Hardness of Exact Distance Queries in Sparse Graphs Through Hub Labeling

A distance labeling scheme is an assignment of bit-labels to the vertices of an undirected, unweighted graph such that the distance between any pair of vertices can be decoded solely from their labels. An important class of distance labeling schemes is that of hub labelings, where a node $v \in G$ stores its distance to the so-called hubs $S_v \subseteq V$, chosen so that for any $u,v \in V$ there is $w \in S_u \cap S_v$ belonging to some shortest $uv$ path. Notice that for most existing graph classes, the best distance labelling constructions existing use at some point a hub labeling scheme at least as a key building block. Our interest lies in hub labelings of sparse graphs, i.e., those with $|E(G)| = O(n)$, for which we show a lowerbound of $\frac{n}{2^{O(\sqrt{\log n})}}$ for the average size of the hubsets. Additionally, we show a hub-labeling construction for sparse graphs of average size $O(\frac{n}{RS(n)^{c}})$ for some $0 < c < 1$, where $RS(n)$ is the so-called Ruzsa-Szemer{\'e}di function, linked to structure of induced matchings in dense graphs. This implies that further improving the lower bound on hub labeling size to $\frac{n}{2^{(\log n)^{o(1)}}}$ would require a breakthrough in the study of lower bounds on $RS(n)$, which have resisted substantial improvement in the last 70 years. For general distance labeling of sparse graphs, we show a lowerbound of $\frac{1}{2^{O(\sqrt{\log n})}} SumIndex(n)$, where $SumIndex(n)$ is the communication complexity of the Sum-Index problem over $Z_n$. Our results suggest that the best achievable hub-label size and distance-label size in sparse graphs may be $\Theta(\frac{n}{2^{(\log n)^c}})$ for some $0<c < 1$

Comparative performance of airyscan and structured illumination superresolution microscopy in the study of the surface texture and 3D shape of pollen

The visualization of taxonomically diagnostic features of individual pollen grains can be a challenge for many ecologically and phylogenetically important pollen types. The resolution of traditional optical microscopy is limited by the diffraction of light (250 nm), while high resolution tools such as electron microscopy are limited by laborious preparation and imaging workflows. Airyscan confocal superresolution and structured illumination superresolution (SR-SIM) microscopy are powerful new tools for the study of nanoscale pollen morphology and three-dimensional structure that can overcome these basic limitations. This study demonstrates their utility in capturing morphological details below the diffraction limit of light. Using three distinct pollen morphotypes (Croton hirtus, Dactylis glomerata, and Helianthus sp.) and contrast-enhancing fluorescent staining, we were able to assess the effectiveness of the Airyscan and SR-SIM. We further demonstrate that these new superresolution methods can be easily applied to the study of fossil pollen material