Optimal algorithms for semi-online preemptive scheduling problems on two uniform machines

This paper investigates two preemptive semi-online scheduling problems to minimize makespan on two uniform machines. In the first semi-online problem, we know in advance that all jobs have their processing times in between p and rp . In the second semi-online problem, we know the size of the largest job in advance. We design an optimal semi-online algorithm which is optimal for every combination of machine speed ratio and job processing time ratio for the first problem, and an optimal semi-online algorithm for every machine speed ratio for the second problem.Received: 2 December 2003, Published online: 16 January 2004This research is supported by the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of MOE, China, and National Natural Science Foundation of China (10271110).     

Constant-time distributed dominating set approximation

Finding a small dominating set is one of the most fundamental problems of classical graph theory. In this paper, we present a new fully distributed approximation algorithm based on LP relaxation techniques. For an arbitrary, possibly constant parameter k and maximum node degree , our algorithm computes a dominating set of expected size in rounds. Each node has to send messages of size . This is the first algorithm which achieves a non-trivial approximation ratio in a constant number of rounds.Received: 9 September 2003, Accepted: 2 September 2004, Published online: 13 January 2005The work presented in this paper was supported (in part) by the National Competence Center in Research on Mobile Information and Communication Systems (NCCR-MICS), a center supported by the Swiss National Science Foundation under grant number 5005-67322.     

Coincidental extension of scattered context languages

All the results given in the paper hold true. In the proof of Theorem 1, change steps IV, V, and VI to IV. for every , add to P; V. for every , add , , , , , to P; VI. for every , add to P.     

Extending the accommodating function

The applicability of the accommodating function, a relatively new measure for the quality of on-line algorithms, is extended.The standard quality measure for on-line algorithms is the competitive ratio, which is, roughly speaking, the worst case ratio of the on-line performance to the optimal off-line performance. However, for many on-line problems, the competitive ratio gives overly pessimistic results and/or fails to distinguish between algorithms that are known to perform very differently in practice. Many researchers have proposed variations on the competitive ratio to obtain more realistic results. These variations are often tailor-made for specific on-line problems.The concept of the accommodating function applies to any on-line problem with some limited resource, such as bins, seats in a train, or pages in a cache. If a limited amount n of some resource is available, the accommodating function is the competitive ratio when input sequences are restricted to those for which the amount of resources suffices for an optimal off-line algorithm. For all resource bounded problems, the standard competitive ratio is .The accommodating function was originally used only for . We focus on , observe that the function now appears interesting for a greater variety of problems, and use it to make new distinctions between known algorithms and to find new ones.Received: 17 September 2002 / 12 June 2003Supported in part by the IST Programme of the EU under contract number IST-1999-14186 (ALCOM-FT) and in part by the Danish Natural Science Research Council (SNF).A preliminary version of this paper appeared in the Eighth Annual International Computing and Combinatorics Conference, Lecture Notes in Computer Science 2387: 87-96, Springer-Verlag, 2002.     

Element distinctness on one-tape Turing machines: a complete solution

We give a complete characterization of the complexity of the element distinctness problem for n elements of bits each on deterministic and nondeterministic one-tape Turing machines. We present an algorithm running in time for deterministic machines and nondeterministic solutions that are of time complexity . For elements of logarithmic size , on nondeterministic machines, these results close the gap between the known lower bound and the previous upper bound . Additional lower bounds are given to show that the upper bounds are optimal for all other possible relations between m and n. The upper bounds employ hashing techniques, while the lower bounds make use of the communication complexity of set disjointness.Received: 23 April 2001, Published online: 2 September 2003Holger Petersen: Supported by Deutsche Akademie der Naturforscher Leopoldina, grant number BMBF-LPD 9901/8-1 of Bundesministerium für Bildung und Forschung.     

