Fast computation by population protocols with a leader

The standard population protocol model assumes that when two agents interact, each observes the entire state of the other agent. We initiate the study of $\textit{message complexity}$ for population protocols, where the state of an agent is divided into an externally-visible $\textit{message}$ and an internal component, where only the message can be observed by the other agent in an interaction. We consider the case of $O(1)$ message complexity. When time is unrestricted, we obtain an exact characterization of the stably computable predicates based on the number of internal states $s(n)$: If $s(n) = o(n)$ then the protocol computes semilinear predicates (unlike the original model, which can compute non-semilinear predicates with $s(n) = O(\log n)$), and otherwise it computes a predicate decidable by a nondeterministic $O(n \log s(n))$-space-bounded Turing machine. We then introduce novel $O(\mathrm{polylog}(n))$ expected time protocols for junta/leader election and general purpose broadcast correct with high probability, and approximate and exact population size counting correct with probability 1. Finally, we show that the main constraint on the power of bounded-message-size protocols is the size of the internal states: with unbounded internal states, any computable function can be computed with probability 1 in the limit by a protocol that uses only $\textit{1-bit}$ messages.

Consider an n-vertex graph G = (V,E) of maximum degree Δ, and suppose that each vertex v ∈ V hosts a processor. The processors are allowed to communicate only with their neighbors in G. The communication is synchronous, i.e., it proceeds in discrete rounds.

The standard population protocol model assumes that when two agents interact, each observes the entire state of the other agent. We initiate the study of $\textit{message complexity}$ for population protocols, where the state of an agent is divided into an externally-visible $\textit{message}$ and an internal component, where only the message can be observed by the other agent in an interaction. We consider the case of $O(1)$ message complexity. When time is unrestricted, we obtain an exact characterization of the stably computable predicates based on the number of internal states $s(n)$: If $s(n) = o(n)$ then the protocol computes semilinear predicates (unlike the original model, which can compute non-semilinear predicates with $s(n) = O(\log n)$), and otherwise it computes a predicate decidable by a nondeterministic $O(n \log s(n))$-space-bounded Turing machine. We then introduce novel $O(\mathrm{polylog}(n))$ expected time protocols for junta/leader election and general purpose broadcast correct with high probability, and approximate and exact population size counting correct with probability 1. Finally, we show that the main constraint on the power of bounded-message-size protocols is the size of the internal states: with unbounded internal states, any computable function can be computed with probability 1 in the limit by a protocol that uses only $\textit{1-bit}$ messages.

We study the fundamental problem of distributed clock synchronization in a basic probabilistic communication setting. We consider a synchronous fully-connected network of $n$ agents, where each agent has a local clock, that is, a counter increasing by one modulo $T$ in each round. The clocks have arbitrary values initially, and they must all indicate the same time eventually. We assume a pull communication model, where in every round each agent receives an $\ell$-bit message from a random agent. We devise several fast synchronization algorithms that use small messages and are self-stabilizing, that is, the complete initial state of each agent (not just its clock value) can be arbitrary. We first provide a surprising algorithm for synchronizing a binary clock ($T=2$) using $1$-bit messages ($\ell = 1$). This is a variant of the voter model and converges in $O(\log n)$ rounds w.h.p., unlike the voter model which needs polynomial time. Next we present an elegant extension of our algorithm that synchronizes a modulo $T=4$ clock, with $\ell = 1$, in $O(\log n)$ rounds. Using these two algorithms, we refine an algorithm of Boczkowski et al.~(SODA'17), that synchronizes a modulo $T$ clock in polylogarithmic time (in $n$ and $T$). The original algorithm uses $\ell=3$ bit messages, and each agent receives messages from two agents per round. Our algorithm reduces the message size to $\ell = 2$, and the number of messages received to one per round, without increasing the running time. Finally, we present two algorithms that simulate our last algorithm achieving $\ell < 2$, without hurting the asymptotic running time. The first algorithm uses a message space of size 3, i.e., $\ell =\log_2(3)$. The second requires a rough upper bound on $\log n$, and uses just 1-bit messages. More generally, our constructions can simulate any self-stabilizing algorithm that requires a shared clock, without increasing the message size and by only increasing the running time by a constant factor and a polylogarithmic term.

