Sublinear Bounds for Randomized Leader Election

This paper concerns randomized leader election in synchronous distributed networks. A distributed leader election algorithm is presented for complete n-node networks that runs in O(1) rounds and (with high probability) takes only O(n-vlog3/2n) messages to elect a unique leader (with high probability). This algorithm is then extended to solve leader election on any connected non-bipartiten-node graph G in O(t(G)) time and O(t(G)n-vlog3/2n) messages, where t(G) is the mixing time of a random walk on G. The above result implies highly efficient (sublinear running time and messages) leader election algorithms for networks with small mixing times, such as expanders and hypercubes. In contrast, previous leader election algorithms had at least linear message complexity even in complete graphs. Moreover, super-linear message lower bounds are known for time-efficientdeterministic leader election algorithms. Finally, an almost-tight lower bound is presented for randomized leader election, showing that O(n-v) messages are needed for any O(1) time leader election algorithm which succeeds with high probability. It is also shown that O(n 1/3) messages are needed by any leader election algorithm that succeeds with high probability, regardless of the number of the rounds. We view our results as a step towards understanding the randomized complexity of leader election in distributed networks.

Sublinear bounds for randomized leader election

This paper concerns randomized leader election in synchronous distributed networks. A distributed leader election algorithm is presented for complete n-node networks that runs in O(1) rounds and (with high probability) uses only O(√ √nlog^3/2n) messages to elect a unique leader (with high probability). When considering the "explicit" variant of leader election where eventually every node knows the identity of the leader, our algorithm yields the asymptotically optimal bounds of O(1) rounds and O(. n) messages. This algorithm is then extended to one solving leader election on any connected non-bipartite n-node graph G in O(τ(. G)) time and O(τ(G)n√log^3/2n) messages, where τ(. G) is the mixing time of a random walk on G. The above result implies highly efficient (sublinear running time and messages) leader election algorithms for networks with small mixing times, such as expanders and hypercubes. In contrast, previous leader election algorithms had at least linear message complexity even in complete graphs. Moreover, super-linear message lower bounds are known for time-efficient deterministic leader election algorithms. Finally, we present an almost matching lower bound for randomized leader election, showing that Ω(n) messages are needed for any leader election algorithm that succeeds with probability at least 1/. e+. ε, for any small constant ε. >. 0. We view our results as a step towards understanding the randomized complexity of leader election in distributed networks.

Bounds for the signless laplacian energy

The energy of a graph G is the sum of the absolute values of the eigenvalues of the adjacency matrix of G. The Laplacian (respectively, the signless Laplacian) energy of G is the sum of the absolute values of the differences between the eigenvalues of the Laplacian (respectively, signless Laplacian) matrix and the arithmetic mean of the vertex degrees of the graph. In this paper, among some results which relate these energies, we point out some bounds to them using the energy of the line graph of G. Most of these bounds are valid for both energies, Laplacian and signless Laplacian. However, we present two new upper bounds on the signless Laplacian which are not upper bounds for the Laplacian energy. © 2010 Elsevier Inc. All rights reserved.

Hausdorff dimension bounds for smoothness of holonomies for codimension 1 hyperbolic dynamics

We prove that the stable holonomies of a proper codimension 1 attractor Λ, for a Cr diffeomorphism f of a surface, are not C1+θ for θ greater than the Hausdorff dimension of the stable leaves of f intersected with Λ. To prove this result we show that there are no diffeomorphisms of surfaces, with a proper codimension 1 attractor, that are affine on a neighbourhood of the attractor and have affine stable holonomies on the attractor.

Bounds on edit metric codes with combinatorial DNA constraints

The design of a large and reliable DNA codeword library is a key problem in DNA based computing. DNA codes, namely sets of fixed length edit metric codewords over the alphabet {A, C, G, T}, satisfy certain combinatorial constraints with respect to biological and chemical restrictions of DNA strands. The primary constraints that we consider are the reverse--complement constraint and the fixed GC--content constraint, as well as the basic edit distance constraint between codewords. We focus on exploring the theory underlying DNA codes and discuss several approaches to searching for optimal DNA codes. We use Conway's lexicode algorithm and an exhaustive search algorithm to produce provably optimal DNA codes for codes with small parameter values. And a genetic algorithm is proposed to search for some sub--optimal DNA codes with relatively large parameter values, where we can consider their sizes as reasonable lower bounds of DNA codes. Furthermore, we provide tables of bounds on sizes of DNA codes with length from 1 to 9 and minimum distance from 1 to 9.

