31926003
The Minimum-Area Spanning Tree Problem
Motivated by optimization problems in sensor coverage, we formulate and study the Minimum-Area Spanning Tree (mast) problem: Given a set $\mathcal{P}$ of n points in the plane, find a spanning tree of $\mathcal{P}$ of minimum “area,” where the area of a spanning tree $\mathcal{T}$ is the area of the union of the n–1 disks whose diameters are the edges in $\mathcal{T}$. We prove that the Euclidean minimum spanning tree of $\mathcal{P}$ is a constant-factor approximation for mast. We then apply this result to obtain constant-factor approximations for the Minimum-Area Range Assignment (mara) problem, for the Minimum-Area Connected Disk Graph (macdg) problem, and for the Minimum-Area Tour (mat) problem. The first problem is a variant of the power assignment problem in radio networks, the second problem is a related natural problem, and the third problem is a variant of the traveling salesman problem.
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.
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-
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.
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.
13929198
Analysis of Abuse-Free Contract Signing
Optimistic contract signing protocols may involve subprotocols that allow a contract to be signed normally or aborted or resolved by a third party. Since there are many ways these subprotocols might interact, protocol analysis involves consideration of a number of complicated cases. With the help of Murk?, a finite-state verification tool, we analyze the abuse-free optimistic contract signing protocol of Garay, Jakobsson, and MacKenzie. In addition to verifying a nmnber of subtle properties, we discover an attack in which negligence or corruption of the trusted third party may allow abuse or unfairness. Contrary to the intent of the protocol, the cheated party is not able to hold the third party accountable. In addition to analyzing a modification to the protocol that avoids these problems, we discuss issues involved in the application of finite-state analysis to fair exchange protocols, in particular models of fairness guarantees, abuse, and corrupt protocol participants.
42803318
File Servers
This chapter discusses file servers that are accessible via some form of local network or internetwork. It characterizes file services from the client point of view along a number of dimensions; distributivity of the service itself, the network environment of the service, the file model supported by the service, the kind(s) of atomic update provided by the service, the techniques used for controlling the allocation of files. It then describes the client interfaces to three existing file services. Finally, the impact of these features on design and implementation is discussed. An appendix contains characterizations of a variety of existing network-based file services using the criteria developed in the chapter.
31091998
Tree Automata with One Memory, Set Constraints, and Ping-Pong Protocols
We introduce a class of tree automata that perform tests on a memory that is updated using function symbol application and projection. The language emptiness problem for this class of tree automata is shown to be in DEXPTIME. We also introduce a class of set constraints with equality tests and prove its decidability by completion techniques and a reduction to tree automata with one memory. Set constraints with equality tests may be used to decide secrecy for a class of cryptographic protocols that properly contains a class of memoryless "ping-pong protocols" introduced by Dolev and Yao.
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.
52847645
Standard ML-NJ Weak Polymorphism and Imperative Constructs
Standard ML of New Jersey (SML-NJ) uses weak-type variables to restrict the polymorphic use of functions that may allocate reference cells, manipulate continuations, or use exceptions. However, the type system used in the SML-NJ compiler has not been presented in a form other than source code and has not been proved correct. A type system, in the form of typing rules and an equivalent algorithm, that appears to subsume the implemented algorithm is presented. Both use type variables of only a slightly more general nature than the compiler. One insight in the analysis is that the indexed type of a free variable is used in two ways, once in describing the applicative behavior of the variable itself and once in describing the larger term containing the variable. Taking this into account, an application rule that is more general than SML-NJ is formulated for applications of polymorphic functions to imperative arguments. The soundness of the type system is proved for imperative code using operational semantics. >