Sublinear Distance Labeling

A distance labeling scheme labels the $n$ nodes of a graph with binary strings such that, given the labels of any two nodes, one can determine the distance in the graph between the two nodes by looking only at the labels. A $D$-preserving distance labeling scheme only returns precise distances between pairs of nodes that are at distance at least $D$ from each other. In this paper we consider distance labeling schemes for the classical case of unweighted graphs with both directed and undirected edges. We present a $O(\frac{n}{D}\log^2 D)$ bit $D$-preserving distance labeling scheme, improving the previous bound by Bollob\'as et. al. [SIAM J. Discrete Math. 2005]. We also give an almost matching lower bound of $\Omega(\frac{n}{D})$. With our $D$-preserving distance labeling scheme as a building block, we additionally achieve the following results: 1. We present the first distance labeling scheme of size $o(n)$ for sparse graphs (and hence bounded degree graphs). This addresses an open problem by Gavoille et. al. [J. Algo. 2004], hereby separating the complexity from distance labeling in general graphs which require $\Omega(n)$ bits, Moon [Proc. of Glasgow Math. Association 1965]. 2. For approximate $r$-additive labeling schemes, that return distances within an additive error of $r$ we show a scheme of size $O\left ( \frac{n}{r} \cdot\frac{\operatorname{polylog} (r\log n)}{\log n} \right )$ for $r \ge 2$. This improves on the current best bound of $O\left(\frac{n}{r}\right)$ by Alstrup et. al. [SODA 2016] for sub-polynomial $r$, and is a generalization of a result by Gawrychowski et al. [arXiv preprint 2015] who showed this for $r=2$.Comment: A preliminary version of this paper appeared at ESA'1

Near-Optimal Induced Universal Graphs for Bounded Degree Graphs

A graph $U$ is an induced universal graph for a family $F$ of graphs if every graph in $F$ is a vertex-induced subgraph of $U$. For the family of all undirected graphs on $n$ vertices Alstrup, Kaplan, Thorup, and Zwick [STOC 2015] give an induced universal graph with $O\!\left(2^{n/2}\right)$ vertices, matching a lower bound by Moon [Proc. Glasgow Math. Assoc. 1965]. Let $k= \lceil D/2 \rceil$. Improving asymptotically on previous results by Butler [Graphs and Combinatorics 2009] and Esperet, Arnaud and Ochem [IPL 2008], we give an induced universal graph with $O\!\left(\frac{k2^k}{k!}n^k \right)$ vertices for the family of graphs with $n$ vertices of maximum degree $D$. For constant $D$, Butler gives a lower bound of $\Omega\!\left(n^{D/2}\right)$. For an odd constant $D\geq 3$, Esperet et al. and Alon and Capalbo [SODA 2008] give a graph with $O\!\left(n^{k-\frac{1}{D}}\right)$ vertices. Using their techniques for any (including constant) even values of $D$ gives asymptotically worse bounds than we present. For large $D$, i.e. when $D = \Omega\left(\log^3 n\right)$, the previous best upper bound was ${n\choose\lceil D/2\rceil} n^{O(1)}$ due to Adjiashvili and Rotbart [ICALP 2014]. We give upper and lower bounds showing that the size is ${\lfloor n/2\rfloor\choose\lfloor D/2 \rfloor}2^{\pm\tilde{O}\left(\sqrt{D}\right)}$. Hence the optimal size is $2^{\tilde{O}(D)}$ and our construction is within a factor of $2^{\tilde{O}\left(\sqrt{D}\right)}$ from this. The previous results were larger by at least a factor of $2^{\Omega(D)}$. As a part of the above, proving a conjecture by Esperet et al., we construct an induced universal graph with $2n-1$ vertices for the family of graphs with max degree $2$. In addition, we give results for acyclic graphs with max degree $2$ and cycle graphs. Our results imply the first labeling schemes that for any $D$ are at most $o(n)$ bits from optimal

A genetically modified minipig model for Alzheimer's disease with SORL1 haploinsufficiency

The established causal genes in Alzheimer’s disease (AD), APP, PSEN1, and PSEN2, are functionally characterized using biomarkers, capturing an in vivo profile reflecting the disease’s initial preclinical phase. Mutations in SORL1, encoding the endosome recycling receptor SORLA, are found in 2%–3% of individuals with early-onset AD, and SORL1 haploinsufficiency appears to be causal for AD. To test whether SORL1 can function as an AD causal gene, we use CRISPR-Cas9-based gene editing to develop a model of SORL1 haploinsufficiency in Göttingen minipigs, taking advantage of porcine models for biomarker investigations. SORL1 haploinsufficiency in young adult minipigs is found to phenocopy the preclinical in vivo profile of AD observed with APP, PSEN1, and PSEN2, resulting in elevated levels of β-amyloid (Aβ) and tau preceding amyloid plaque formation and neurodegeneration, as observed in humans. Our study provides functional support for the theory that SORL1 haploinsufficiency leads to endosome cytopathology with biofluid hallmarks of autosomal dominant AD

Abstract 13369: Sympathovagal Imbalance Decades After Atrial Septal Defect Repair

Introduction: Heart rate variability (HRV), a measure of the autonomic nervous system activity, is a morbidity and mortality predictor. HRV is decreased in children with atrial septal defects (ASD) indicating parasympathetic withdrawal and sympathetic predominance, and increased right atrial end-diastolic pressure is believed to influence the sympathovagal balance. Despite the belief of a benign long-term outcome after closure, ASD patients have increased mortality rates after the age of 30 yrs. This may correlate to HRV, why we study HRV in adults after closure. Hypothesis: ASD patients have impaired HRV after closure when compared with controls. Methods: Surgically closed ASD patients (n=17, mean age 32±9 yrs., mean time since closure 19±8 yrs.), percutaneously closed ASD patients (n=18, mean age 28±7 yrs., mean time since closure 15±5 yrs.) and age- and gender-matched controls (n=15, mean age 30±9 yrs.) underwent a 48-hr Holter monitoring. Inclusion criteria were an age of minimum 2 yrs. at the time of diagnosis and minimum 3 yrs. must have passed since ASD closure. The following time-domain HRV measures were analyzed: SD of NN intervals (SDNN), SD of the average NN interval for each 5-minute segment, mean of the SDs of all NN intervals for each 5-minute (SDNNi), root mean square of successive RR interval differences (RMSSD), percentage of successive RR intervals that differ by &gt;50 ms (pNN50), and triangular index. Results: Surgically closed ASD patients have an impairment of all time-domain parameters, while transcatheter closed patients have an impairment of half the parameters. Conclusion: These novel findings demonstrate a cardiac sympathovagal imbalance in adult patients several yrs. after ASD closure and may potentially explain why we observe a long-term morbidity and mortality increase in ASD patients. </jats:p

Optimal induced universal graphs and adjacency labeling for trees

We show that there exists a graph $G$ with $O(n)$ nodes, where any forest of $n$ nodes is a node-induced subgraph of $G$. Furthermore, for constant arboricity $k$, the result implies the existence of a graph with $O(n^k)$ nodes that contains all $n$-node graphs as node-induced subgraphs, matching a $\Omega(n^k)$ lower bound. The lower bound and previously best upper bounds were presented in Alstrup and Rauhe (FOCS'02). Our upper bounds are obtained through a $\log_2 n +O(1)$ labeling scheme for adjacency queries in forests. We hereby solve an open problem being raised repeatedly over decades, e.g. in Kannan, Naor, Rudich (STOC 1988), Chung (J. of Graph Theory 1990), Fraigniaud and Korman (SODA 2010).Comment: A preliminary version of this paper appeared at FOCS'1