Exact Distance Oracles for Planar Graphs with Failing Vertices

We consider exact distance oracles for directed weighted planar graphs in the presence of failing vertices. Given a source vertex $u$, a target vertex $v$ and a set $X$ of $k$ failed vertices, such an oracle returns the length of a shortest $u$-to-$v$ path that avoids all vertices in $X$. We propose oracles that can handle any number $k$ of failures. More specifically, for a directed weighted planar graph with $n$ vertices, any constant $k$, and for any $q \in [1,\sqrt n]$, we propose an oracle of size $\tilde{\mathcal{O}}(\frac{n^{k+3/2}}{q^{2k+1}})$ that answers queries in $\tilde{\mathcal{O}}(q)$ time. In particular, we show an $\tilde{\mathcal{O}}(n)$-size, $\tilde{\mathcal{O}}(\sqrt{n})$-query-time oracle for any constant $k$. This matches, up to polylogarithmic factors, the fastest failure-free distance oracles with nearly linear space. For single vertex failures ($k=1$), our $\tilde{\mathcal{O}}(\frac{n^{5/2}}{q^3})$-size, $\tilde{\mathcal{O}}(q)$-query-time oracle improves over the previously best known tradeoff of Baswana et al. [SODA 2012] by polynomial factors for $q = \Omega(n^t)$, $t \in (1/4,1/2]$. For multiple failures, no planarity exploiting results were previously known

A study in the general population about sadness to disentangle the continuum from well-being to depressive disorders

International audienceSadness is both a common experience in general population and one of the main criteria of major depressive disorder (MDD). We tested the hypothesis of a depressive continuum using sadness as an intermediate experience between well-being and disorder.A French cross-sectional Mental Health survey in General Population interviewed 38,694 individuals. We examined prevalences and compared sociodemographic correlates and psychiatric disorders of individuals in 3 independent groups 1) MDD, 2) sadness without MDD, and 3) controls.The prevalence of sadness was of 29.8% in the whole sample and of 93% in subjects suffering from MDD (n = 4976). The "sadness" group shared the same sociodemographic patterns as the "MDD" group. All psychiatric disorders assessed (i.e. bipolar disorder, anxiety disorder, alcohol use disorder, psychotic disorder and suicide attempts) were significantly associated with both "sadness" and "MDD" groups compared to "controls". Individuals with sadness, compared to those with MDD, were significantly less likely to meet the criteria for all psychiatric disorders. MDD's sensitivity of sadness was 94,2%.Even though we used a quota sampling method, the sample was not strictly representative of the general population.Sadness validates the depressive continuum hypothesis, since it is more frequent in the general population than MDD itself and at the same time shares with MDD the same sociodemographic and clinical correlates. A gradual association from controls to MDD was observed for psychiatric comorbidities. Finally, the high sensitivity of sadness may suggest its use to screen at-risk individuals converting from well-being to full psychiatric disorders