Distribution-Free Bounds for Serial Correlation Coefficients in Heteroskedastic Symmetric Time Series

We consider the problem of testing whether the observations X1, ..., Xn of a time series are independent with unspecified (possibly nonidentical) distributions symmetric about a common known median. Various bounds on the distributions of serial correlation coefficients are proposed: exponential bounds, Eaton-type bounds, Chebyshev bounds and Berry-Esséen-Zolotarev bounds. The bounds are exact in finite samples, distribution-free and easy to compute. The performance of the bounds is evaluated and compared with traditional serial dependence tests in a simulation experiment. The procedures proposed are applied to U.S. data on interest rates (commercial paper rate).

Extension of Wu-Peters bounds to Catmull-Clark and 4-8 subdivision

La méthode de subdivision Catmull-Clark ainsi que la méthode de subdivision Loop sont des normes industrielle de facto. D'autre part, la méthode de subdivision 4-8 est bien adaptée à la subdivision adaptative, parce que cette méthode augmente le nombre de faces ou de sommets par seulement un facteur de 2 à chaque raffinement. Cela promet d'être plus pratique pour atteindre un niveau donné de précision. Dans ce mémoire, nous présenterons une méthode permettant de paramétrer des surfaces de subdivision de la méthode Catmull-Clark et de la méthode 4-8. Par conséquent, de nombreux algorithmes mis au point pour des surfaces paramétriques pourrant être appliqués aux surfaces de subdivision Catmull-Clark et aux surfaces de subdivision 4-8. En particulier, nous pouvons calculer des bornes garanties et réalistes sur les patches, un peu comme les bornes correspondantes données par Wu-Peters pour la méthode de subdivision Loop.

«School choice with controlled choice constraints: hard bounds versus soft bounds»

Controlled choice over public schools attempts giving options to parents while maintaining diversity, often enforced by setting feasibility constraints with hard upper and lower bounds for each student type. We demonstrate that there might not exist assignments that satisfy standard fairness and non-wastefulness properties; whereas constrained non-wasteful assignments which are fair for same type students always exist. We introduce a "controlled" version of the deferred acceptance algorithm with an improvement stage (CDAAI) that finds a Pareto optimal assignment among such assignments. To achieve fair (across all types) and non-wasteful assignments, we propose the control constraints to be interpreted as soft bounds-flexible limits that regulate school priorities. In this setting, a modified version of the deferred acceptance algorithm (DAASB) finds an assignment that is Pareto optimal among fair assignments while eliciting true preferences. CDAAI and DAASB provide two alternative practical solutions depending on the interpretation of the control constraints. JEL C78, D61, D78, I20.

Characterization of Probability Distributions by Variance Bounds and its Applications

Department of Statistics, Cochin University of Science and Technology

Lower Bounds for Nonforgetting Restarting Automata and CD-Systems of Restarting Automata

The nonforgetting restarting automaton is a generalization of the restarting automaton that, when executing a restart operation, changes its internal state based on the current state and the actual contents of its read/write window instead of resetting it to the initial state. Another generalization of the restarting automaton is the cooperating distributed system (CD-system) of restarting automata. Here a finite system of restarting automata works together in analyzing a given sentence, where they interact based on a given mode of operation. As it turned out, CD-systems of restarting automata of some type X working in mode =1 are just as expressive as nonforgetting restarting automata of the same type X. Further, various types of determinism have been introduced for CD-systems of restarting automata called strict determinism, global determinism, and local determinism, and it has been shown that globally deterministic CD-systems working in mode =1 correspond to deterministic nonforgetting restarting automata. Here we derive some lower bound results for some types of nonforgetting restarting automata and for some types of CD-systems of restarting automata. In this way we establish separations between the corresponding language classes, thus providing detailed technical proofs for some of the separation results announced in the literature.

Risk Bounds for Mixture Density Estimation

In this paper we focus on the problem of estimating a bounded density using a finite combination of densities from a given class. We consider the Maximum Likelihood Procedure (MLE) and the greedy procedure described by Li and Barron. Approximation and estimation bounds are given for the above methods. We extend and improve upon the estimation results of Li and Barron, and in particular prove an $O(\\frac{1}{\\sqrt{n}})$ bound on the estimation error which does not depend on the number of densities in the estimated combination.

Lower Bounds for Achieving Synchronous Early Stopping Consensus with Orderly Crash Failures

In this paper, we discuss the consensus problem for synchronous distributed systems with orderly crash failures. For a synchronous distributed system of n processes with up to t crash failures and f failures actually occur, first, we present a bivalency argument proof to solve the open problem of proving the lower bound, min (t + 1, f + 2) rounds, for early-stopping synchronous consensus with orderly crash failures, where t < n - 1. Then, we extend the system model with orderly crash failures to a new model in which a process is allowed to send multiple messages to the same destination process in a round and the failing processes still respect the order specified by the protocol in sending messages. For this new model, we present a uniform consensus protocol, in which all non-faulty processes always decide and stop immediately by the end of f + 1 rounds. We prove that the lower bound of early stopping protocols for both consensus and uniform consensus are f + 1 rounds under the new model, and our proposed protocol is optimal.