scholarly journals Running Time Analysis of Broadcast Consensus Protocols

2021 ◽  
pp. 164-183
Author(s):  
Philipp Czerner ◽  
Stefan Jaax
Keyword(s):  
Turing Machine ◽  
Polynomial Time ◽  
Execution Time ◽  
Time Analysis ◽  
Asymptotically Optimal ◽  
Population Protocols ◽  
Running Time ◽  
Running Time Analysis ◽  
Finite State ◽  
State Agents

AbstractBroadcast consensus protocols (BCPs) are a model of computation, in which anonymous, identical, finite-state agents compute by sending/receiving global broadcasts. BCPs are known to compute all number predicates in $$\mathsf {NL}=\mathsf {NSPACE}(\log n)$$ NL = NSPACE ( log n ) where n is the number of agents. They can be considered an extension of the well-established model of population protocols. This paper investigates execution time characteristics of BCPs. We show that every predicate computable by population protocols is computable by a BCP with expected $$\mathcal {O}(n \log n)$$ O ( n log n ) interactions, which is asymptotically optimal. We further show that every log-space, randomized Turing machine can be simulated by a BCP with $$\mathcal {O}(n \log n \cdot T)$$ O ( n log n · T ) interactions in expectation, where T is the expected runtime of the Turing machine. This allows us to characterise polynomial-time BCPs as computing exactly the number predicates in $$\mathsf {ZPL}$$ ZPL , i.e. predicates decidable by log-space, randomised Turing machine with zero-error in expected polynomial time where the input is encoded as unary.

scholarly journals Towards efficient verification of population protocols

2021 ◽  
Author(s):  
Michael Blondin ◽  
Javier Esparza ◽  
Stefan Jaax ◽  
Philipp J. Meyer
Keyword(s):  
Linear Constraints ◽  
Computational Power ◽  
High Complexity ◽  
Reachability Problem ◽  
Population Protocols ◽  
Finite State ◽  
State Agents ◽  
Efficient Verification ◽  
Primitive Recursive ◽  
Very High

AbstractPopulation protocols are a well established model of computation by anonymous, identical finite-state agents. A protocol is well-specified if from every initial configuration, all fair executions of the protocol reach a common consensus. The central verification question for population protocols is the well-specification problem: deciding if a given protocol is well-specified. Esparza et al. have recently shown that this problem is decidable, but with very high complexity: it is at least as hard as the Petri net reachability problem, which is -hard, and for which only algorithms of non-primitive recursive complexity are currently known. In this paper we introduce the class $${ WS}^3$$ WS 3 of well-specified strongly-silent protocols and we prove that it is suitable for automatic verification. More precisely, we show that $${ WS}^3$$ WS 3 has the same computational power as general well-specified protocols, and captures standard protocols from the literature. Moreover, we show that the membership and correctness problems for $${ WS}^3$$ WS 3 reduce to solving boolean combinations of linear constraints over $${\mathbb {N}}$$ N . This allowed us to develop the first software able to automatically prove correctness for all of the infinitely many possible inputs.

Running-time Analysis of Ant System Algorithms with Upper-bound Comparison

2017 ◽  
Vol 8 (4) ◽  
pp. 1-17
Author(s):  
Han Huang ◽  
Hongyue Wu ◽  
Yushan Zhang ◽  
Zhiyong Lin ◽  
Zhifeng Hao
Keyword(s):  
Optimal Solution ◽  
Numerical Results ◽  
Ant System ◽  
Time Analysis ◽  
Traveling Salesman Problems ◽  
Running Time ◽  
Cycle System ◽  
Running Time Analysis ◽  
Global Optimal

Running-time analysis of ant colony optimization (ACO) is crucial for understanding the power of the algorithm in computation. This paper conducts a running-time analysis of ant system algorithms (AS) as a kind of ACO for traveling salesman problems (TSP). The authors model the AS algorithm as an absorbing Markov chain through jointly representing the best-so-far solutions and pheromone matrix as a discrete stochastic status per iteration. The running-time of AS can be evaluated by the expected first-hitting time (FHT), the least number of iterations needed to attain the global optimal solution on average. The authors derive upper bounds of the expected FHT of two classical AS algorithms (i.e., ant quantity system and ant-cycle system) for TSP. They further take regular-polygon TSP (RTSP) as a case study and obtain numerical results by calculating six RTSP instances. The RTSP is a special but real-world TSP where the constraint of triangle inequality is stringently imposed. The numerical results derived from the comparison of the running time of the two AS algorithms verify our theoretical findings.

scholarly journals Running Time Analysis of the ( $$1+1$$ 1 + 1 )-EA for OneMax and LeadingOnes Under Bit-Wise Noise

Algorithmica ◽  
2018 ◽  
Vol 81 (2) ◽  
pp. 749-795 ◽  
Author(s):  
Chao Qian ◽  
Chao Bian ◽  
Wu Jiang ◽  
Ke Tang
Keyword(s):  
Time Analysis ◽  
Running Time ◽  
Running Time Analysis

scholarly journals A General Approach to Running Time Analysis of Multi-objective Evolutionary Algorithms

2018 ◽  
Author(s):  
Chao Bian ◽  
Chao Qian ◽  
Ke Tang
Keyword(s):  
Evolutionary Algorithms ◽  
Optimization Problems ◽  
Theoretical Aspect ◽  
Combinatorial Problems ◽  
Time Analysis ◽  
Running Time ◽  
Asymptotic Bounds ◽  
Multi Objective ◽  
Running Time Analysis ◽  
Empirical Performance

Evolutionary algorithms (EAs) have been widely applied to solve multi-objective optimization problems. In contrast to great practical successes, their theoretical foundations are much less developed, even for the essential theoretical aspect, i.e., running time analysis. In this paper, we propose a general approach to estimating upper bounds on the expected running time of multi-objective EAs (MOEAs), and then apply it to diverse situations, including bi-objective and many-objective optimization as well as exact and approximate analysis. For some known asymptotic bounds, our analysis not only provides their leading constants, but also improves them asymptotically. Moreover, our results provide some theoretical justification for the good empirical performance of MOEAs in solving multi-objective combinatorial problems.

scholarly journals Partial Quicksort and Quickpartitionsort

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Conrado Martínez ◽  
Uwe Rösler
Keyword(s):  
Lower Bound ◽  
Time Analysis ◽  
Information Theoretic ◽  
Running Time ◽  
Running Time Analysis ◽  
International Audience

International audience Partial Quicksort sorts the $l$ smallest elements in a list of length $n$. We provide a complete running time analysis for this combination of Find and Quicksort. Further we give some optimal adapted versions, called Partition Quicksort, with an asymptotic running time $c_1l\ln l+c_2l+n+o(n)$. The constant $c_1$ can be as small as the information theoretic lower bound $\log_2 e$.

scholarly journals Running time analysis of the (1+1)-EA for robust linear optimization

2020 ◽  
Vol 843 ◽  
pp. 57-72
Author(s):  
Chao Bian ◽  
Chao Qian ◽  
Ke Tang ◽  
Yang Yu
Keyword(s):  
Linear Optimization ◽  
Time Analysis ◽  
Running Time ◽  
Running Time Analysis
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