A program correctness checker is an algorithm for checking the output of a computation. That is, given a program and an instance on which the program is run, the checker certifies whether the output of the program on that instance is correct. This paper defines the concept of a program checker. It designs program checkers for a few specific and carefully chosen problems in the class FP of functions computable in polynomial time. Problems in FP for which checkers are presented in this paper include Sorting, Matrix Rank and GCD. It also applies methods of modern cryptography, especially the idea of a probabilistic interactive proof, to the design of program checkers for group theoretic computations. Two strucural theorems are proven here. One is a characterization of problems that can be checked. The other theorem establishes equivalence classes of problems such that whenever one problem in a class is checkable, all problems in the class are checkable. Supported by NSF Grant #CCR88-136...

This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NP-completeness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NP-Completeness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, cross-references will be given to that book and the list of problems (NP-complete and harder) presented there. Readers who have results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.) or open problems they would like publicized, should

Interactive environments for dynamically deforming objects play an important role in surgery simulation and entertainment technology. These environments require fast deformable models and very efficient collision handling techniques. While collision detection for rigid bodies is well-investigated, collision detection for deformable objects introduces additional challenging problems. This paper focusses on these aspects and summarizes recent research in the area of deformable collision detection. Various approaches based on bounding volume hierarchies, distance fields, and spatial partitioning are discussed. Further, image-space techniques and stochastic methods are considered. Applications in cloth modeling and surgical simulation are presented.

This is the preliminary version of a chapter that will appear in the Handbook on Computational Geometry, edited by J.-R. Sack and J. Urrutia. A comprehensive overview is given of algorithms and data structures for proximity problems on point sets in IR D . In particular, the closest pair problem, the exact and approximate post-office problem, and the problem of constructing spanners are discussed in detail. Contents 1 Introduction 1 2 The static closest pair problem 4 2.1 Preliminary remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Algorithms that are optimal in the algebraic computation tree model . 5 2.2.1 An algorithm based on the Voronoi diagram . . . . . . . . . . . 5 2.2.2 A divide-and-conquer algorithm . . . . . . . . . . . . . . . . . . 5 2.2.3 A plane sweep algorithm . . . . . . . . . . . . . . . . . . . . . . 6 2.3 A deterministic algorithm that uses indirect addressing . . . . . . . . . 7 2.3.1 The degraded grid . . . . . . . . . . . . . . . . . . ...

One of the basic geometric operations involves determining whether a pair of convex objects intersect. This problem is well understood in a model of computation in which the objects are given as input and their intersection is returned as output. For many applications, however, it may be assumed that the objects already exist within the computer and that the only output desired is a single piece of data giving a common point if the objects intersect or reporting no intersection if they are disjoint. For this problem, none of the previous lower bounds are valid and algorithms are proposed requiring sublinear time for their solution in two and three dimensions.

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In this paper we systematically derive randomized algorithms (both sequential and parallel) for sorting and selection from basic principles and fundamental techniques like random sampling. We prove several sampling lemmas which will find independent applications. The new algorithms derived here are the most efficient known. From among other results, we have an efficient algorithm for sequential sorting. The problem of sorting has attracted so much attention because of its vital importance. Sorting with as few comparisons as possible while keeping the storage size minimum is a long standing open problem. This problem is referred to as ‘the minimum storage sorting ’ [10] in the literature. The previously best known minimum storage sorting algorithm is due to Frazer and McKellar [10]. The expected number of comparisons made by this algorithm is n log n + O(n log log n). The algorithm we derive in this paper makes only an expected n log n + O(n ω(n)) number of comparisons, for any function ω(n) that tends to infinity. A variant of this algorithm makes no more than n log n + O(n log log n) comparisons on any input of size n with overwhelming probability. We also prove high probability bounds for several randomized algorithms for which only expected bounds have been proven so far.

Abstract, Finding a minimum weighted complete matching on a set of vertices in which the distances satisfy the triangle inequality is of general interest and of particular importance when drawing graphs on a mechanical plotter. The "greedy " heuristic of repeatedly matching the two closest unmatched points can be implemented in worst-case time O(n log n), a reasonable savings compared to the general minimum weighted matching algorithm which requires time proportional to n to find the minimum cost matching in a weighted graph. We show that, for an even number n of vertices whose distances satisfy the triangle inequality, the ratio of the cost of the matching produced by this greedy heuristic to the cost of the minimal lg matching is at most 3n 1, lg 0.58496, and there are examples that achieve this bound. We conclude that this greedy heuristic, although desirable because of its simplicity, would be a poor choice for this problem. Key words, graph algorithms, matching, greedy heuristic, analysis of algorithms Introduction, We begin with some motivation, the connection of which to our central topic will become clear later. The problem of drawing a graph G (V, E) on a mechanical plotter with prespecified vertex locations arises in numerous applications [7]. For example, in the

In the last few years, theoretical study of quantum systems serving as computational devices has achieved tremendous progress. We now have strong theoretical evidence that quantum computers, if built, might be used as a dramatically powerful computational tool, capable of performing tasks which seem intractable for classical computers. This review is about to tell the story of theoretical quantum computation. I left out the developing topic of experimental realizations of the model, and neglected other closely related topics which are quantum information and quantum communication. As a result of narrowing the scope of this paper, I hope it has gained the benefit of being an almost self contained introduction to the exciting field of quantum computation. The review begins with background on theoretical computer science, Turing machines and Boolean circuits. In light of these models, I define quantum computers, and discuss the issue of universal quantum gates. Quantum algorithms, including Shor’s factorization algorithm and Grover’s algorithm for searching databases, are explained. I will devote much attention to understanding what the origins of the quantum computational power are, and what the limits of this power are. Finally, I describe the recent theoretical results which show that quantum computers maintain their complexity power even in the presence of noise, inaccuracies and finite precision. This question cannot be separated from that of quantum complexity, because any realistic model will inevitably be subject to such inaccuracies. I tried to put all results in their context, asking what the implications to other issues in computer science and physics are. In the end of this review I make these connections explicit, discussing the possible implications of quantum computation on fundamental physical questions, such as the transition from quantum to classical physics. 1

The mathematical technique of randomization yielding probabilistic algorithms is shown, for the first time, through a physical interpretation based on statistical thermodynamics, to be a basis for energy savings in computing. Concretely, at the fundamental limit, it is shown that the energy needed to compute a single probabilistic bit or PBIT is proportional to the probability p of computing a PBIT accurately. This result is established through the introduction of an idealized switch, for computing a PBIT, using which a network of switches can be constructed. Interesting examples of such networks including AND, OR and NOT gates (or as functions, boolean conjunction, disjunction and negation respectively), are constructed and the potential for energy savings through randomization is established. To quantify these savings, novel measures of "technology independent" energy complexity are introduced---these parallel conventional machine-independent measures of computational complexity such as the algorithm's running time. Networks of switches can be shown to be equivalent to Turing machines and to boolean circuits, both of which are widely-known and well-understood models of computation. These savings are realized using a novel way of representing a PBIT in the physical domain through a group of classical microstates. A measurement and thus detection of a microstate yields the value of the PBIT. While the eventual goal of this work is to lead to the physical realization of these theoretical constructs through the innovation of randomized (CMOS based) devices, the current goal is to rigorously establish the potential for energy savings through probabilistic computing at a fundamental physical level, based on the canonical thermodynamic models of idealized monoa...