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146
PolynomialTime Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
 SIAM J. on Computing
, 1997
"... A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. ..."
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Cited by 882 (2 self)
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A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and which have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored.
DavenportSchinzel Sequences and Their Geometric Applications
, 1998
"... An (n; s) DavenportSchinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly noncontiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \ ..."
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Cited by 425 (121 self)
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An (n; s) DavenportSchinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly noncontiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \Delta \Delta a \Delta \Delta \Delta b \Delta \Delta \Delta of length s + 2 between two distinct symbols a and b. The close relationship between DavenportSchinzel sequences and the combinatorial structure of lower envelopes of collections of functions make the sequences very attractive because a variety of geometric problems can be formulated in terms of lower envelopes. A nearlinear bound on the maximum length of DavenportSchinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems.
Simulating Physics with Computers
 SIAM Journal on Computing
, 1982
"... A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time of at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. ..."
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Cited by 393 (1 self)
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A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time of at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored. AMS subject classifications: 82P10, 11Y05, 68Q10. 1 Introduction One of the first results in the mathematics of computation, which underlies the subsequent development of much of theoretical computer science, was the distinction between computable and ...
Orienting Polygonal Parts without Sensors
, 1992
"... In manufacturing, it is often necessary to orient parts prior to packing or assembly. We say that a planar part is polygonal if its convex hull is a polygon. We consider the following problem: given a list of n vertices describing a polygonal part whose initial orientation is unknown, find the short ..."
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Cited by 207 (41 self)
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In manufacturing, it is often necessary to orient parts prior to packing or assembly. We say that a planar part is polygonal if its convex hull is a polygon. We consider the following problem: given a list of n vertices describing a polygonal part whose initial orientation is unknown, find the shortest sequence of mechanical gripper actions that is guaranteed to orient the part up to symmetry in its convex hull. We show that such a sequence exists for any polygonal part by giving an O#n log n# algorithm for finding the sequence. Since the gripper actions do not require feedback, this result implies that any polygonal part can be oriented without sensors.
Geometric Shortest Paths and Network Optimization
 Handbook of Computational Geometry
, 1998
"... Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of t ..."
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Cited by 147 (12 self)
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Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of the edges that comprise it. Efficient algorithms are well known for this problem, as briefly summarized below. The shortest path problem takes on a new dimension when considered in a geometric domain. In contrast to graphs, where the encoding of edges is explicit, a geometric instance of a shortest path problem is usually specified by giving geometric objects that implicitly encode the graph and its edge weights. Our goal in devising efficient geometric algorithms is generally to avoid explicit construction of the entire underlying graph, since the full induced graph may be very large (even exponential in the input size, or infinite). Computing an optimal
Motion Planning in Dynamic Environments using Velocity Obstacles
 International Journal of Robotics Research
, 1998
"... Abstract!llis paper presents a new approach for rot)ot]notion planning in dynamic environments, based on the concept of Velocity Obstacle.,4 velocity obstacle defines the set of robot velocities that would result in a collision between tlie robot and an obstacle moving at a given velocity. The avo ..."
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Cited by 111 (3 self)
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Abstract!llis paper presents a new approach for rot)ot]notion planning in dynamic environments, based on the concept of Velocity Obstacle.,4 velocity obstacle defines the set of robot velocities that would result in a collision between tlie robot and an obstacle moving at a given velocity. The avoidance maneuver a { a specific time is thus computed by selecting robot’s velocities out of that set. ~’lle set [If all avoid inp, velocities is redllced to the dynamica]]y fcasib]e maneuvers by considering the robot’s acceleration constraints. This computation is re])eatecl at regular time intro vals to account for genera] obstacle trajectories. The t rajtxtory from start to goal can be com])uted by searching a tree of feasible avoidance maneuvers C.olnputecl at discrete time ini ervals. An exhaustive search of the tree yields nearoptima] trajectories that either minimize distance or motion time. A heuristic search of t}le tree yields trajectories that satisfy a l)rioritized list of objectives, such as reaching t}~e goal, maximizing speed, and achieving a desired trajectory structure. The heuristic approach is computational]y ctlicientl a])plic.al)le to online planning of industrial robots, performing assembly tasks cm lnoving conveyers, and to intelligent vehicles negotiating freeway traffic. The method is demonstrated for planning the trajectory of an automated vehicle in an Intelligent Vehicle Highway System scenario. 1.