Lower bounds for sense of direction in regular graphs

A graph G with n vertices and maximum degree cannot be given weak sense of direction using less than colours. It is known that n colours are always sufficient, and it was conjectured that just are really needed, that is, one more colour is sufficient. Nonetheless, it has been shown [3] that for sufficiently large n there are graphs requiring more colours than . In this paper, using recent results in asymptotic graph enumeration, we show that (surprisingly) the same bound holds for regular graphs. We also show that colours are necessary, where d _G is the degree of G.Received: April 2002, Accepted: April 2003, Sebastiano Vigna: Partially supported by the Italian MURST (Finanziamento di iniziative di ricerca diffusa condotte da parte di giovani ricercatori).The results of this paper appeared in a preliminary form in Distributed Computing. 14th International Conference, DISC 2000, Springer-Verlag, 2000.     

Condition-based consensus solvability: a hierarchy of conditions and efficient protocols

The condition-based approach for consensus solvability consists of identifying sets of input vectors, called conditions, for which there exists an asynchronous protocol solving consensus despite the occurrence of up to f process crashes. This paper investigates , the largest set of conditions which allow us to solve the consensus problem in an asynchronous shared memory system.The first part of the paper shows that is made up of a hierarchy of classes of conditions, where d is a parameter (called degree of the condition), starting with and ending with d = 0, where . We prove that each one is strictly contained in the previous one: . Various properties of the hierarchy are also derived. It is shown that a class can be characterized in two equivalent but complementary ways: one is convenient for designing protocols while the other is for analyzing the class properties. The paper also defines a linear family of conditions that can be used to derive many specific conditions. In particular, for each d, two natural conditions are presented.The second part of the paper is devoted to the design of efficient condition-based protocols. A generic condition-based protocol is presented. This protocol can be instantiated with any condition C, , and requires at most shared memory read/write operations per process in the synchronization part of the protocol. Thus, the value (f-d) represents the difficulty of the class . An improvement of the protocol for the conditions in is also presented.Received: 15 November 2001, Accepted: 15 April 2003, Published online: 6 February 2004Parts of it have previously appeared in [23] and [25].     

Computational challenges in bounded model checking

We describe several observations regarding the completeness and the complexity of bounded model checking and propose techniques to solve some of the associated computational challenges. We begin by defining the completeness threshold ( ) problem: for every finite model M and an LTL property , there exists a number such that if there is no counterexample to in M of length or less, then M . Finding this number, if it is sufficiently small, offers a practical method for making bounded model checking complete. We describe how to compute an overapproximation to for a general LTL property using Büchi automata, following the Vardi–Wolper LTL model checking framework. This computation is based on finding the initialized diameter and initialized recurrence-diameter (the longest loop-free path from an initial state) of the product automaton. We show a method for finding a recurrence diameter with a formula of size O(klogk) (or O(k(logk)²) in practice), where k is the attempted depth, which is an improvement compared to the previously known method that requires a formula of size in O(k²). Based on the value of , we prove that the complexity of standard SAT-based BMC is doubly exponential and that, consequently, there is a complexity gap of an exponent between this procedure and standard LTL model checking. We discuss ways to bridge this gap.     

Querying high-dimensional data in single-dimensional space

In this paper, we propose a new tunable index scheme, called iMinMax( ), that maps points in high-dimensional spaces to single-dimensional values determined by their maximum or minimum values among all dimensions. By varying the tuning knob, , we can obtain different families of iMinMax structures that are optimized for different distributions of data sets. The transformed data can then be indexed using existing single-dimensional indexing structures such as the B ⁺ -trees. Queries in the high-dimensional space have to be transformed into queries in the single-dimensional space and evaluated there. We present efficient algorithms for evaluating window queries as range queries on the single-dimensional space. We conducted an extensive performance study to evaluate the effectiveness of the proposed schemes. Our results show that iMinMax( ) outperforms existing techniques, including the Pyramid scheme and VA-file, by a wide margin. We then describe how iMinMax could be used in approximate K-nearest neighbor (KNN) search, and we present a comparative study against the recently proposed iDistance, a specialized KNN indexing method.Received: 21 May 2000, Revised: 14 March 2002, Published online: 8 April 2004Edited by: M. Kitsuregawa.     

