Showing papers in "Journal of the ACM in 2004"

A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries

Abstract: We present a polynomial-time randomized algorithm for estimating the permanent of an arbitrary n × n matrix with nonnegative entries. This algorithm---technically a "fully-polynomial randomized approximation scheme"---computes an approximation that is, with high probability, within arbitrarily small specified relative error of the true value of the permanent.

On clusterings: Good, bad and spectral

Abstract: We motivate and develop a natural bicriteria measure for assessing the quality of a clustering that avoids the drawbacks of existing measures. A simple recursive heuristic is shown to have poly-logarithmic worst-case guarantees under the new measure. The main result of the article is the analysis of a popular spectral algorithm. One variant of spectral clustering turns out to have effective worst-case guarantees; another finds a "good" clustering, if one exists.

The random oracle methodology, revisited

Abstract: We take a critical look at the relationship between the security of cryptographic schemes in the Random Oracle Model, and the security of the schemes that result from implementing the random oracle by so called "cryptographic hash functions".The main result of this article is a negative one: There exist signature and encryption schemes that are secure in the Random Oracle Model, but for which any implementation of the random oracle results in insecure schemes. In the process of devising the above schemes, we consider possible definitions for the notion of a "good implementation" of a random oracle, pointing out limitations and challenges.

Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time

Abstract: We introduce the smoothed analysis of algorithms, which continuously interpolates between the worst-case and average-case analyses of algorithms. In smoothed analysis, we measure the maximum over inputs of the expected performance of an algorithm under small random perturbations of that input. We measure this performance in terms of both the input size and the magnitude of the perturbations. We show that the simplex algorithm has smoothed complexity polynomial in the input size and the standard deviation of Gaussian perturbations.

Fast monte-carlo algorithms for finding low-rank approximations

Abstract: We consider the problem of approximating a given m × n matrix A by another matrix of specified rank k, which is smaller than m and n. The Singular Value Decomposition (SVD) can be used to find the "best" such approximation. However, it takes time polynomial in m, n which is prohibitive for some modern applications. In this article, we develop an algorithm that is qualitatively faster, provided we may sample the entries of the matrix in accordance with a natural probability distribution. In many applications, such sampling can be done efficiently. Our main result is a randomized algorithm to find the description of a matrix D* of rank at most k so that holds with probability at least 1 − δ (where v·vF is the Frobenius norm). The algorithm takes time polynomial in k,1/e, log(1/δ) only and is independent of m and n. In particular, this implies that in constant time, it can be determined if a given matrix of arbitrary size has a good low-rank approximation.

Quantum lower bounds for the collision and the element distinctness problems

Abstract: Given a function f as an oracle, the collision problem is to find two distinct indexes i and j such that f(i) = f(j), under the promise that such indexes exist. Since the security of many fundamental cryptographic primitives depends on the hardness of finding collisions, our lower bounds provide evidence for the existence of cryptographic primitives that are immune to quantum cryptanalysis. We prove that any quantum algorithm for finding a collision in an r-to-one function must evaluate the function Ω((n/r)1/3) times, where n is the size of the domain and r|n. This matches an upper bound of Brassard, Hoyer, and Tapp. No lower bound better than constant was previously known. Our result also implies a quantum lower bound of Ω(n2/3) queries for the element distinctness problem, which is to determine whether n integers are all distinct. The best previous lower bound was Ω(√n) queries.

Approximating extent measures of points

Abstract: We present a general technique for approximating various descriptors of the extent of a set P of n points in Rd when the dimension d is an arbitrary fixed constant. For a given extent measure μ and a parameter e > 0, it computes in time O(n + 1/eO(1)) a subset Q ⊆ P of size 1/eO(1), with the property that (1 − e)μ(P) ≤ μ(Q) ≤ μ(P). The specific applications of our technique include e-approximation algorithms for (i) computing diameter, width, and smallest bounding box, ball, and cylinder of P, (ii) maintaining all the previous measures for a set of moving points, and (iii) fitting spheres and cylinders through a point set P. Our algorithms are considerably simpler, and faster in many cases, than previously known algorithms.