Introduction . Polynomial phase signals frequently appear in radar, sonar, communication and technical applications. Therefore, estimation of polynomial phase coefficients of such signals is an urgent problem in signal theory. Currently, a large number of estimation algorithms have been proposed. The best way is the maximum likelihood (ML) method. However, its implementation is associated with a multidimensional retrieval, which makes the method unsuitable for practical implementation. A number of alternative strategies have been developed to circumvent the ML difficulties. These strategies are very close to optimal. Among them one can single out the HAF-algorithm based on the computation of the High order Ambiguity Function and the CPF algorithm, which uses the computation of the Cubic Phase Function and produces very accurate estimates for signals with the quadratic frequency modulation. However, both algorithms have obvious drawbacks. The HAF algorithm pro-duces a large number of combinatorial noise components. The CPF algorithm is limited in its implementation to the third order polynomial signals and does not use fast algorithms, such as the Fast Fourier Transform. Aim. Synthesis of an estimation algorithm that produces a small number of noise combinatorial components and uses the Fast Fourier Transform computation algorithms to find coefficient estimates of an arbitrary order phase polynomial. Materials and methods. In the paper a concept of a decisive function was introduced. It was calculated so that its phase contained only a first-order monomial with a coefficient equal to the highest coefficient of the signal phase polynomial. Results. A new estimation algorithm was proposed able to use Fast Fourier Transform computation algorithms to find estimates. Each polynomial coefficient was estimated on the basis of a unified procedure, which reduced the number of combinatorial noise components in an estimate search. Conclusions. The synthesized algorithm gives asymptotically efficient estimates for lower signal-to-noise ratios in comparison with the HAF-algorithm.

The dominating set problem asks for a small subset D of nodes in a graph such that every node is either in D or adjacent to a node in D. This problem arises in a number of distributed network applications, where it is important to locate a small number of centers in the network such that every node is nearby at least one center. Finding a dominating set of minimum size is NP-complete, and the best known approximation is logarithmic in the maximum degree of the graph and is provided by the same simple greedy approach that gives the well-known logarithmic approximation result for the closely related set cover problem.We describe and analyze new randomized distributed algorithms for the dominating set problem that run in polylogarithmic time, independent of the diameter of the network, and that return a dominating set of size within a logarithmic factor from optimal, with high probability. In particular, our best algorithm runs in O(log n log Δ) rounds with high probability, where n is the number of nodes, Δ is one plus the maximum degree of any node, and each round involves a constant number of message exchanges among any two neighbors; the size of the dominating set obtained is within O (log Δ) of the optimal in expectation and within O(log n) of the optimal with high probability. We also describe generalizations to the weighted case and the case of multiple covering requirements.

A population protocol is a sequence of pairwise interactions of n agents. During one interaction, two randomly selected agents update their states by applying a deterministic transition function. The goal is to stabilize the system at a desired output property. The main performance objectives in designing such protocols are small number of states per agent and fast stabilization time. We present a fast population protocol for the exact-majority problem, which uses Theta(log n) states (per agent) and stabilizes in O(log^{5/3} n) parallel time (i.e., in O(n log^{5/3} n) interactions) in expectation and with high probability. Alistarh et al. [SODA 2018] showed that exact-majority protocols which stabilize in expected O(n^{1-Omega(1)}) parallel time and have the properties of monotonicity and output dominance require Omega(log n) states. Note that the properties mentioned above are satisfied by all known population protocols for exact majority, including ours. They also showed an O(log^2 n)-time exact-majority protocol with O(log n) states, which, prior to our work, was the fastest exact-majority protocol with polylogarithmic number of states. The standard design framework for majority protocols is based on O(log n) phases and requires that all agents are well synchronized within each phase, leading naturally to upper bounds of the order of log^2 n because of Theta(log n) synchronization time per phase. We show how this framework can be tightened with weak synchronization to break the O(log^2 n) upper bound of previous protocols.

