8125193
On simultaneous planar graph embeddings
We consider the problem of simultaneous embedding of planar graphs. There are two variants of this problem, one in which the mapping between the vertices of the two graphs is given and another in which the mapping is not given. We present positive and negative results for the two versions of the problem. Among the positive results with given mapping, we show that we can embed two paths on an nxn grid, and two caterpillar graphs on a 3nx3n grid. Among the negative results with given mapping, we show that it is not always possible to simultaneously embed three paths or two general planar graphs. If the mapping is not given, we show that any number of outerplanar graphs can be embedded simultaneously on an O(n)xO(n) grid, and an outerplanar and general planar graph can be embedded simultaneously on an O(n^2)xO(n^2) grid.
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.
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.
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.
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".
30142201
The Implication Problem for Functional and Inclusion Dependencies
There are two implication problems for functional dependencies and inclusion dependencies: general implication and finite implication. Given a set of dependencies Σ ∪ { σ }, the problems are to determine whether σ holds in all databases satisfying Σ or all finite databases satisfying Σ . Contrary to the possibility suggested in Casanova, Fagin, and Papadimitriou (“Proceedings, 1st ACM Conf. on Principles of Database System,” pp. 171–176, 1982), there is a natural, complete axiom system for general implication. However, a simple observation shows that both implication problems are recursively unsolvable. It follows that there is no recursively enumerable set of axioms for finite implication.
7468814
Coupled optimization in protein docking
Here, we introduce a new approach to macromolecular docking which combines the continuous global optimization algorithm CGU [6, 21] with the grid-based conformational search engine DOT [23]. We also detail the use of this method to dock acetylcholine into the fasciculin-acetylcholinesterase complex. Fasciculin blocks access to the long, narrow pathway leading to the active site, which is located in the interior of the acetylcholinesterase. For this reason, flnding the location computationally is rather non-trivial. We not only successfully identifled the optimal docking conflguration, but were able to do so in a matter of hours.
17480240
A compositional logic for protocol correctness
Abstract: We present a specialized protocol logic that is built around a process language for describing the actions of a protocol. In general terms, the relation between logic and protocol is like the relation between assertions in Floyd-Hoare logic and standard imperative programs. Like Floyd-Hoare logic, our logic contains axioms and inference rules for each of the main protocol actions and proofs are protocol-directed, meaning that the outline of a proof of correctness follows the sequence of actions in the protocol. We prove that the protocol logic is sound, in a specific sense: each provable assertion about an action or sequence of actions holds in any run of the protocol, under attack, in which the given actions occur. This approach lets us prove properties of protocols that hold in all runs, while explicitly reasoning only about the sequence of actions needed to achieve this property. In particular, no explicit reasoning about the potential actions of an attacker is required.
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.