Motion Planning in the Presence of Moving Obstacles
, 1985
"... This paper investigates the computational complexity of planning the motion of a body B in 2D or 3D space, so as to avoid collision with moving obstacles of known, easily computed, trajectories. Dynamic movement problems are of fundamental importance to robotics, but their computational compl ..."
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Cited by 109 (9 self)
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This paper investigates the computational complexity of planning the motion of a body B in 2D or 3D space, so as to avoid collision with moving obstacles of known, easily computed, trajectories. Dynamic movement problems are of fundamental importance to robotics, but their computational complexity has not previously been investigated. We provide evidence that the 3D dynamic movement problem is intractable even if B has only a constant number of degrees of freedom of movement. In particular, we prove the problem is PSPACEhard if B is given a velocity modulus bound on its movements and is NP hard even if B has no velocity modulus bound, where in both cases B has 6 degrees of freedom. To prove these results we use a unique method of simulation of a Turing machine which uses time to encode configurations (whereas previous lower bound proofs in robotic motion planning used the system position to encode configurations and so required unbounded number of degrees of freedom)...
Efficient algorithms for geometric optimization
 ACM Comput. Surv
, 1998
"... We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear progra ..."
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Cited by 94 (12 self)
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We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, pruneandsearch techniques for linear programming and related problems, and LPtype problems and their efficient solution. We then describe a variety of applications of these and other techniques to numerous problems in geometric optimization, including facility location, proximity problems, statistical estimators and metrology, placement and intersection of polygons and polyhedra, and ray shooting and other querytype problems.
Towards Exact Geometric Computation
, 1994
"... Exact computation is assumed in most algorithms in computational geometry. In practice, implementors perform computation in some fixedprecision model, usually the machine floatingpoint arithmetic. Such implementations have many wellknown problems, here informally called "robustness issues". To rec ..."
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Cited by 90 (6 self)
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Exact computation is assumed in most algorithms in computational geometry. In practice, implementors perform computation in some fixedprecision model, usually the machine floatingpoint arithmetic. Such implementations have many wellknown problems, here informally called "robustness issues". To reconcile theory and practice, authors have suggested that theoretical algorithms ought to be redesigned to become robust under fixedprecision arithmetic. We suggest that in many cases, implementors should make robustness a nonissue by computing exactly. The advantages of exact computation are too many to ignore. Many of the presumed difficulties of exact computation are partly surmountable and partly inherent with the robustness goal. This paper formulates the theoretical framework for exact computation based on algebraic numbers. We then examine the practical support needed to make the exact approach a viable alternative. It turns out that the exact computation paradigm encomp...
Kinodynamic Motion Planning
, 1993
"... : Kinodynamic planning attempts to solve a robot motion problem subject to simultaneous kinematic and dynamics constraints. In the general problem, given a robot system, we must find a minimaltime trajectory that goes from a start position and velocity to a goal position and velocity while avoidin ..."
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Cited by 85 (6 self)
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: Kinodynamic planning attempts to solve a robot motion problem subject to simultaneous kinematic and dynamics constraints. In the general problem, given a robot system, we must find a minimaltime trajectory that goes from a start position and velocity to a goal position and velocity while avoiding obstacles by a safety margin and respecting constraints on velocity and acceleration. We consider the simplified case of a point mass under Newtonian mechanics, together with velocity and acceleration bounds. The point must be flown from a start to a goal, amidst polyhedral obstacles in 2D or 3D. While exact solutions to this problem are not known, we provide the first provably good approximation algorithm, and show that it runs in polynomial time. 1 An early version of this work appeared as [CDRX] 2 Computer Science Department, Cornell University, Ithaca, NY 14853 3 Computer Science Division, University of California, Berkeley, CA 94720 4 Computer Science Department, Duke Universit...