Containment and equivalence for a fragment of XPath

Abstract: XPath is a language for navigating an XML document and selecting a set of element nodes. XPath expressions are used to query XML data, describe key constraints, express transformations, and reference elements in remote documents. This article studies the containment and equivalence problems for a fragment of the XPath query language, with applications in all these contexts.In particular, we study a class of XPath queries that contain branching, label wildcards and can express descendant relationships between nodes. Prior work has shown that languages that combine any two of these three features have efficient containment algorithms. However, we show that for the combination of features, containment is coNP-complete. We provide a sound and complete algorithm for containment that runs in exponential time, and study parameterized PTIME special cases. While we identify one parameterized class of queries for which containment can be decided efficiently, we also show that even with some bounded parameters, containment remains coNP-complete. In response to these negative results, we describe a sound algorithm that is efficient for all queries, but may return false negatives in some cases.

Compact oracles for reachability and approximate distances in planar digraphs

Abstract: It is shown that a planar digraph can be preprocessed in near-linear time, producing a near-linear space oracle that can answer reachability queries in constant time. The oracle can be distributed as an O(log n) space label for each vertex and then we can determine if one vertex can reach another considering their two labels only.The approach generalizes to give a near-linear space approximate distances oracle for a weighted planar digraph. With weights drawn from l0, …, Nr, it approximates distances within a factor (1 + e) in O(log log (nN) + 1/e) time. Our scheme can be extended to find and route along correspondingly short dipaths.

A new approach to dynamic all pairs shortest paths

Abstract: We study novel combinatorial properties of graphs that allow us to devise a completely new approach to dynamic all pairs shortest paths problems. Our approach yields a fully dynamic algorithm for general directed graphs with non-negative real-valued edge weights that supports any sequence of operations in O(n2log3n) amortized time per update and unit worst-case time per distance query, where n is the number of vertices. We can also report shortest paths in optimal worst-case time. These bounds improve substantially over previous results and solve a long-standing open problem. Our algorithm is deterministic, uses simple data structures, and appears to be very fast in practice.

New lattice-based cryptographic constructions

Abstract: We introduce the use of Fourier analysis on lattices as an integral part of a lattice-based construction. The tools we develop provide an elegant description of certain Gaussian distributions around lattice points. Our results include two cryptographic constructions that are based on the worst-case hardness of the unique shortest vector problem. The main result is a new public key cryptosystem whose security guarantee is considerably stronger than previous results (O(n1.5) instead of O(n7)). This provides the first alternative to Ajtai and Dwork's original 1996 cryptosystem. Our second result is a family of collision resistant hash functions with an improved security guarantee in terms of the unique shortest vector problem. Surprisingly, both results are derived from one theorem that presents two indistinguishable distributions on the segment [0, 1). It seems that this theorem can have further applications; as an example, we use it to solve an open problem in quantum computation related to the dihedral hidden subgroup problem.

Number-theoretic constructions of efficient pseudo-random functions

Abstract: We describe efficient constructions for various cryptographic primitives in private-key as well as public-key cryptography. Our main results are two new constructions of pseudo-random functions. We prove the pseudo-randomness of one construction under the assumption that factoring (Blum integers) is hard while the other construction is pseudo-random if the decisional version of the Diffie--Hellman assumption holds. Computing the value of our functions at any given point involves two subset products. This is much more efficient than previous proposals. Furthermore, these functions have the advantage of being in TC0 (the class of functions computable by constant depth circuits consisting of a polynomial number of threshold gates). This fact has several interesting applications. The simple algebraic structure of the functions implies additional features such as a zero-knowledge proof for statements of the form "y = fs(x)" and "y ≠ fs(x)" given a commitment to a key s of a pseudo-random function fs.

Solving convex programs by random walks

Abstract: Minimizing a convex function over a convex set in n-dimensional space is a basic, general problem with many interesting special cases. Here, we present a simple new algorithm for convex optimization based on sampling by a random walk. It extends naturally to minimizing quasi-convex functions and to other generalizations.

