17335559
Finding Optimal Bipartitions of Points and Polygons
We give efficient algorithms to compute an optimal bipartition of a set of points or a set of simple polygons in the plane. We examine various criteria involving the perimeter and the area of the convex hulls of the two subsets.
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.
1836645
Computing the visibility graph of points within a polygon
We study the problem of computing the visibility graph defined by a set <i>P</i> of <i>n</i> points inside a polygon <i>Q</i>: two points <i>p</i>,<i>q</i> ε <i>P</i> are joined by an edge if the segment ‾<i><sub>pq</sub></i> ⊂ <i>Q</i>. Efficient output-sensitive algorithms are known for the case in which <i>P</i> is the set of all vertices of <i>Q</i>. We examine the general case in which <i>P</i> is an arbitrary set of points, interior or on the boundary of <i>Q</i> and study a variety of algorithmic questions. We give an output-sensitive algorithm, which is nearly optimal, when <i>Q</i> is a simple polygon. We introduce a notion of "fat" or "robust" visibility, and give a nearly optimal algorithm for computing visibility graphs according to it, in polygons <i>Q</i> that may have holes. Other results include an algorithm to detect if there are any visible pairs among <i>P</i>, and algorithms for output-sensitive computation of visibility graphs with distance restrictions, <i>invisibility</i> graphs, and rectangle visibility graphs.
11133086
Query-sensitive ray shooting
Ray shooting is the problem of determining the first intersection of a ray with a set of obstacies. We assume that obstacles are modeled as polygons in the plane, or generally polyhedra in higher dimensions. Since many ray shooting queries may be applied to a given set of obstacles, it is desirable to preprocess the obstacles so that query processing is as efficient as possible. The problem of ray shooting in a simple polygon with n vertices haa been studied by [7, 6, 10]. In all cases, with linear space, ray shooting queries can be answered in O(log n) time. Ray shooting amidst a set of polygonal obstacles has been considered by [1, 6, 8, 10, 15]. In all these cases, space is within a polylogarithmic factor of linear, and the best query time is O(fi log n) query time. One of the shortcomings of these approaches is that they fail to account for the geometric structure of the obstacle set and query ray. Analyses based solely upon input size cannot account for the fact that, for a fixed in-
8247618
An efficiently computable metric for comparing polygonal shapes
Abstract : Model-based recognition is concerned with comparing a shape A, which is stored as a model for some particular object, with a shape B, which is found to exist in an image. If A and B are close to being the same shape, then a vision system should report a match and return a measure of how good that match is. To be useful this measure should satisfy a number of properties, including: (1) it should be a metric, (2) it should be invariant under translation, rotation, and change-of-scale, (3) it should be reasonably easy to compute, and (4) it should match our intuition (i.e., answers should be similar to those that a person might give). We develop a method for comparing polygons that has these properties. The method works for both convex and nonconvex polygons and runs in time O(mn log mn) where m is the number of vertices in one polygon and n is the number of vertices in the other. Some examples are presented that show the method produces answers that are intuitively reasonable.
57900050
Computational Geometry Column 42
A compendium of thirty previously published open problems in computational geometry is presented.
12706653
Securing Java RMI-based distributed applications
Both Java RMI and Jini use a proxy-based architecture. In this architecture, a client interacts with a service through a proxy, which is code downloaded from a directory and installed on the client's machine. An attacker who controls the communication channels or the directory may compromise the confidentiality and integrity of the client and of the service. We present a security architecture that protects both clients and services in distributed proxy-based computing. In this architecture, the service registers a signed authentication proxy with the directory. The client, after downloading a signed authentication proxy from the directory, verifies the signature on the proxy, authenticates itself to the service through the proxy, and receives a dedicated session proxy for the service over a secure channel. We also describe a Java-based toolkit that implements the security architecture. This toolkit enables developers to add security to Java RMI-based applications with minimal implementation effort.
3062362
Managing Digital Rights using Linear Logic
Digital music players protect songs by enforcing licenses that convey specific rights for individual songs or groups of songs. For licenses specified in industry, we show that deciding whether a license authorizes a sequence of actions is NP-complete, with a restricted version of the problem solvable efficiently using a reduction to maximum network flow. The authorization algorithm used in industry is online, deciding which rights to exercise as actions occur, but we show that all online algorithms are necessarily non-monotonic: each allows actions under one license that it does not allow under a more flexible license. In one approach to achieving monotonicity, we exhibit the unique maximal set of licenses on which there exists a monotonic online algorithm. This set of well-behaved licenses induces an approximation algorithm by replacing each license with a well-behaved license. In a second approach, we consider allowing the player to revise its past decisions about which rights to exercise while still ensuring compliance with the license. We propose an efficient algorithm based on linear logic, with linear negation used to revise past decisions. We prove our algorithm monotonic, live, and sound with respect to the semantics of licenses
16701916
Probabilistic Bisimulation and Equivalence for Security Analysis of Network Protocols
Using a probabilistic polynomial-time process calculus designed for specifying security properties as observational equivalences, we develop a form of bisimulation that justifies an equational proof system. This proof system is sufficiently powerful to derive the semantic security of El Gamal encryption from the Decision Diffie-Hellman (DDH) assumption. The proof system can also derive the converse: if El Gamal is secure, then DDH holds. While these are not new cryptographic results, these example proofs show the power of probabilistic bisimulation and equational reasoning for protocol security.
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.
5553551
Decision Problems for Propositional Linear Logic
Abstract Linear logic, introduced by Girard, is a refinement of classical logic with a natural, intrinsic accounting of resources. This accounting is made possible by removing the ‘structural’ rules of contraction and weakening, adding a modal operator and adding finer versions of the propositional connectives. Linear logic has fundamental logical interest and applications to computer science, particularly to Petri nets, concurrency, storage allocation, garbage collection and the control structure of logic programs. In addition, there is a direct correspondence between polynomial-time computation and proof normalization in a bounded form of linear logic. In this paper we show that unlike most other propositional (quantifier-free) logics, full propositional linear logic is undecidable. Further, we prove that without the modal storage operator, which indicates unboundedness of resources, the decision problem becomes PSPACE-complete. We also establish membership in NP for the multiplicative fragment, NP-completeness for the multiplicative fragment extended with unrestricted weakening, and undecidability for fragments of noncommutative propositional linear logic.