Towards Tight Bounds for the Streaming Set Cover Problem

We consider the classic Set Cover problem in the data stream model. For $n$ elements and $m$ sets ($m\geq n$) we give a $O(1/\delta)$-pass algorithm with a strongly sub-linear $\tilde{O}(mn^{\delta})$ space and logarithmic approximation factor. This yields a significant improvement over the earlier algorithm of Demaine et al. [DIMV14] that uses exponentially larger number of passes. We complement this result by showing that the tradeoff between the number of passes and space exhibited by our algorithm is tight, at least when the approximation factor is equal to $1$. Specifically, we show that any algorithm that computes set cover exactly using $({1 \over 2\delta}-1)$ passes must use $\tilde{\Omega}(mn^{\delta})$ space in the regime of $m=O(n)$. Furthermore, we consider the problem in the geometric setting where the elements are points in $\mathbb{R}^2$ and sets are either discs, axis-parallel rectangles, or fat triangles in the plane, and show that our algorithm (with a slight modification) uses the optimal $\tilde{O}(n)$ space to find a logarithmic approximation in $O(1/\delta)$ passes. Finally, we show that any randomized one-pass algorithm that distinguishes between covers of size 2 and 3 must use a linear (i.e., $\Omega(mn)$) amount of space. This is the first result showing that a randomized, approximate algorithm cannot achieve a space bound that is sublinear in the input size. This indicates that using multiple passes might be necessary in order to achieve sub-linear space bounds for this problem while guaranteeing small approximation factors.Comment: A preliminary version of this paper is to appear in PODS 201

Factores de modelado que afectan la elección del teletrabajo y su impacto en la demanda en las redes de transporte

This research estimates the extent of using teleworking to mean the feasibility and appropriateness of this method of work for employees and professors according to their characteristics and features of career. The study population included university staff and professors in Tehran and data collection was carried out through 447 questionnaires. A logistic regression model was used to investigate the transport demand caused by teleworking. The results showed that various factors including history and percentage of telework and after that, the time delay of home-to-work and trave distance affected the model of transportation demand of professor’s members. For the staffing community, it had the greatest impact on teleworking, history and percentage of telework, followed by travel distances, latency from work to home, and latency from home to work