A Linear-Size Logarithmic Stretch Path-Reporting Distance Oracle for General Graphs

In 2001 Thorup and Zwick devised a distance oracle, which given an $n$-vertex undirected graph and a parameter $k$, has size $O(k n^{1+1/k})$. Upon a query $(u,v)$ their oracle constructs a $(2k-1)$-approximate path $\Pi$ between $u$ and $v$. The query time of the Thorup-Zwick's oracle is $O(k)$, and it was subsequently improved to $O(1)$ by Chechik. A major drawback of the oracle of Thorup and Zwick is that its space is $\Omega(n \cdot \log n)$. Mendel and Naor devised an oracle with space $O(n^{1+1/k})$ and stretch $O(k)$, but their oracle can only report distance estimates and not actual paths. In this paper we devise a path-reporting distance oracle with size $O(n^{1+1/k})$, stretch $O(k)$ and query time $O(n^\epsilon)$, for an arbitrarily small $\epsilon > 0$. In particular, our oracle can provide logarithmic stretch using linear size. Another variant of our oracle has size $O(n \log\log n)$, polylogarithmic stretch, and query time $O(\log\log n)$. For unweighted graphs we devise a distance oracle with multiplicative stretch $O(1)$, additive stretch $O(\beta(k))$, for a function $\beta(\cdot)$, space $O(n^{1+1/k} \cdot \beta)$, and query time $O(n^\epsilon)$, for an arbitrarily small constant $\epsilon >0$. The tradeoff between multiplicative stretch and size in these oracles is far below girth conjecture threshold (which is stretch $2k-1$ and size $O(n^{1+1/k})$). Breaking the girth conjecture tradeoff is achieved by exhibiting a tradeoff of different nature between additive stretch $\beta(k)$ and size $O(n^{1+1/k})$. A similar type of tradeoff was exhibited by a construction of $(1+\epsilon,\beta)$-spanners due to Elkin and Peleg. However, so far $(1+\epsilon,\beta)$-spanners had no counterpart in the distance oracles' world. An important novel tool that we develop on the way to these results is a {distance-preserving path-reporting oracle}

Budgeting for Growth and Prosperity: A Long-Term Plan to Balance the Budget, Grow the Economy, and Strengthen the Middle Class

Proposes reducing the deficit by investing in education, infrastructure, and technology; spending more efficiently; bolstering the social safety net; containing healthcare costs; simplifying the tax code; and raising gas and financial transaction taxes

The role of visual and mechanosensory cues in structuring forward flight in Drosophila melanogaster

It has long been known that many flying insects use visual cues to orient with respect to the wind and to control their groundspeed in the face of varying wind conditions. Much less explored has been the role of mechanosensory cues in orienting insects relative to the ambient air. Here we show that Drosophila melanogaster, magnetically tethered so as to be able to rotate about their yaw axis, are able to detect and orient into a wind, as would be experienced during forward flight. Further, this behavior is velocity dependent and is likely subserved, at least in part, by the Johnston's organs, chordotonal organs in the antennae also involved in near-field sound detection. These wind-mediated responses may help to explain how flies are able to fly forward despite visual responses that might otherwise inhibit this behavior. Expanding visual stimuli, such as are encountered during forward flight, are the most potent aversive visual cues known for D. melanogaster flying in a tethered paradigm. Accordingly, tethered flies strongly orient towards a focus of contraction, a problematic situation for any animal attempting to fly forward. We show in this study that wind stimuli, transduced via mechanosensory means, can compensate for the aversion to visual expansion and thus may help to explain how these animals are indeed able to maintain forward flight

Free-flight responses of Drosophila melanogaster to attractive odors

Many motile organisms localize the source of attractive odorants by following plumes upwind. In the case of D. melanogaster, little is known of how individuals alter their flight trajectories after encountering and losing a plume of an attractive odorant. We have characterized the three-dimensional flight behavior of D. melanogaster in a wind tunnel under a variety of odor conditions. In the absence of olfactory cues, hungry flies initiate flight and display anemotactic orientation. Following contact with a narrow ribbon plume of an attractive odor, flies reduce their crosswind velocity while flying faster upwind, resulting in a surge directed toward the odor source. Following loss of odor contact due to plume truncation, flies frequently initiate a stereotyped crosswind casting response, a behavior rarely observed in a continuous odor plume. Similarly, within a homogeneous odor cloud, flies move fast while maintaining an upwind heading. These results indicate both similarities and differences between the behavior of D. melanogaster and the responses of male moths to pheromone plumes, suggesting possible differences in underlying neural mechanisms

Do Physicians Respond to Liability Standards?

In this paper, we explore the sensitivity in the clinical decisions of physicians to the standards of care expected of them under the law, drawing on the abandonment by states over time of rules holding physicians to standards determined by local customs and the contemporaneous adoption of national-standard rules. Using data on broad rates of surgical interventions at the county-by-year level from the Area Resource File, we find that local surgery rates converge towards national surgery rates upon the adoption of national-standard rules. Moreover, we find that these effects are more pronounced among rural counties