Lower bound for scalable Byzantine Agreement

We consider the problem of computing Byzantine Agreement in a synchronous network with n processors, each with a private random string, where each pair of processors is connected by a private communication line. The adversary is malicious and non-adaptive, i.e., it must choose the processors to corrupt at the start of the algorithm. Byzantine Agreement is known to be computable in this model in an expected constant number of rounds. We consider a scalable model where in each round each uncorrupt processor can send to any set of log n other processors and listen to any set of log n processors. We define the loss of an execution to be the number of uncorrupt processors whose output does not agree with the output of the majority of uncorrupt processors. We show that if there are t corrupt processors, then any randomised protocol which has probability at least 1/2 + 1/ logn of loss less than requires at least f rounds. This also shows that lossless protocols require both rounds, and for at least one uncorrupt processor to send messages during the protocol.     

Computing Population Variance and Entropy under Interval Uncertainty: Linear-Time Algorithms

In statistical analysis of measurement results it is often necessary to compute the range of the population variance when we only know the intervals of possible values of the x _i . While can be computed efficiently, the problem of computing is, in general, NP-hard. In our previous paper “Population Variance under Interval Uncertainty: A New Algorithm” (Reliable Computing 12 (4) (2006), pp. 273–280) we showed that in a practically important case we can use constraints techniques to compute in time O(n · log(n)). In this paper we provide new algorithms that compute (in all cases) and (for the above case) in linear time O(n). Similar linear-time algorithms are described for computing the range of the entropy when we only know the intervals of possible values of probabilities p _i . In general, a statistical characteristic ƒ can be more complex so that even computing ƒ can take much longer than linear time. For such ƒ, the question is how to compute the range in as few calls to ƒ as possible. We show that for convex symmetric functions ƒ, we can compute in n calls to ƒ.     

Translational lemmas for DLOGTIME-uniform circuits,alternating TMs,and PRAMs

We present translational lemmas for the three standard models of parallel computation, and apply them to obtain tight hierarchy results. It is shown that, for arbitrarily small rational constant , (i) there is a language which can be accepted by a -uniform circuit family of depth and size but not by any -uniform circuit family of depth and size , (ii) there is a language which can be accepted by a -time -space ATM with l worktapes but not by any -time -space ATM with the same l worktapes if the number of tape symbols is fixed, and (iii) there is a language which can be accepted by a -time PRAM with processors but not by any -time PRAM with processors. Here, c > 0, d ≥ 1, r ₁ > 1, and r ₂ ≥ 1 are arbitrary rational constants, and l ≥ 2 is an arbitrary integer. Preliminary versions of different parts of this paper appeared in Proc. MCU 2004 (LNCS 3354) and Proc. FCT 2005 (LNCS 3623).     

Optimal Deterministic Broadcasting in Known Topology Radio Networks

We consider deterministic broadcasting in radio networks whose nodes have full topological information about the network. The aim is to design a polynomial algorithm, which, given a graph G with source s, produces a fast broadcast scheme in the radio network represented by G. The problem of finding a fastest broadcast scheme for a given graph is NP-hard, hence it is only possible to get an approximation algorithm. We give a deterministic polynomial algorithm which produces a broadcast scheme of length , for every n-node graph of diameter D, thus improving a result of Gąsieniec et al. (PODC 2005) [17] and solving a problem stated there. Unless the inclusion NP BPTIME( holds, the length of a polynomially constructible deterministic broadcast scheme is optimal.A preliminary version of this paper (with a weaker result) appeared in the Proc. 7th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX’2004), August 2004, Harvard University, Cambridge, USA, LNCS 3122, 171–182. Research of the second author supported in part by NSERC discovery grant and by the Research Chair in Distributed Computing of the Université du Québec en Outaouais. Part of this work was done during the second author’s visit at the Max-Planck-Institut für Informatik.     

Fuzzy logic models in a category of fuzzy relations

We investigate interpretations of formulas ψ in a first order fuzzy logic in models which are based on objects of a category SetR(Ω) which consists of Ω-sets, i.e. sets with similarity relations with values in a complete MV-algebra Ω and with morphisms defined as special fuzzy relations between Ω-sets. The interpretations are then morphisms in a category SetR(Ω) from some Ω-set to the object . We define homomorphisms between models in a category SetR(Ω) and we prove that if is a (special) homomorphism of models in a category SetR(Ω) then there is a relation between interpretations of a formula ψ in models . Supported by MSM6198898701, grant 201/07/0191 of GAČR and grant 1M0572.     

