35 Nov. 1993
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Breaking the /spl Theta/(nlog/sup 2/ n) barrier for sorting with faults
Publication Year: 1993, Page(s):734  743We study the problem of constructing a sorting circuit, network, or PRAM algorithm that is tolerant to faults. For the most part, we focus on fault patterns that are random, e.g., where the result of each comparison is independently faulty with probability upperbounded by some constant. All previous faulttolerant sorting circuits, networks, and parallel algorithms require /spl Omega/(log/sup 2/ ... View full abstract»

The hardness of approximate optima in lattices, codes, and systems of linear equations
Publication Year: 1993, Page(s):724  733
Cited by: Papers (38)We prove the following about the Nearest Lattice Vector Problem (in any l/sub p/ norm), the Nearest Codeword Problem for binary codes, the problem of learning a halfspace in the presence of errors, and some other problems. 1. Approximating the optimum within any constant factor is NPhard. 2. If for some /spl epsiv/>0 there exists a polynomial time algorithm that approximates the optimum withi... View full abstract»

Externalmemory computational geometry
Publication Year: 1993, Page(s):714  723
Cited by: Papers (28)  Patents (2)In this paper we give new techniques for designing efficient algorithms for computational geometry problems that are too large to be solved in internal memory. We use these techniques to develop optimal and practical algorithms for a number of important largescale problems. We discuss our algorithms primarily in the context of single processor/single disk machines, a domain in which they are not ... View full abstract»

Efficient outofcore algorithms for linear relaxation using blocking covers
Publication Year: 1993, Page(s):704  713
Cited by: Papers (5)When a numerical computation fails to fit in the primary memory of a serial or parallel computer, a socalled "outofcore" algorithm must be used which moves data between primary and secondary memories. In this paper, we study outofcore algorithms for sparse linear relaxation problems in which each iteration of the algorithm updates the state of every vertex in a graph with a linear combination... View full abstract»

On choosing a dense subgraph
Publication Year: 1993, Page(s):692  701
Cited by: Papers (18)This paper concerns the problem of computing the densest kvertex subgraph of a given graph, namely, the subgraph with the most edges, or with the highest edgestovertices ratio. A sequence of approximation algorithms is developed for the problem, with each step yielding a better ratio at the cost of a more complicated solution. The approximation ratio of our final algorithm is O/spl tilde/(n/sup... View full abstract»

A Chernoff bound for random walks on expander graphs
Publication Year: 1993, Page(s):680  691
Cited by: Papers (7)We consider a finite random walk on a weighted graph G; we show that the sample average of visits to a set of vertices A converges to the stationary probability /spl pi/(A) with error probability exponentially small in the length of the random walk and the square of the size of the deviation from /spl pi/(A). The exponential bound is in terms of the expansion of G and improves previous results. We... View full abstract»

Eavesdropping games: a graphtheoretic approach to privacy in distributed systems
Publication Year: 1993, Page(s):670  679
Cited by: Papers (2)We initiate a graphtheoretic approach to study the (informationtheoretic) maintenance of privacy in distributed environments in the presence of a bounded number of mobile eavesdroppers ("bugs"). For two fundamental privacy problemssecure message transmission and distributed database maintenancewe assume an adversary is "playing eavesdropping games," coordinating the movement of the bugs among ... View full abstract»

A sublinear time distributed algorithm for minimumweight spanning trees
Publication Year: 1993, Page(s):659  668
Cited by: Papers (6)This paper considers the question of identifying the parameters governing the behavior of fundamental global network problems. Many papers on distributed network algorithms consider the task of optimizing the running time successful when an O(n) bound is achieved on an nvertex network. We propose that a more sensitive parameter is the network's diameter Diam. This is demonstrated in the paper by ... View full abstract»