Efficient low-contention asynchronous consensus with the value-oblivious adversary scheduler

We consider asynchronous consensus in the shared-memory setting. We present the first efficient low-contention consensus algorithm for the weak-adversary-scheduler model. The algorithm achieves consensus in total work and (hot-spot) contention, both expected and with high probability. The algorithm assumes the value-oblivious scheduler, which is defined in the paper. Previous efficient consensus algorithms for weak adversaries suffer from memory contention.Yonatan Aumann: This work was partially completed while theauthor was at Harvard University, supported in part by ONRcontract ONR-N00014-91-J-1981.Michael A. Bender: This work was supported inpart by HRL Laboratories, Sandia National Laboratories, and NSF GrantsACI-032497, CCR-0208670, and EIA-0112849. This work was partiallycompleted while the author was at Harvard University, supported inpart by NSF grants CCR-9700365, CCR-9504436, and CCR-9313775.An early version of this paper was presented in the 23rd International Colloquium on Automata, Languages, and Programming (ICALP 96) [8].     

The Expected Size of the Rule k Dominating Set

Dai, Li, and Wu proposed Rule k, a localized approximation algorithm that attempts to find a small connected dominating set in a graph. In this paper we consider the "average-case" performance of two closely related versions of Rule k for the model of random unit disk graphs constructed from n random points in an square. We show that if and then for both versions of Rule k, the expected size of the Rule k dominating set is as It follows that, for in a suitable range, the expected size of the Rule k dominating sets are within a constant factor of the optimum.     

Cardinal interpolation with periodic polysplines on strips

Abstract  We obtain a multivariate extension of a classical result of Schoenberg on cardinal spline interpolation. Specifically, we prove the existence of a unique function in , polyharmonic of order p on each strip , , and periodic in its last n variables, whose restriction to the parallel hyperplanes , , coincides with a prescribed sequence of n-variate periodic data functions satisfying a growth condition in . The constructive proof is based on separation of variables and on Micchelli’s theory of univariate cardinal -splines. Keywords: cardinal -splines, polyharmonic functions, multivariable interpolation Mathematics Subject Classification (2000): 41A05, 41A15, 41A63     

Safety assurance via on-line monitoring

This paper proposes a new approach and new techniques for on-line monitoring of concurrent programs to ensure that some of their safety properties are not violated. The techniques modify erroneous systems, which violate a certain safety property, into new systems which satisfy the safety property. It does so by adding a new layer that controls the scheduling of steps in the system. We formally characterize the relationship between the erroneous and the new system. Safety monitors for mutual-exclusion, -exclusion, and the producer-consumer tasks are presented. Proofs for the mutual-exclusion task and the -exclusion task are presented to demonstrate the applicability of our approach.Received: May 2001, Accepted: December 2002, An extended abstract of this work appears in the Proceedings of the fifth International Symposium on Autonomous Decentralized Systems (ISADS) 2001. Part of this work was done while the first author visited Wayne State University.     

Congruences et Automorphismes des Automates Finis

Summary We study a class of congruences of strongly connected finite automata, called the group congruences, which may be defined in this way: every element fixing any class of the congruence induces a permutation on this class. These congruences form an ideal of the lattice of all congruences of the automaton and we study the group associated with the maximal group congruence (maximal induced group) with respect to the Suschkevitch group of the transition monoid of . The transitivity equivalence of the subgroups of the automorphism group of are found to be the group congruences associated with regular groups, which form also in ideal of the lattice of congruences of . We then characterize the automorphism group of with respect to the maximal induced group. As an application, we show that, given a group G and an automaton , there exists an automaton whose automorphism group is isomorphic to G and such that the quotient by the automorphism congruence is .     