Quantum Error Correction of Time-correlated Errors

The complexity of the error correction circuitry forces us to design quantum error correction codes capable of correcting a single error per error correction cycle. Yet, time-correlated error are common for physical implementations of quantum systems; an error corrected during the previous cycle may reoccur later due to physical processes specific for each physical implementation of the qubits. In this paper, we study quantum error correction for a restricted class of time-correlated errors in a spin-boson model. The algorithm we propose allows the correction of two errors per error correction cycle, provided that one of them is time-correlated. The algorithm can be applied to any stabilizer code when the two logical qubits and are entangled states of 2 ⁿ basis states in .      

On the Relation between States and Maps in Infinite Dimensions

Relations between states and maps, which are known for quantum systems in finitedimensional Hilbert spaces, are formulated rigorously in geometrical terms with no use of coordinate (matrix) interpretation. In a tensor product realization they are represented simply by a permutation of factors. This leads to natural generalizations for infinite-dimensional Hilbert spaces and a simple proof of a generalized Choi Theorem. The natural framework is based on spaces of Hilbert-Schmidt operators and the corresponding tensor products of Hilbert spaces. It is proved that the corresponding isomorphisms cannot be naturally extended to compact (or bounded) operators, nor reduced to the trace-class operators. On the other hand, it is proven that there is a natural continuous map from trace-class operators on (with the nuclear norm) into compact operators mapping the space of all bounded operators on into trace class operators on (with the operator-norm). Also in the infinite-dimensional context, the Schmidt measure of entanglement and multipartite generalizations of state-maps relations are considered in the paper.     

Uniform stabilization for linear systems with persistency of excitation: the neutrally stable and the double integrator cases

Consider the controlled system dx/dt = Ax + α(t)Bu where the pair (A, B) is stabilizable and α(t) takes values in [0, 1] and is persistently exciting, i.e., there exist two positive constants μ, T such that, for every t ≥ 0, ${\int_t^{t+T}\alpha(s){\rm d}s \geq \mu}Consider the controlled system dx/dt = Ax + α(t)Bu where the pair (A, B) is stabilizable and α(t) takes values in [0, 1] and is persistently exciting, i.e., there exist two positive constants μ, T such that, for every t ≥ 0, . In particular, when α(t) becomes zero the system dynamics switches to an uncontrollable system. In this paper, we address the following question: is it possible to find a linear time-invariant state-feedback u = Kx, with K only depending on (A, B) and possibly on μ, T, which globally asymptotically stabilizes the system? We give a positive answer to this question for two cases: when A is neutrally stable and when the system is the double integrator. Notation  A continuous function is of class , if it is strictly increasing and is of class if it is continuous, non-increasing and tends to zero as its argument tends to infinity. A function is said to be a class -function if, for any t ≥ 0, and for any s ≥ 0. We use |·| for the Euclidean norm of vectors and the induced L ₂-norm of matrices.     

Strong non-standard completeness for fuzzy logics

In this paper we are going to introduce the notion of strong non-standard completeness (SNSC) for fuzzy logics. This notion naturally arises from the well known construction by ultraproduct. Roughly speaking, to say that a logic is strong non-standard complete means that, for any countable theory Γ over and any formula φ such that , there exists an evaluation e of -formulas into a -algebra such that the universe of is a non-Archimedean extension of the real unit interval [0,1], e is a model for Γ, but e(φ) < 1. Then we will apply SNSC to prove that various modal fuzzy logics allowing to deal with simple and conditional probability of infinite-valued events are complete with respect to classes of models defined starting from non-standard measures, that is measures taking value in .     

A simple population protocol for fast robust approximate majority

We describe and analyze a 3-state one-way population protocol to compute approximate majority in the model in which pairs of agents are drawn uniformly at random to interact. Given an initial configuration of x’s, y’s and blanks that contains at least one non-blank, the goal is for the agents to reach consensus on one of the values x or y. Additionally, the value chosen should be the majority non-blank initial value, provided it exceeds the minority by a sufficient margin. We prove that with high probability n agents reach consensus in O(n log n) interactions and the value chosen is the majority provided that its initial margin is at least . This protocol has the additional property of tolerating Byzantine behavior in of the agents, making it the first known population protocol that tolerates Byzantine agents.     

