2767172
Fast rendering of irregular grids
We propose a fast algorithm for rendering general irregular grids. Our method uses a sweep-plane approach to accelerate ray casting, and can handle disconnected and nonconvex (even with holes) unstructured irregular grids with a rendering cost that decreases as the "disconnectedness" decreases. The algorithm is carefully tailored to exploit spatial coherence even if the image resolution differs substantially from the object space resolution. In this paper, we establish the practicality of our method through experimental results based on our implementation, and we also provide theoretical results, both lower and upper bounds, on the complexity of ray casting of irregular grids.
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.
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.
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.
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.
6833897
A Probabilistic Polynomial-time Calculus For Analysis of Cryptographic Protocols (Preliminary Report)
We prove properties of a process calculus that is designed for an- alyzing security protocols. Our long-term goal is to develop a form of protocol analysis, consistent with standard cryptographic assumptions, that provides a language for expressing probabilistic polynomial-time protocol steps, a spec- iflcation method based on a compositional form of equivalence, and a logical basis for reasoning about equivalence. The process calculus is a variant of CCS, with bounded replication and probabilistic polynomial-time expressions allowed in messages and boolean tests. To avoid inconsistency between security and nondeterminism, messages are scheduled probabilistically instead of nondeterministically. We prove that evaluation of any process expression halts in probabilistic polynomial time and deflne a form of asymptotic protocol equivalence that allows security proper- ties to be expressed using observational equivalence, a standard relation from programming language theory that involves quantifying over possible environ- ments that might interact with the protocol. We develop a form of probabilistic bisimulation and use it to establish the soundness of an equational proof system based on observational equivalences. The proof system is illustrated by a formation derivation of the assertion, well-known in cryptography, that ElGamal encryption's semantic security is equivalent to the (computational) Decision Di-e-Hellman assumption. This example demonstrates the power of probabilistic bisimulation and equational reasoning for protocol security.
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.
1653333
A derivation system for security protocols and its logical formalization
Many authentication and key exchange protocols are built using an accepted set of standard concepts such as Diffie-Hellman key exchange, nonces to avoid replay, certificates from an accepted authority, and encrypted or signed messages. We introduce a basic framework for deriving security protocols from such simple components. As a case study, we examine the structure of a family of key exchange protocols that includes station-to-station (STS), ISO-9798-3, just fast keying (JFK), IKE and related protocols, deriving all members of the family from two basic protocols using a small set of refinements and protocol transformations. As initial steps toward associating logical derivations with protocol derivations, we extend a previous security protocol logic with preconditions and temporal assertions. Using this logic, we prove the security properties of the standard signature based challenge-response protocol and the Diffie-Hellman key exchange protocol. The ISO-9798-3 protocol is then proved correct by composing the correctness proofs of these two simple protocols.
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.
6372262
Algorithmic aspects of type inference with subtypes
We study the complexity of type inference for programming languages with subtypes. There are three language variations that effect the problem: (i) basic functions may have polymorphic or more limited types, (ii) the subtype hierarchy may be fixed or vary as a result of subtype declarations within a program, and (iii) the subtype hierarchy may be an arbitrary partial order or may have a more restricted form, such as a tree or lattice. The naive algorithm for infering a most general polymorphic type, undervariable subtype hypotheses, requires deterministic exponential time. If we fix the subtype ordering, this upper bound grows to nondeterministic exponential time. We show that it is NP-hard to decide whether a lambda term has a type with respect to a fixed subtype hierarchy (involving only atomic type names). This lower bound applies to monomorphic or polymorphic languages. We give PSPACE upper bounds for deciding polymorphic typability if the subtype hierarchy has a lattice structure or the subtype hierarchy varies arbitrarily. We also give a polynomial time algorithm for the limited case where there are of no function constants and the type hierarchy is either variable or any fixed lattice.