deterministic algorithm
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2022 ◽  
Vol 18 (1) ◽  
pp. 1-16
Author(s):  
Alessandra Graf ◽  
David G. Harris ◽  
Penny Haxell

An independent transversal (IT) in a graph with a given vertex partition is an independent set consisting of one vertex in each partition class. Several sufficient conditions are known for the existence of an IT in a given graph and vertex partition, which have been used over the years to solve many combinatorial problems. Some of these IT existence theorems have algorithmic proofs, but there remains a gap between the best existential bounds and the bounds obtainable by efficient algorithms. Recently, Graf and Haxell (2018) described a new (deterministic) algorithm that asymptotically closes this gap, but there are limitations on its applicability. In this article, we develop a randomized algorithm that is much more widely applicable, and demonstrate its use by giving efficient algorithms for two problems concerning the strong chromatic number of graphs.


2022 ◽  
pp. 285-296
Author(s):  
Charles Bouillaguet ◽  
Claire Delaplace ◽  
Monika Trimoska

2021 ◽  
Vol 8 (4) ◽  
pp. 1-25
Author(s):  
Laurent Feuilloley ◽  
Pierre Fraigniaud

We carry on investigating the line of research questioning the power of randomization for the design of distributed algorithms. In their seminal paper, Naor and Stockmeyer [STOC 1993] established that, in the context of network computing in which all nodes execute the same algorithm in parallel, any construction task that can be solved locally by a randomized Monte-Carlo algorithm can also be solved locally by a deterministic algorithm. This result, however, holds only for distributed tasks such that the correctness of their solutions can be locally checked by a deterministic algorithm. In this article, we extend the result of Naor and Stockmeyer to a wider class of tasks. Specifically, we prove that the same derandomization result holds for every task such that the correctness of their solutions can be locally checked using a 2-sided error randomized Monte-Carlo algorithm.


2021 ◽  
Vol 8 (4) ◽  
pp. 1-26
Author(s):  
Prasad Jayanti ◽  
Siddhartha Jayanti

The abortable mutual exclusion problem, proposed by Scott and Scherer in response to the needs in real-time systems and databases, is a variant of mutual exclusion that allows processes to abort from their attempt to acquire the lock. Worst-case constant remote memory reference algorithms for mutual exclusion using hardware instructions such as Fetch&Add or Fetch&Store have long existed for both cache coherent (CC) and distributed shared memory multiprocessors, but no such algorithms are known for abortable mutual exclusion. Even relaxing the worst-case requirement to amortized, algorithms are only known for the CC model. In this article, we improve this state of the art by designing a deterministic algorithm that uses Fetch&Store to achieve amortized O (1) remote memory reference in both the CC and distributed shared memory models. Our algorithm supports Fast Abort (a process aborts within six steps of receiving the abort signal) and has the following additional desirable properties: it supports an arbitrary number of processes of arbitrary names, requires only O (1) space per process, and satisfies a novel fairness condition that we call Airline FCFS . Our algorithm is short with fewer than a dozen lines of code.


2021 ◽  
Vol 17 (12) ◽  
pp. e1009713
Author(s):  
Jesse Kreger ◽  
Natalia L. Komarova ◽  
Dominik Wodarz

To study viral evolutionary processes within patients, mathematical models have been instrumental. Yet, the need for stochastic simulations of minority mutant dynamics can pose computational challenges, especially in heterogeneous systems where very large and very small sub-populations coexist. Here, we describe a hybrid stochastic-deterministic algorithm to simulate mutant evolution in large viral populations, such as acute HIV-1 infection, and further include the multiple infection of cells. We demonstrate that the hybrid method can approximate the fully stochastic dynamics with sufficient accuracy at a fraction of the computational time, and quantify evolutionary end points that cannot be expressed by deterministic models, such as the mutant distribution or the probability of mutant existence at a given infected cell population size. We apply this method to study the role of multiple infection and intracellular interactions among different virus strains (such as complementation and interference) for mutant evolution. Multiple infection is predicted to increase the number of mutants at a given infected cell population size, due to a larger number of infection events. We further find that viral complementation can significantly enhance the spread of disadvantageous mutants, but only in select circumstances: it requires the occurrence of direct cell-to-cell transmission through virological synapses, as well as a substantial fitness disadvantage of the mutant, most likely corresponding to defective virus particles. This, however, likely has strong biological consequences because defective viruses can carry genetic diversity that can be incorporated into functional virus genomes via recombination. Through this mechanism, synaptic transmission in HIV might promote virus evolvability.