Polynomial-time data reduction for dominating set

Abstract: Dealing with the NP-complete Dominating Set problem on graphs, we demonstrate the power of data reduction by preprocessing from a theoretical as well as a practical side. In particular, we prove that Dominating Set restricted to planar graphs has a so-called problem kernel of linear size, achieved by two simple and easy-to-implement reduction rules. Moreover, having implemented our reduction rules, first experiments indicate the impressive practical potential of these rules. Thus, this work seems to open up a new and prospective way how to cope with one of the most important problems in graph theory and combinatorial optimization.

Automatic information extraction from large websites

Abstract: Information extraction from websites is nowadays a relevant problem, usually performed by software modules called wrappers. A key requirement is that the wrapper generation process should be automated to the largest extent, in order to allow for large-scale extraction tasks even in presence of changes in the underlying sites. So far, however, only semi-automatic proposals have appeared in the literature.We present a novel approach to information extraction from websites, which reconciles recent proposals for supervised wrapper induction with the more traditional field of grammar inference. Grammar inference provides a promising theoretical framework for the study of unsupervised---that is, fully automatic---wrapper generation algorithms. However, due to some unrealistic assumptions on the input, these algorithms are not practically applicable to Web information extraction tasks.The main contributions of the article stand in the definition of a class of regular languages, called the prefix mark-up languages, that abstract the structures usually found in HTML pages, and in the definition of a polynomial-time unsupervised learning algorithm for this class. The article shows that, differently from other known classes, prefix mark-up languages and the associated algorithm can be practically used for information extraction purposes.A system based on the techniques described in the article has been implemented in a working prototype. We present some experimental results on known Websites, and discuss opportunities and limitations of the proposed approach.

Satisfiability of word equations with constants is in PSPACE

Abstract: We prove that the satisfiability problem for word equations is in PSPACE. The satisfiability problem for word equations has a simple formulation: find out whether or not an input word equation has a solution. The decidability of the problem was proved by G.S. Makanin (1977). His decision procedure is one of the most complicated algorithms existing in the literature. We propose an alternative algorithm. The full version of the algorithm requires only a proof of the upper bound for index of periodicity of a minimal solution (A. Koscielski and L. Pacholski, see Journal of ACM, vol.43, no.4. p.670-84). Our algorithm is the first one which is proved to work in polynomial space.

Self-stabilizing clock synchronization in the presence of Byzantine faults

Abstract: We initiate a study of bounded clock synchronization under a more severe fault model than that proposed by Lamport and Melliar-Smith [1985]. Realistic aspects of the problem of synchronizing clocks in the presence of faults are considered. One aspect is that clock synchronization is an on-going task, thus the assumption that some of the processors never fail is too optimistic. To cope with this reality, we suggest self-stabilizing protocols that stabilize in any (long enough) period in which less than a third of the processors are faulty. Another aspect is that the clock value of each processor is bounded. A single transient fault may cause the clock to reach the upper bound. Therefore, we suggest a bounded clock that wraps around when appropriate.We present two randomized self-stabilizing protocols for synchronizing bounded clocks in the presence of Byzantine processor failures. The first protocol assumes that processors have a common pulse, while the second protocol does not. A new type of distributed counter based on the Chinese remainder theorem is used as part of the first protocol.

Concurrent zero-knowledge

Abstract: Concurrent executions of a zero-knowledge protocol by a single prover (with one or more verifiers) may leak information and may not be zero-knowledge in toto. In this article, we study the problem of maintaining zero-knowledge.We introduce the notion of an (α, β) timing constraint: for any two processors P1 and P2, if P1 measures α elapsed time on its local clock and P2 measures β elapsed time on its local clock, and P2 starts afterP1 does, then P2 will finish after P1 does. We show that if the adversary is constrained by an (α, β) assumption then there exist four-round almost concurrent zero-knowledge interactive proofs and perfect concurrent zero-knowledge arguments for every language in NP. We also address the more specific problem of Deniable Authentication, for which we propose several particularly efficient solutions. Deniable Authentication is of independent interest, even in the sequential case; our concurrent solutions yield sequential solutions without recourse to timing, that is, in the standard model.

