Data-Collection for the Sloan Digital Sky Survey: a Network-Flow Heuristic

The goal of the Sloan Digital Sky Survey is ``to map in detail one-quarter of the entire sky, determining the positions and absolute brightnesses of more than 100 million celestial objects''. The survey will be performed by taking ``snapshots'' through a large telescope. Each snapshot can capture up to 600 objects from a small circle of the sky. This paper describes the design and implementation of the algorithm that is being used to determine the snapshots so as to minimize their number. The problem is NP-hard in general; the algorithm described is a heuristic, based on Lagriangian-relaxation and min-cost network flow. It gets within 5-15% of a naive lower bound, whereas using a ``uniform'' cover only gets within 25-35%.Comment: proceedings version appeared in ACM-SIAM Symposium on Discrete Algorithms (1998

The Random Bit Complexity of Mobile Robots Scattering

We consider the problem of scattering $n$ robots in a two dimensional continuous space. As this problem is impossible to solve in a deterministic manner, all solutions must be probabilistic. We investigate the amount of randomness (that is, the number of random bits used by the robots) that is required to achieve scattering. We first prove that $n \log n$ random bits are necessary to scatter $n$ robots in any setting. Also, we give a sufficient condition for a scattering algorithm to be random bit optimal. As it turns out that previous solutions for scattering satisfy our condition, they are hence proved random bit optimal for the scattering problem. Then, we investigate the time complexity of scattering when strong multiplicity detection is not available. We prove that such algorithms cannot converge in constant time in the general case and in $o(\log \log n)$ rounds for random bits optimal scattering algorithms. However, we present a family of scattering algorithms that converge as fast as needed without using multiplicity detection. Also, we put forward a specific protocol of this family that is random bit optimal ($n \log n$ random bits are used) and time optimal ($\log \log n$ rounds are used). This improves the time complexity of previous results in the same setting by a $\log n$ factor. Aside from characterizing the random bit complexity of mobile robot scattering, our study also closes its time complexity gap with and without strong multiplicity detection (that is, $O(1)$ time complexity is only achievable when strong multiplicity detection is available, and it is possible to approach it as needed otherwise)

Tolerance analysis approach based on the classification of uncertainty (aleatory / epistemic)

Uncertainty is ubiquitous in tolerance analysis problem. This paper deals with tolerance analysis formulation, more particularly, with the uncertainty which is necessary to take into account into the foundation of this formulation. It presents: a brief view of the uncertainty classification: Aleatory uncertainty comes from the inherent uncertain nature and phenomena, and epistemic uncertainty comes from the lack of knowledge, a formulation of the tolerance analysis problem based on this classification, its development: Aleatory uncertainty is modeled by probability distributions while epistemic uncertainty is modeled by intervals; Monte Carlo simulation is employed for probabilistic analysis while nonlinear optimization is used for interval analysis.“AHTOLA” project (ANR-11- MONU-013