The Effect of Faults on Network Expansion

We study the problem of how resilient networks are to node faults. Specifically, we investigate the question of how many faults a network can sustain and still contain a large (i.e., linear-sized) connected component with approximately the same expansion as the original fault-free network. We use a pruning technique that culls away those parts of the faulty network that have poor expansion. The faults may occur at random or be caused by an adversary. Our techniques apply in either case. In the adversarial setting we prove that for every network with expansion a large connected component with basically the same expansion as the original network exists for up to a constant times faults. We show this result is tight in the sense that every graph G of size n and uniform expansion can be broken into components of size o(n) with faults. Unlike the adversarial case, the expansion of a graph gives a very weak bound on its resilience to random faults. While it is the case, as before, that there are networks of uniform expansion that are not resilient against a fault probability of a constant times it is also observed that there are networks of uniform expansion that are resilient against a constant fault probability. Thus, we introduce a different parameter, called the span of a graph, which gives us a more precise handle on the maximum fault probability. We use the span to show the first known results for the effect of random faults on the expansion of d-dimensional meshes.     

On the Power of Unambiguity in Alternating Machines

Unambiguity in alternating Turing machines has received considerable attention in the context of analyzing globally unique games by Aida et al. [ACRW] and in the design of efficient protocols involving globally unique games by Crasmaru et al. [CGRS]. This paper explores the power of unambiguity in alternating Turing machines in the following settings: 1. We show that unambiguity-based hierarchies-AUPH, UPH, and UPH-are infinite in some relativized world. For each , we construct another relativized world where the unambiguity-based hierarchies collapse so that they have exactly k distinct levels and their k-th levels coincide with PSPACE. These results shed light on the relativized power of the unambiguity-based hierarchies, and parallel the results known for the case of the polynomial hierarchy. 2. For every , we define the bounded-level unambiguous alternating solution class UAS(k) as the class of all sets L for which there exists a polynomial-time alternating Turing machine N, which need not be unambiguous on every input, with at most k alternations such that if and only if x is accepted unambiguously by N. We construct a relativized world where, for all and . 3. Finally, we show that robustly k-level unambiguous alternating polynomial-time Turing machines, i.e., polynomial-time alternating Turing machines that for every oracle have k alternating levels and are unambiguous, accept languages that are computable in , for every oracle A. This generalizes a result of Hartmanis and Hemachandra [HH].     

Representing and Building Models for Decidable Subclasses of Equational Clausal Logic

We develop new techniques for the automated construction of models for subclasses of equational clausal logic. The models are represented by some specific class of rewrite rules. We show that the evaluation of arbitrary first-order formulae in these interpretations is a decidable problem. As an example of an application, we consider the class , a decidable subclass of first-order clausal logic without equality. In the equational case, is undecidable, but it is known to be decidable if all the equational literals are ground. We extend this result to a class of clause sets possibly containing nonground equational literals. The algorithms for extracting models from positively disconnected saturated sets proposed by Fermüller and Leitsch are extended in order to handle the full ordering restrictions of the resolution/paramodulation calculus.     

GCD of Random Linear Combinations

We show that for arbitrary positive integers with probability the gcd of two linear combinations of these integers with rather small random integer coefficients coincides with This naturally leads to a probabilistic algorithm for computing the gcd of several integers, with probability via just one gcd of two numbers with about the same size as the initial data (namely the above linear combinations). This algorithm can be repeated to achieve any desired confidence level.     

Self-embedded context-free grammars with regular counterparts

In general, it is undecidable if an arbitrary context-free grammar has a regular solution. Past work has focused on special cases, such as one-letter grammars, non self-embedded grammars and the finite-language grammars, for which regular counterparts have been proven to exist. However, little is known about grammars with the self-embedded property. Using systems of equations, we highlight a number of subclasses of grammars, with self-embeddedness terms, such as and , that can still have regular languages as solutions. Constructive proofs that allow these subclasses of context-free grammars to be transformed to regular expressions are provided. We also point out a subclass of context-free grammars that is inherently non-regular. Our latest results can help demarcate more precisely the known boundaries between the regular and non-regular languages, within the context-free domain.Received: 17 January 2003, Published online: 17 February 2004Stefan Andrei: stefan@infoiasi.ro