Abstract
Restarting is a technique frequently employed in randomized algorithms. After some number of computation steps, the state of the algorithm is reinitialized with a new, independent random seed. Luby et al. (Inf. Process. Lett. 47(4), 173–180, 1993) introduced a universal restart strategy. They showed that their strategy is an optimal universal strategy in the worst case. However, the optimality result has only been shown for discrete processes. In this work, it is shown that their result does not translate into a continuous setting. Furthermore, we show that there are no (asymptotically) optimal strategies in a continuous setting. Nevertheless, we obtain an optimal universal strategy on a restricted class of continuous probability distributions. Furthermore, as a side result, we show that the expected value under restarts for the lognormal distribution tends towards 0. Finally, the results are illustrated using simulations.
Highlights
Numerous algorithms use restarts to boost their performance: If the algorithm fails to find a solution, the algorithm’s state is reset to its initial state
Stochastic local search algorithms restart if the number of local search steps exceeds a certain threshold and backtracking algorithms commence a restart after a certain number of backtracking steps
In settings such as these, the results by Luby et al [12] can be applied. They studied the case of Las Vegas algorithms, where restarts can be triggered at discrete time steps
Summary
Numerous algorithms use restarts to boost their performance: If the algorithm fails to find a solution, the algorithm’s state is reset to its initial state. They proved that restarts potentially help to improve the expected runtime of this type of algorithms In their work, they introduced two of the most important restart strategies: The fixed-cutoff strategy and Luby’s (universal) strategy. Luby’s strategy is beneficial in other contexts, such as (1 + 1) evolutionary algorithms [5] Albeit both the fixed-cutoff strategy and Luby’s strategy have only been introduced for discrete quantities, the restart paradigm can be applied to continuous quantities to minimize the expected value of this quantity. Restarts can help to minimize continuous resources such as money, amount of fuel, traveled distances, emitted greenhouse gases, and so forth In such scenarios, it seems natural to use established strategies from algorithmics such as the fixed-cutoff and Luby’s strategy. As a spin-off result, we show that the expected value under restarts of the lognormal distribution approaches zero under certain circumstances
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