Monadic datalog and the expressive power of languages for Web information extraction

Abstract: Research on information extraction from Web pages (wrapping) has seen much activity recently (particularly systems implementations), but little work has been done on formally studying the expressiveness of the formalisms proposed or on the theoretical foundations of wrapping. In this paper, we first study monadic datalog over trees as a wrapping language. We show that this simple language is equivalent to monadic second order logic (MSO) in its ability to specify wrappers. We believe that MSO has the right expressiveness required for Web information extraction and propose MSO as a yardstick for evaluating and comparing wrappers. Along the way, several other results on the complexity of query evaluation and query containment for monadic datalog over trees are established, and a simple normal form for this language is presented. Using the above results, we subsequently study the kernel fragment Elog− of the Elog wrapping language used in the Lixto system (a visual wrapper generator). Curiously, Elog− exactly captures MSO, yet is easier to use. Indeed, programs in this language can be entirely visually specified.

Spatial gossip and resource location protocols

Abstract: The dynamic behavior of a network in which information is changing continuously over time requires robust and efficient mechanisms for keeping nodes updated about new information. Gossip protocols are mechanisms for this task in which nodes communicate with one another according to some underlying deterministic or randomized algorithm, exchanging information in each communication step. In a variety of contexts, the use of randomization to propagate information has been found to provide better reliability and scalability than more regimented deterministic approaches.In many settings, such as a cluster of distributed computing hosts, new information is generated at individual nodes, and is most "interesting" to nodes that are nearby. Thus, we propose distance-based propagation bounds as a performance measure for gossip mechanisms: a node at distance d from the origin of a new piece of information should be able to learn about this information with a delay that grows slowly with d, and is independent of the size of the network.For nodes arranged with uniform density in Euclidean space, we present natural gossip mechanisms, called spatial gossip, that satisfy such a guarantee: new information is spread to nodes at distance d, with high probability, in O(log1 + ed) time steps. Such a bound combines the desirable qualitative features of uniform gossip, in which information is spread with a delay that is logarithmic in the full network size, and deterministic flooding, in which information is spread with a delay that is linear in the distance and independent of the network size. Our mechanisms and their analysis resolve a conjecture of Demers et al. [1987].We further show an application of our gossip mechanisms to a basic resource location problem, in which nodes seek to rapidly learn the location of the nearest copy of a resource in a network. This problem, which is of considerable practical importance, can be solved by a very simple protocol using Spatial Gossip, whereas we can show that no protocol built on top of uniform gossip can inform nodes of their approximately nearest resource within poly-logarithmic time. The analysis relies on an additional useful property of spatial gossip, namely that information travels from its source to sinks along short paths not visiting points of the network far from the two nodes.

A unified theory of shared memory consistency

Abstract: The traditional assumption about memory is that a read returns the value written by the most recent write. However, in a shared memory multiprocessor several processes independently and simultaneously submit reads and writes resulting in a partial order of memory operations. In this partial order, the definition of most recent write may be ambiguous. Memory consistency models have been developed to specify what values may be returned by a read given that memory operations may only be partially ordered. Before this work, consistency models were defined independently. Each model followed a set of rules which was separate from the rules of every other model. In our work, we have defined a set of four consistency properties. Any subset of the four properties yields a set of rules which constitute a consistency model. Every consistency model previously described in the literature can be defined based on our four properties. Therefore, we present these properties as a unfied theory of shared memory consistency.Our unified theory provides several benefits. First, we claim that these four properties capture the underlying structure of memory consistency. That is, the goal of memory consistency is to ensure certain declarative properties which can be intuitively understood by a programmer, and hence allow him or her to write a correct program. Our unified theory provides a uniform, formal definition of all previously described consistency models, and in addition some combinations of properties produce new models that have not yet been described. We believe these new models will prove to be useful because they are based on declarative properties which programmers desire to be enforced. Finally, we introduce the idea of selecting a consistency model as an on-line activity. Before our work, a shared memory program would run start to finish under a single consistency model. Our unified theory allows the consistency model to change as the program runs while maintaining a consistent definition of what values may be returned by each read.