Fast algorithms for constructing tspanners and paths with stretch t
Publication Year: 1993, Page(s):648  658
Cited by: Papers (9)The distance between two vertices in a weighted graph is the weight of a minimumweight path between them. A path has stretch t if its weight is at most t times the distance between its end points. We consider a weighted undirected graph G=(V, E) and present algorithms that compute paths with stretch 2/spl les/t/spl les/log n. We present a O/spl tilde/((m+k)n/sup (2+/spl epsiv///t)) time randomize... View full abstract»

Nearlinear cost sequential and distributed constructions of sparse neighborhood covers
Publication Year: 1993, Page(s):638  647
Cited by: Papers (2)This paper introduces the first nearlinear (specifically, O(Elog n+nlog/sup 2/ n)) time algorithm for constructing a sparse neighborhood cover in sequential and distributed environments. This automatically implies analogous improvements (from quadratic to nearlinear) to all the results in the literature that rely on network decompositions, both in sequential and distributed domains, including ad... View full abstract»

An O(nlog/sup 3/ n) algorithm for the real root problem
Publication Year: 1993, Page(s):626  635Given a univariate complex polynomial f(x) of degree n with rational coefficients expressed as a ratio of two integers <2/sup m/, the root problem is to find all the roots of f(x) up to specified precision 2/sup /spl mu//. In this paper we assume the arithmetic model for computation. We give an algorithm for the real root problem: where all the roots of the polynomial are real. Our real root a... View full abstract»

Counting rational points on curves over finite fields
Publication Year: 1993, Page(s):616  625We consider the problem of counting the number of points on a plane curve, given by a homogeneous polynomial F/spl isin/F/sub p/[x, y, z], which is rational over the ground field F/sub p/. More precisely, we show that if we are given a projective plane curve C of degree n, and if C has only ordinary multiple points, then one can compute the number of F/sub p/rational points on C in randomized tim... View full abstract»

Directed vs. undirected monotone contact networks for threshold functions
Publication Year: 1993, Page(s):604  613We consider the problem of computing threshold functions using directed and undirected monotone contact networks. Our main results are the following. First, we show that there exist directed monotone contact networks that compute T/sub k//sup n/, 2/spl les/k/spl les/n1, of size O(k(nk+2)log(nk+2)). This bound is almost optimal for small thresholds, since there exists an /spl Omega/(knlog (n/(k... View full abstract»

Signal propagation, with application to a lower bound on the depth of noisy formulas
Publication Year: 1993, Page(s):594  603
Cited by: Papers (9)We study the decay of an information signal propagating through a series of noisy channels. We obtain exact bounds on such decay, and as a result provide a new lower bound on the depth of formulas with noisy components. This improves upon previous work of N. Pippenger (1988) and significantly decreases the gap between his lower bound and the classical upper bound of von Neumann. We also discuss co... View full abstract»

Refining a triangulation of a planar straightline graph to eliminate large angles
Publication Year: 1993, Page(s):583  592
Cited by: Papers (1)We show that any planar straight line graph (PSLG) with v vertices can be triangulated with no angle larger than 7/spl pi//8 by adding O(v/sup 2/log v) Steiner points in O(v/sup 2/log/sup 2/ v) time. We first triangulate the PSLG with an arbitrary constrained triangulation and then refine that triangulation by adding additional vertices and edges. We follow a lazy strategy of starting from an obtu... View full abstract»

A compact piecewiselinear Voronoi diagram for convex sites in the plane
Publication Year: 1993, Page(s):573  582
Cited by: Patents (2)In the plane, the postoffice problem, which asks for the closest site to a query site, and retraction motion planning, which asks for a onedimensional retract of the free space of a robot, are both classically solved by computing a Voronoi diagram. When the sites are k disjoint convex sets, we give a compact representation of the Voronoi diagram, using O(k) line segments, that is sufficient for ... View full abstract»

A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed
Publication Year: 1993, Page(s):566  572
Cited by: Papers (5)We prove that for any dimension d there exists a polynomial time algorithm for counting integral points in polyhedra in the ddimensional Euclidean space. Previously such algorithms were known for dimensions d=1,2,3, and 4 only.<<ETX>> View full abstract»

