A Multi-Objective Optimization Problem on Evacuating 2 Robots from the Disk in the Face-to-Face Model; Trade-Offs between Worst-Case and Average-Case Analysis †
Abstract
:1. Introduction
1.1. Related Work
1.2. Discussion on Closely Related Literature and Improvements
1.3. Outline of Our Results and Paper Organization
2. Preliminaries
2.1. Problem Definition and Main Results
2.2. Computing Evacuation Times
2.3. Trajectory Description
Robot | Phase # | Trajectory | Duration |
1 | |||
2 | |||
⋮ | ⋮ |
3. Two Benchmark Algorithms and Motivation
Robot | Phase # | Trajectory | Duration |
1 | |||
2 | |||
3 | |||
4 |
4. New Evacuation Algorithms
Robot | Phase # | Trajectory | Duration |
1 | |||
2 | |||
3 |
Robot | Phase # | Trajectory | Duration |
1 | |||
2 | |||
3 |
Robot | Phase # | Trajectory | Duration |
1 | |||
2 | |||
3 | |||
1 | |||
2 | |||
3 |
5. Worst-Case Performance Analysis
6. Average-Case Performance Analysis and the Efficient Frontier
- -
- for all , is -efficient, and ,
- -
- for all , is -efficient, and ,
- -
- for all , is -efficient, and .
7. Conclusions and Open Problems
- -
- Prove lower bounds for Evac, for any w. Is any of our algorithms, for any w optimal?
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- For the value , we designed algorithm for Evac which for a proper value of a has worst-case performance exactly w, while its average-case performance is strictly less than . Is it feasible to attain worst-case performance strictly less than w, while having average-case performance at most ?
- -
- The bound to the efficient (pareto) frontier we derived for problem Evac is indeed continuous, with respect to parameter w, but not differentiable. Is the optimal pareto frontier smooth, or is there any other family of algorithms that improves upon our results and gives a smooth transition between families of evacuation algorithms?
- -
- The algorithmic families we derived for Evac exhibit the following property. is a natural extension to . Similarly, is a natural extension to . Finally, is a natural extension to . However, and have different behavior (there are no values of their parameters that induce the same evacuation protocol), even though for a proper choice of their parameters, they induce algorithms with the same worst-case and average-case performance.
- -
- Observe that the average-case performance of is . All our evacuation algorithms induce average cost at least . We conjecture that even in the wireless model, as well as for any number of robots, is tight lower bound for the average performance of evacuation algorithm.
- -
- Our algorithms can also be interpreted as randomized algorithms that have access to infinitely many bits (or enough many bits, in order to simulate a uniformly random deployment point on the circle). What if the algorithm has access only to a limited number of random bits?
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- To the best of our knowledge, the current paper is the first attempt to study multi-objective optimization search-type problems. It was followed by [48,49] who considered time energy trade-offs for a search problems on the line. This line of research admits many future directions based on any combination of multiple objectives, e.g., worst-case, average-case and competitive cost, time, energy and any other efficiency measure, or even trade-offs involving number of faults or even complexity resources, e.g., memory, communication or randomness.
Author Contributions
Funding
Conflicts of Interest
References
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Chuangpishit, H.; Georgiou, K.; Sharma, P. A Multi-Objective Optimization Problem on Evacuating 2 Robots from the Disk in the Face-to-Face Model; Trade-Offs between Worst-Case and Average-Case Analysis. Information 2020, 11, 506. https://doi.org/10.3390/info11110506
Chuangpishit H, Georgiou K, Sharma P. A Multi-Objective Optimization Problem on Evacuating 2 Robots from the Disk in the Face-to-Face Model; Trade-Offs between Worst-Case and Average-Case Analysis. Information. 2020; 11(11):506. https://doi.org/10.3390/info11110506
Chicago/Turabian StyleChuangpishit, Huda, Konstantinos Georgiou, and Preeti Sharma. 2020. "A Multi-Objective Optimization Problem on Evacuating 2 Robots from the Disk in the Face-to-Face Model; Trade-Offs between Worst-Case and Average-Case Analysis" Information 11, no. 11: 506. https://doi.org/10.3390/info11110506