60004
When can you fold a map?
We explore the following problem: given a collection of creases on a piece of paper, each assigned a folding direction of mountain or valley, is there a fiat folding by a sequence of simple folds? There are several models of simple folds; the simplest one-layer simple fold rotates a portion of paper about a crease in the paper by ±180°. We first consider the analogous questions in one dimension lower--bending a segment into a flat object--which lead to interesting problems on strings. We develop efficient algorithms for the recognition of simply foldable 1D crease patterns, and reconstruction of a sequence of simple folds. Indeed, we prove that a 1D crease pattern is flat-foldable by any means precisely if it is by a sequence of one-layer simple folds. 
 
Next we explore simple foldability in two dimensions, and find a surprising contrast: "map" folding and variants are polynomial, but slight generalizations are NP-complete. Specifically, we develop a linear-time algorithm for deciding foldability of an orthogonal crease pattern on a rectangular piece of paper, and prove that it is (weakly) NP-complete to decide foldability of (1) an orthogonal crease pattern on a orthogonal piece of paper, (2) a crease pattern of axis-parallel and diagonal (45-degree) creases on a square piece of paper, and (3) crease patterns without a mountain/valley assignment.
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.
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.
15882954
The Development of Type Systems for Object-Oriented Languages
This paper, which is partly tutorial in nature, summarizes some basic research goals in the study and development of typed object-oriented programming languages. These include both immediate repairs to problems with existing languages and the long-term development of more flexible and expressive, yet type-safe, approaches to program organization and design. The technical part of the paper is a summary and comparison of three object models from the literature. We conclude by discussing approaches to selected research problems, including changes in the type of a method from super class to sub class and the use of types that give information about the implementations as well as the interfaces of objects. Such implementation types seem essential for adequate typing of binary operations on objects, for example.
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
21271034
Conflict and combination in privacy policy languages
Many modern enterprises require methods for guaranteeing compliance with privacy legislation and announced privacy policies. IBM has proposed a formal language, the Enterprise Privacy Authorization Language (EPAL), for describing privacy policies rigorously. In this paper, we identify four desirable properties of a privacy policy language: guaranteed consistency, guaranteed safety, admitting local reasoning, and closure under combination. While EPAL achieves only one of these four goals, an extended language framework allows us to achieve three out of four, while retaining the basic EPAL framework of restricting access and imposing obligations on users of confidential information.
23474411
Finite-state analysis of two contract signing protocols
Optimistic contract signing protocols allow two parties to commit to a previously agreed upon contract, relying on a third party to abort or confirm the contract if needed. These protocols are relatively subtle, since there may be interactions between the subprotocols used for normal signing without the third party, aborting the protocol through the third party, or requesting confirmation from the third party. With the help of Mur, a finite-state verification tool, we analyze two related contract signing protocols: the optimistic contract signing protocol of Asokan, Shoup, and Waidner, and the abuse-free contract signing protocol of Garay, Jakobsson, and MacKenzie. For the first protocol, we discover that a malicious participant can produce inconsistent versions of the contract or mount a replay attack. For the second protocol, we discover that negligence or corruption of the trusted third party may allow abuse or unfairness. In this case, contrary to the intent of the protocol, the cheated party is not able to hold the third party accountable. We present and analyze modifications to the protocols that avoid these problems and discuss the basic challenges involved in formal analysis of fair exchange protocols.
6701720
Reconstructing Trust Management
We present a trust management kernel that clearly separates authorization and structured distributed naming. Given an access request and supporting credentials, the kernel determines whether the request is authorized. We prove soundness and completeness of the authorization system without names and prove that naming is orthogonal to authorization in a precise sense. The orthogonality theorem gives us simple soundness and completeness proofs for the entire kernel. The kernel is formally verified in PVS, allowing for the automatic generation of a verified implementation of a reference monitor. By separating naming and authorization primitives, we arrive at a compositional model and avoid concepts such as “speaks-for” that have led to anomalies in logical characterizations of other trust management systems. 
 
Supported in part by DARPA contract N66001-00-C-8015 and ONR grant N00014-01-1-0795. 
 
Supported in part by DARPA contract N66001-00-C-8015 and ONR grant N00014-01-1-0837.