The NC equivalence of planar integer linear programming and Euclidean GCD
Publication Year: 1993, Page(s):557  564We show NCreduction of integer linear programming with two variables to the evaluation of the remainder sequence arising in the application of the Euclidean algorithm to two positive integers. Due to the previous result of X. Deng (1989), this implies NCequivalence of both of these problems, whose membership in NC, as well as Pcompleteness, remain unresolved open problems.<<ETX>> View full abstract»

On bounded queries and approximation
Publication Year: 1993, Page(s):547  556
Cited by: Papers (2)This paper investigates the computational complexity of approximating NPoptimization problems using the number of queries to an NP oracle as a complexity measure. The results show a tradeoff between the closeness of the approximation and the number of queries required. For an approximation factor k(n), loglog/sub k(n/) n queries to an NP oracle can be used to approximate the maximum clique size ... View full abstract»

What can we sort in o(nlog n) time?
Publication Year: 1993, Page(s):538  546We define two conditions on a random access machine (RAM) with arithmetic and Boolean instructions and possible bounds on word and memory sizes. One condition asserts that we either restrict attention to short words or allow nonuniform programs. The second asserts that we either allow a large memory or a doubleprecision multiplication. Our main theorem shows that the RAM can sort in o(nlog n) ti... View full abstract»

Better lower bounds on detecting affine and spherical degeneracies
Publication Year: 1993, Page(s):528  536
Cited by: Papers (2)We show that in the worst case, /spl Omega/(n/sup d/) sidedness queries are required to determine whether a set of n points in R/sup d/ is affinely degenerate, i.e., whether it contains d+1 points on a common hyperplane. This matches known upper bounds. We give a straightforward adversary argument, based on the explicit construction of a point set containing /spl Omega/(n/sup d/) "collapsible" sim... View full abstract»

The union of convex polyhedra in three dimensions
Publication Year: 1993, Page(s):518  527
Cited by: Papers (9)We show that the number of vertices, edges, and faces of the union of k convex polyhedra in 3space, having a total of n faces, is O(k/sup 3/+knlog/sup 2/ k). This bound is almost tight in the worst case. We also describe a rather simple randomized incremental algorithm for computing the boundary of the union in O(k/sup 3/+knlog/sup 3/ k) expected time.<<ETX>> View full abstract»

Efficient computation of Euclidean shortest paths in the plane
Publication Year: 1993, Page(s):508  517
Cited by: Papers (13)We propose a new algorithm for a classical problem in plane computational geometry: computing a shortest path between two points in the presence of polygonal obstacles. Our algorithm runs in worstcase time O(nlog/sup 2/ n) and requires O(nlog n) space, where n is the total number of vertices in the obstacle polygons. Our algorithm actually computes a planar map that encodes shortest paths from a ... View full abstract»

Almost tight upper bounds for lower envelopes in higher dimensions
Publication Year: 1993, Page(s):498  507
Cited by: Papers (2)We show that the combinatorial complexity of the lower envelope of n surfaces or surface patches in dspace (d/spl ges/3), all algebraic of constant maximum degree, and bounded by algebraic surfaces of constant maximum degree, is O(n/sup d1+/spl epsi//), for any /spl epsi/>0; the constant of proportionality depends on /spl epsi/, d, and the shape and degree of the surface patches and of their ... View full abstract»

A weak version of the Blum, Shub and Smale model
Publication Year: 1993, Page(s):486  495
Cited by: Papers (4)We propose a weak version of the BlumShubSmale model (1989) of computation over the real numbers. In this weak model only a "moderate" usage of multiplications and divisions is allowed. The class of languages recognizable in polynomial time as shown to be the complexity class P/poly. This implies under a standard complexitytheoretic assumption that P/spl ne/NP in the weak model, and that proble... View full abstract»