The present thesis studies some important random walk-based algorithms, which are randomized rumor spreading and balanced allocation protocols on networks. In the rst part of the thesis, we study the Push and the Push-Pull protocols introduced by [DGG+87], which are basic randomized protocols for information dissemination on networks. In Chapter 2, we propose a new model where the number of calls of each node in every round is chosen independently according to a probability distribution R with bounded mean determined at the beginning of the process. In addition to the model being a natural extension of the standard protocols, it also serves as a more realistic model for rumor spreading in a network whose entities are not completely uniform and may have di erent levels of power. We provide both lower and upper bounds on the rumor spreading time depending on statistical properties of R such as the mean or the variance. While it is well-known that the standard protocols need Θ(log n) rounds to spread a rumor on a complete network with n nodes, we show that, if R follows a power law distribution with exponent β ∈ (2, 3), then the Push-Pull protocol spreads a rumor in Θ(log log n) rounds. Moreover, when β = 3, we show a runtime of Θ ( logn log logn ) . In Chapter 3, we analyze the behavior of the standard Push-Pull protocol on a class of random graphs, called random k-trees for every integer k > 2, that are suitable to model poorly connected, small-world and scale free networks. Here, we show that the Push-Pull protocol propagates a rumor from a randomly chosen informed node to almost all nodes of a random k-tree with n nodes in O((log n)1+ck) rounds with high probability, where 0 3 can be any integer. Let l be a given positive integer. In each round t, 1 6 t 6 n, ball t picks a node of G uniformly at random and performs a nonbacktracking random walk of length l from the chosen node. Then it deterministically selects a subset of the visited nodes as the potential choices and allocates itself on one of the choices with minimum load (ties are broken uniformly at random). Provided G has a su ciently large girth, we establish an upper bound for the maximum number of balls at any bin after allocating n balls by the algorithm. We also show that the upper bound is tight up to a O(log log n) factor. In particular, we show that if we set l = b(log n) 1+ 2 c, for any constant ∈ (0, 1], and G has girth at least ω(l), then the maximum load is bounded by O(1/ ) with high probability.

By employing the virtual machines (VMs) consolidation technique at a virtualized data center, optimal mapping of VMs to physical machines (PMs) can be performed. The type of optimization approach and the policy of detecting the appropriate time to implement the consolidation process are influential in the performance of the consolidation technique. In a majority of researches, the consolidation approach merely focuses on the management of underloaded or overloaded PMs, while a number of VMs could also be in an underload or overload state. Managing an abnormal state of VM results in the postponement of PM getting into an abnormal state as well and affects the implementation time of the consolidation process. For the aim of optimal VM consolidation in this research, a self-adaptive architecture is presented to detect and manage underloaded and overloaded VMs /PMs in reaction to workload changes in the data center. The goal of consolidation process is employing the minimum number of active VMs and PMs, while guaranteeing the quality of service (QoS). Assessment criteria of QoS are two parameters including average number of requests in the PM buffer and average waiting time in the VM. To evaluate these two parameters, a probabilistic model of the data center is proposed by applying the queuing theory. The assessment results of the probabilistic model form a basis for decision-making in the modules of the proposed architecture. Numerical results obtained from the assessment of the probabilistic model via discrete-event simulator under various parameter settings confirm the efficiency of the proposed architecture in achieving the aims of the consolidation process.