On the complexity of graph self-assembly in accretive systems

We study the complexity of the Accretive Graph Assembly Problem (). An instance of consists of an edge-weighted graph G, a seed vertex in G, and a temperature τ. The goal is to determine if the graph G can be assembled by a sequence of vertex additions starting from the seed vertex. The edge weights model the forces of attraction and repulsion, and determine which vertices can be added to a partially assembled graph at the given temperature. A vertex can be added when the total weight to its already built neighbors in the graph is at least τ. The assembly process is sequential meaning that only one vertex can be added at a time. Our first result is that is NP-complete even on planar graphs with maximum degree 3 when edges have only two different types of weights. This resolves the complexity of in the sense that the problem is poly-time solvable when either the maximum degree is at most 2 or the number of distinct edge weights is one, and is NP-complete otherwise. Our second result is a dichotomy theorem that completely characterizes the complexity of on graphs with maximum degree 3 and two distinct weights: w _p and w _n . We give a simple system of linear constraints on w _p , w _n , and τ that determines whether the problem is NP-complete or is poly-time solvable. In the process of establishing this dichotomy, we give a poly-time algorithm to solve a non-trivial class of Finally, we consider the optimization version of where the goal is to assemble a largest-possible induced subgraph of the given input graph. We show that even on graphs that can be assembled and have maximum degree 3, it is NP-hard to assemble a (1/n ^1-ε)-fraction of the input graph for any here n denotes the number of vertices in G.     

A new bound for the D0L sequence equivalence problem

The D0L sequence equivalence problem consists of deciding, given two morphisms , and a word , whether or not g ⁱ (w) = h ⁱ (w) for all i ≥ 0. We show that in case of smooth and loop-free morphisms, to decide the D0L sequence equivalence problem, it suffices to consider the terms of the sequences with , where n is the cardinality of X.     

A Novel Class of Symmetric and Nonsymmetric Periodizing Variable Transformations for Numerical Integration

Variable transformations for numerical integration have been used for improving the accuracy of the trapezoidal rule. Specifically, one first transforms the integral via a variable transformation that maps [0,1] to itself, and then approximates the resulting transformed integral by the trapezoidal rule. In this work, we propose a new class of symmetric and nonsymmetric variable transformations which we denote , where r and s are positive scalars assigned by the user. A simple representative of this class is . We show that, in case , or but has algebraic (endpoint) singularities at x = 0 and/or x = 1, the trapezoidal rule on the transformed integral produces exceptionally high accuracies for special values of r and s. In particular, when and we employ , the error in the approximation is (i) O(h ^r ) for arbitrary r and (ii) O(h ^2r ) if r is a positive odd integer at least 3, h being the integration step. We illustrate the use of these transformations and the accompanying theory with numerical examples.      

Many-to-Many Communication in Radio Networks

Radio networks model wireless data communication when the bandwidth is limited to one wave frequency. The key restriction of such networks is mutual interference of packets arriving simultaneously at a node. The many-to-many (m2m) communication primitive involves p participant nodes from among n nodes in the network, where the distance between any pair of participants is at most d. The task is to have all the participants get to know all the input messages. We consider three cases of the m2m communication problem. In the ad-hoc case, each participant knows only its name and the values of n, p and d. In the partially centralized case, each participant knows the topology of the network and the values of p and d, but does not know the names of the other participants. In the centralized case, each participant knows the topology of the network and the names of all the participants. For the centralized m2m problem, we give deterministic protocols, for both undirected and directed networks, working in time, which is provably optimal. For the partially centralized m2m problem, we give a randomized protocol for undirected networks working in time with high probability (whp), and we show that any deterministic protocol requires time. For the ad-hoc m2m problem, we develop a randomized protocol for undirected networks that works in time whp. We show two lower bounds for the ad-hoc m2m problem. One lower bound states that any randomized protocol for the m2m ad hoc problem requires expected time. Another lower bound states that for any deterministic protocol for the m2m ad hoc problem, there is a network on which the protocol requires time when n−p(n)=Ω(n) and d>1, and that it requires Ω(n) time when n−p(n)=o(n). The results of this paper appeared in a preliminary form in “On many-to-many communication in packet radio networks” in Proceedings of the 10th Conference on Principles of Distributed Systems (OPODIS), Bordeaux, France, 2006, Lecture Notes in Computer Science 4305, Springer, Heidelberg, pp. 258–272. The work of B.S. Chlebus was supported by NSF Grant 0310503.     

Efficient Solution of A, x ( k )?= b ( k ) Using A ?1

In this work, we consider the problem of solving , , where b ^(k+1) = f(x ^(k)). We show that when A is a full matrix and , where depends on the specific software and hardware setup, it is faster to solve for by explicitly evaluating the inverse matrix A ⁻¹ rather than through the LU decomposition of A. We also show that the forward error is comparable in both methods, regardless of the condition number of A.     

Adaptive sorting: an information theoretic perspective

We present two algorithms that are near optimal with respect to the number of inversions present in the input. One of the algorithms is a variation of insertion sort, and the other is a variation of merge sort. The number of comparisons performed by our algorithms, on an input sequence of length n that has I inversions, is at most . Moreover, both algorithms have implementations that run in time . All previously published algorithms require at least comparisons for some c > 1. M. L. Fredman was supported in part by NSF grant CCR-9732689.