Lenses in arrangements of pseudo-circles and their applications

Abstract: A collection of simple closed Jordan curves in the plane is called a family of pseudo-circles if any two of its members intersect at most twice. A closed curve composed of two subarcs of distinct pseudo-circles is said to be an empty lens if the closed Jordan region that it bounds does not intersect any other member of the family. We establish a linear upper bound on the number of empty lenses in an arrangement of n pseudo-circles with the property that any two curves intersect precisely twice. We use this bound to show that any collection of nx-monotone pseudo-circles can be cut into O(n8/5) arcs so that any two intersect at most once; this improves a previous bound of O(n5/3) due to Tamaki and Tokuyama. If, in addition, the given collection admits an algebraic representation by three real parameters that satisfies some simple conditions, then the number of cuts can be further reduced to O(n3/2(log n)O(α(s(n))), where α(n) is the inverse Ackermann function, and s is a constant that depends on the the representation of the pseudo-circles. For arbitrary collections of pseudo-circles, any two of which intersect exactly twice, the number of necessary cuts reduces still further to O(n4/3). As applications, we obtain improved bounds for the number of incidences, the complexity of a single level, and the complexity of many faces in arrangements of circles, of pairwise intersecting pseudo-circles, of arbitrary x-monotone pseudo-circles, of parabolas, and of homothetic copies of any fixed simply shaped convex curve. We also obtain a variant of the Gallai--Sylvester theorem for arrangements of pairwise intersecting pseudo-circles, and a new lower bound on the number of distinct distances under any well-behaved norm.

Segmentation problems

Abstract: We study a novel genre of optimization problems, which we call segmentation problems, motivated in part by certain aspects of clustering and data mining. For any classical optimization problem, the corresponding segmentation problem seeks to partition a set of cost vectors into several segments, so that the overall cost is optimized. We focus on two natural and interesting (but MAXSNP-complete) problems in this class, the hypercube segmentation problem and the catalog segmentation problem, and present approximation algorithms for them. We also present a general greedy scheme, which can be specialized to approximate any segmentation problem.

Nonclairvoyant scheduling to minimize the total flow time on single and parallel machines

Abstract: Scheduling a sequence of jobs released over time when the processing time of a job is only known at its completion is a classical problem in CPU scheduling in time sharing operating systems. A widely used measure for the responsiveness of the system is the average flow time of the jobs, that is, the average time spent by jobs in the system between release and completion.The Windows NT and the Unix operating system scheduling policies are based on the Multilevel Feedback algorithm. In this article, we prove that a randomized version of the Multilevel Feedback algorithm is competitive for single and parallel machine systems, in our opinion providing one theoretical validation of the goodness of an idea that has proven effective in practice along the last two decades.The randomized Multilevel Feedback algorithm (RMLF) was first proposed by Kalyanasundaram and Pruhs for a single machine achieving an O(log n log log n) competitive ratio to minimize the average flow time against the on-line adaptive adversary, where n is the number of jobs that are released. We present a version of RMLF working for any number m of parallel machines. We show for RMLF a first O(log n log n/m) competitiveness result against the oblivious adversary on parallel machines. We also show that the same RMLF algorithm surprisingly achieves a tight O(log n) competitive ratio against the oblivious adversary on a single machine, therefore matching the lower bound for this case.

Almost tight upper bounds for vertical decompositions in four dimensions

Abstract: We show that the complexity of the vertical decomposition of an arrangement of n fixed-degree algebraic surfaces or surface patches in four dimensions is O(n4+e), for any e > 0. This improves the best previously known upper bound for this problem by a near-linear factor, and settles a major problem in the theory of arrangements of surfaces, open since 1989. The new bound can be extended to higher dimensions, yielding the bound O(n2d−4+e), for any e > 0, on the complexity of vertical decompositions in dimensions d ≥ 4. We also describe the immediate algorithmic applications of these results, which include improved algorithms for point location, range searching, ray shooting, robot motion planning, and some geometric optimization problems.

Resolution lower bounds for the weak pigeonhole principle

Abstract: We prove that any Resolution proof for the weak pigeonhole principle, with n holes and any number of pigeons, is of length Ω(2ne), (for some global constant e > 0). One corollary is that a certain propositional formulation of the statement NP ⊄ P/poly does not have short Resolution proofs.