Author(s):  
Tomer Lange ◽  
Joseph (Seffi) Naor ◽  
Gala Yadgar

Flash-based solid state drives (SSDs) have gained a central role in the infrastructure of large-scale datacenters, as well as in commodity servers and personal devices. The main limitation of flash media is its inability to support update-in-place: after data has been written to a physical location, it has to be erased before new data can be written to it. Moreover, SSDs support read and write operations in granularity of pages, while erasures are performed on entire blocks, which often contain hundreds of pages. When erasing a block, any valid data it stores must be rewritten to a clean location. As an SSD eventually wears out with progressing number of erasures, the efficiency of the management algorithm has a significant impact on its endurance. In this paper we first formally define the SSD management problem. We then explore this problem from an algorithmic perspective, considering it in both offline and online settings. In the offline setting, we present a near-optimal algorithm that, given any input, performs a negligible number of rewrites (relative to the input length). We also discuss the hardness of the offline problem. In the online setting, we first consider algorithms that have no prior knowledge about the input. We prove that no deterministic algorithm outperforms the greedy algorithm in this setting, and discuss the possible benefit of randomization. We then augment our model, assuming that each request for a page arrives with a prediction of the next time the page is updated. We design an online algorithm that uses such predictions, and show that its performance improves as the prediction error decreases. We also show that the performance of our algorithm is never worse than that guaranteed by the greedy algorithm, even when the prediction error is large. We complement our theoretical findings with an empirical evaluation of our algorithms, comparing them with the state-of-the-art scheme. The results confirm that our algorithms exhibit an improved performance for a wide range of input traces.


2021 ◽  
Vol Volume 17, Issue 4 ◽  
Author(s):  
Michal Koucký ◽  
Vojtěch Rödl ◽  
Navid Talebanfard

We show that for every $r \ge 2$ there exists $\epsilon_r > 0$ such that any $r$-uniform hypergraph with $m$ edges and maximum vertex degree $o(\sqrt{m})$ contains a set of at most $(\frac{1}{2} - \epsilon_r)m$ edges the removal of which breaks the hypergraph into connected components with at most $m/2$ edges. We use this to give an algorithm running in time $d^{(1 - \epsilon_r)m}$ that decides satisfiability of $m$-variable $(d, k)$-CSPs in which every variable appears in at most $r$ constraints, where $\epsilon_r$ depends only on $r$ and $k\in o(\sqrt{m})$. Furthermore our algorithm solves the corresponding #CSP-SAT and Max-CSP-SAT of these CSPs. We also show that CNF representations of unsatisfiable $(2, k)$-CSPs with variable frequency $r$ can be refuted in tree-like resolution in size $2^{(1 - \epsilon_r)m}$. Furthermore for Tseitin formulas on graphs with degree at most $k$ (which are $(2, k)$-CSPs) we give a deterministic algorithm finding such a refutation.


2021 ◽  
Vol 47 (6) ◽  
Author(s):  
Craig Gross ◽  
Mark A. Iwen ◽  
Lutz Kämmerer ◽  
Toni Volkmer

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3022
Author(s):  
Marta Bellés-Muñoz ◽  
Barry Whitehat ◽  
Jordi Baylina ◽  
Vanesa Daza ◽  
Jose Luis Muñoz-Tapia

Circuit-based zero-knowledge proofs have arose as a solution to the implementation of privacy in blockchain applications, and to current scalability problems that blockchains suffer from. The most efficient circuit-based zero-knowledge proofs use a pairing-friendly elliptic curve to generate and validate proofs. In particular, the circuits are built connecting wires that carry elements from a large prime field, whose order is determined by the number of elements of the pairing-friendly elliptic curve. In this context, it is important to generate an inner curve using this field, because it allows to create circuits that can verify public-key cryptography primitives, such as digital signatures and encryption schemes. To this purpose, in this article, we present a deterministic algorithm for generating twisted Edwards elliptic curves defined over a given prime field. We also provide an algorithm for checking the resilience of this type of curve against most common security attacks. Additionally, we use our algorithms to generate Baby Jubjub, a curve that can be used to implement elliptic-curve cryptography in circuits that can be validated in the Ethereum blockchain.


Author(s):  
Othon Michail ◽  
Paul G. Spirakis ◽  
Michail Theofilatos

We examine the problem of gathering [Formula: see text] agents (or multi-agent rendezvous) in dynamic graphs which may change in every round. We consider a variant of the [Formula: see text]-interval connectivity model [9] in which all instances (snapshots) are always connected spanning subgraphs of an underlying graph, not necessarily a clique. The agents are identical and not equipped with explicit communication capabilities, and are initially arbitrarily positioned on the graph. The problem is for the agents to gather at the same node, not fixed in advance. We first show that the problem becomes impossible to solve if the underlying graph has a cycle. In light of this, we study a relaxed version of this problem, called weak gathering, where the agents are allowed to gather either at the same node, or at two adjacent nodes. Our goal is to characterize the class of 1-interval connected graphs and initial configurations in which the problem is solvable, both with and without homebases. On the negative side we show that when the underlying graph contains a spanning bicyclic subgraph and satisfies an additional connectivity property, weak gathering is unsolvable, thus we concentrate mainly on unicyclic graphs. As we show, in most instances of initial agent configurations, the agents must meet on the cycle. This adds an additional difficulty to the problem, as they need to explore the graph and recognize the nodes that form the cycle. We provide a deterministic algorithm for the solvable cases of this problem that runs in [Formula: see text] number of rounds.


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