goal is to determine a set <i>X</i> of disks given by their centers <i>(t<sub>j</sub>)</i> and radii <i>(r<sub>j</sub>)</i> that cover a given set of demand points <i>Y∈R</i><sup>2</sup> at the smallest possible cost. We consider cost functions of the form Ε<i><sub>j</sub>f(r<sub>j</sub>)</i>, where <i>f(r)=r</i><sup>α</sup> is the cost of transmission to radius <i>r</i>. Special cases arise for α=1 (sum of radii) and α=2 (total area); power consumption models in wireless network design often use an exponent α>2. Different scenarios arise according to possible restrictions on the transmission centers <i>t<sub>j</sub></i>, which may be constrained to belong to a given discrete set or to lie on a line, etc.We obtain several new results, including (a) exact and approximation algorithms for selecting transmission points <i>t<sub>j</sub></i> on a given line in order to cover demand points <i>Y∈R</i><sup>2</sup>; (b) approximation algorithms (and an algebraic intractability result) for selecting an optimal line on which to place transmission points to cover <i>Y</i>; (c) a proof of NP-hardness for a discrete set of transmission points in <i>R<sup>2</sup></i> and any fixed α>1; and (d) a polynomial-time approximation scheme for the problem of computing a <i>minimum cost covering tour</i> (MCCT), in which the total cost is a linear combination of the transmission cost for the set of disks and the length of a tour/path that connects the centers of the disks.
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.
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.
57900050
Computational Geometry Column 42
A compendium of thirty previously published open problems in computational geometry is presented.
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.
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.
18575731
Classifying the World Anti-Doping Agency's 2005 Prohibited List Using the Chemistry Development Kit Fingerprint
We used the freely available Chemistry Development Kit (CDK) fingerprint to classify 5235 representative molecules taken from ten banned classes in the 2005 World Anti-Doping Agency’s (WADA) prohibited list, including molecules taken from the corresponding activity classes in the MDL Drug Data Report (MDDR). We used both Random Forest and k-Nearest Neighbours (kNN) algorithms to generate classifiers. The kNN classifiers withk = 1 gave a very slightly better Matthews Correlation Coefficient than the Random Forest classifiers; the latter, however, predicted fewer false positives. The performance of kNN classifiers tended to decline with increasing k. The performance of the CDK fingerprint is essentially equivalent to that of Unity 2D. Our results suggest that it will be possible to use freely available chemoinformatics tools to aid the fight against drugs in sport, while minimising the risk of wrongfully penalising innocent athletes.
18008261
Optimization Complexity of Linear Logic Proof Games
A class of linear logic proof games is developed, each with a numeric score that depends on the number of preferred axioms used in a complete or partial proof tree. The complexity of these games is analyzed for the NP-complete multiplicative fragment (MLL) extended with additive constants and the PSPACE-complete multiplicative, additive fragment (MALL) of propositional linear logic. In each case, it is shown that it is as hard to compute an approximation of the best possible score as it is to determine the optimal strategy. Furthermore, it is shown that no efficient heuristics exist unless there is an unexpected collapse in the complexity hierarchy.
2518390
A Compositional Logic for Proving Security Properties of Protocols
We present a logic for proving security properties of protocols that use nonces (randomly generated numbers that uniquely identify a protocol session) and public-key cryptography. The logic, designed around a process calculus with actions for each possible protocol step, consists of axioms about protocol actions and inference rules that yield assertions about protocols composed of multiple steps. Although assertions are written using only steps of the protocol, the logic is sound in a stronger sense: each provable assertion about an action or sequence of actions holds in any run of the protocol that contains the given actions and arbitrary additional actions by a malicious attacker. This approach lets us prove security properties of protocols under attack while reasoning only about the sequence of actions taken by honest parties to the protocol. The main security-specific parts of the proof system are rules for reasoning about the set of messages that could reveal secret data and an invariant ride called the "honesty rule".
53232540
Architectural Support for Copy and Tamper Resistant Software
14607433
Lower bounds on type inference with subtypes
We investigate type inference for programming languages with subtypes. As described in previous work, there are several type inference problems for any given expression language, depending on the form of the subtype partial order and the ability to define new subtypes in programs. Our first main result is that for any specific subtype partial order, the problem of determining whether a lambda term is typable is algorithmically (polynomial-time) equivalent to a form of satisfiability problem over the same partial order. This gives the first exact characterization of the problem that is independent of the syntax of expressions. In addition, since this form of satisfiability problem is PSPACE-hard over certain partial orders, this equivalence strengthens the previous lower bound of NP-hard to PSPACE-hard. Our second main result is a lower bound on the length of most general types when the subtype hierarchy may change as a result of additional type declarations within the program. More specifically, given any input expression, a type inference algorithm tries to find a most general (or principal) typing. The property of a most general typing is that it has all other possible typings as instances. However, there are several sound notions of instance in the presence of subtyping. Our lower bound is that no sound definition of instance would allow the set of additional subtyping hypotheses about a term to grow less than linearly in the size of the term.