This paper solves a longstanding open problem in fully dynamic algorithms: We present the first fully dynamic algorithms that maintain connectivity, bipartiteness, and approximate minimum spanning trees in polylogarithmic time per edge insertion or deletion. The algorithms are designed using a new dynamic technique that combines a novel graph decomposition with randomization. They are Las-Vegas type randomized algorithms which use simple data structures and have a small constant factor. Let n denote the number of nodes in the graph. For a sequence of Ω( m 0 ) operations, where m 0 is the number of edges in the initial graph, the expected time for p updates is O ( p log 3 n ) (througout the paper the logarithms are based 2) for connectivity and bipartiteness. The worst-case time for one query is O (log n /log log n ). For the k -edge witness problem (“Does the removal of k given edges disconnect the graph?”) the expected time for p updates is O ( p log 3 n ) and the expected time for q queries is O ( qk log 3 n ). Given a graph with k different weights, the minimum spanning tree can be maintained during a sequence of p updates in expected time O ( pk log 3 n ). This implies an algorithm to maintain a 1 + ε-approximation of the minimum spanning tree in expected time O (( p log 3 n log U )/ε) for p updates, where the weights of the edges are between 1 and U .

We study two fundamental communication primitives: broadcasting and leader election in the classical model of multi-hop radio networks with unknown topology and without collision detection mechanisms. It has been known for almost 20 years that in undirected networks with n nodes and diameter D, randomized broadcasting requires Ω(D log t n/D + log2n) rounds in expectation, assuming that uninformed nodes are not allowed to communicate (until they are informed). Only very recently, Haeupler and Wajc (PODC'2016) showed that this bound can be slightly improved for the model with spontaneous transmissions, providing an O(D(log n log log n)/(log D) + logO(1)n)-time broadcasting algorithm. In this paper, we give a new and faster algorithm that completes broadcasting in O(D(log n)/(log D) + logO(1)n) time, with high probability. This yields the first optimal O(D)-time broadcasting algorithm whenever D is polynomial in n. Furthermore, our approach can be applied to design a new leader election algorithm that matches the performance of our broadcasting algorithm. Previously, all fast randomized leader election algorithms have been using broadcasting as their subroutine and their complexity have been asymptotically strictly bigger than the complexity of broadcasting. In particular, the fastest previously known randomized leader election algorithm of Ghaffari and Haeupler (SODA'2013) requires O(D log n/D min(log log n, log n/D) + logO(1)n-time with high probability. Our new algorithm requires O(D(log n)/(log D) + logO(1)n time with high probability, and it achieves the optimal O(D) time whenever D is polynomial in n.

In wind turbine engineering, stability and control rely on precision. A new approach for discrete-time systems is presented in this study, which makes use of constrained Gramians and frequency weights. Wind turbines with a double-fed induction generator and dynamic rotational speeds can have their model order reduced using the suggested method, which makes use of sophisticated state-space representations. A novel balanced realization method, along with frequency-weighted and limited Gramians, successfully lowers the dimensionality of large state models. Minimizing approximation errors and ensuring stability are both achieved by the resulting lower-order system. This paper makes a significant contribution by offering an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> formula for error boundaries, which allows for more efficient and faster computations. A paradigm shift in improving the accuracy of modeling techniques is marked by this groundbreaking method, which applies frequency-weighted and limited Gramians to real-time systems like wind turbines.

Dynamic 3-sided planar range queries with expected doubly-logarithmic time

A High-Dimensional Pricing Framework for Fi nancial Instruments Valuation

This paper proposes a robust intelligent technique to produce the initial population close to the optimal solution for the job-shop scheduling problem (JSSP). The proposed technique is designed by a new heuristic based on an intelligent skip from the primal point of the solution space to a better one that considers a new classification of jobs on machines. This new classification is named mPlates-Jobs. The main advantages of the proposed technique are its capability to produce any size of the initial population, its proximity to the optimal solution, and its capability to observe the best-known solution in the generated initial population for benchmark datasets. The comparison of the experimental results with those of Kuczapski’s, Yahyaoui’s, Moghaddam and Giffler’s, and Thompson’s initialization techniques, which are considered the four state-of-the-art initialization techniques, proves the abovementioned advantages. In this study, the proposed intelligent initialization technique can be considered a fast and intelligent heuristic algorithm to solve the JSSP based on the quality of its results. DOI: http://dx.doi.org/10.5755/j01.itc.43.3.3536