On sufficient conditions for unsatisfiability of random formulas

Abstract: A descriptive complexity approach to random 3-SAT is initiated. We show that unsatisfiability of any significant fraction of random 3-CNF formulas cannot be certified by any property that is expressible in Datalog. Combined with the known relationship between the complexity of constraint satisfaction problems and expressibility in Datalog, our result implies that any constraint propagation algorithm working with small constraints will fail to certify unsatisfiability almost always. Our result is a consequence of designing a winning strategy for one of the players in the existential pebble game. The winning strategy makes use of certain extension axioms that we introduce and hold almost surely on a random 3-CNF formula. The second contribution of our work is the connection between finite model theory and propositional proof complexity. To make this connection explicit, we establish a tight relationship between the number of pebbles needed to win the game and the width of the Resolution refutations. As a consequence to our result and the known size--width relationship in Resolution, we obtain new proofs of the exponential lower bounds for Resolution refutations of random 3-CNF formulas and the Pigeonhole Principle.

The security of all RSA and discrete log bits

Abstract: We study the security of individual bits in an RSA encrypted message EN(x). We show that given EN(x), predicting any single bit in x with only a nonnegligible advantage over the trivial guessing strategy, is (through a polynomial-time reduction) as hard as breaking RSA. Moreover, we prove that blocks of O(log log N) bits of x are computationally indistinguishable from random bits. The results carry over to the Rabin encryption scheme.Considering the discrete exponentiation function gx modulo p, with probability 1 − o(1) over random choices of the prime p, the analog results are demonstrated. The results do not rely on group representation, and therefore applies to general cyclic groups as well. Finally, we prove that the bits of ax + b modulo p give hard core predicates for any one-way function f.All our results follow from a general result on the chosen multiplier hidden number problem: given an integer N, and access to an algorithm Px that on input a random a i ZN, returns a guess of the ith bit of ax mod N, recover x. We show that for any i, if Px has at least a nonnegligible advantage in predicting the ith bit, we either recover x, or, obtain a nontrivial factor of N in polynomial time. The result also extends to prove the results about simultaneous security of blocks of O(log log N) bits.

Existential second-order logic over graphs: Charting the tractability frontier

Abstract: Fagin's theorem, the first important result of descriptive complexity, asserts that a property of graphs is in NP if and only if it is definable by an existential second-order formula. In this article, we study the complexity of evaluating existential second-order formulas that belong to prefix classses of existential second-order logic, where a prefix class is the collection of all existential second-order formulas in prenex normal form such that the second-order and the first-order quantifiers obey a certain quantifier pattern. We completely characterize the computational complexity of prefix classes of existential second-order logic in three different contexts: (1) over directed graphs, (2) over undirected graphs with self-loops and (3) over undirected graphs without self-loops. Our main result is that in each of these three contexts a dichotomy holds, that is to say, each prefix class of existential second-order logic either contains sentences that can express NP-complete problems, or each of its sentences expresses a polynomial-time solvable problem. Although the boundary of the dichotomy coincides for the first two cases, it changes, as one moves to undirected graphs without self-loops. The key difference is that a certain prefix class, based on the well-known Ackermann class of first-order logic, contains sentences that can express NP-complete problems over graphs of the first two types, but becomes tractable over undirected graphs without self-loops. Moreover, establishing the dichotomy over undirected graphs without self-loops turns out to be a technically challenging problem that requires the use of sophisticated machinery from graph theory and combinatorics, including results about graphs of bounded tree-width and Ramsey's theorem.

Lower bounds on the bounded coefficient complexity of bilinear maps

Abstract: We prove lower bounds of order n log n for both the problem of multiplying polynomials of degree n, and of dividing polynomials with remainder, in the model of bounded coefficient arithmetic circuits over the complex numbers. These lower bounds are optimal up to order of magnitude. The proof uses a recent idea of R. Raz [Proc. 34th STOC 2002] proposed for matrix multiplication. It reduces the linear problem of multiplying a random circulant matrix with a vector to the bilinear problem of cyclic convolution. We treat the arising linear problem by extending J. Morgenstern's bound [J. ACM 20, pp. 305--306, 1973] in a unitarily invariant way. This establishes a new lower bound on the bounded coefficient complexity of linear forms in terms of the singular values of the corresponding matrix. In addition, we extend these lower bounds for linear and bilinear maps to a model of circuits that allows a restricted number of unbounded scalar multiplications.