The power of vertex sparsifiers in dynamic graph algorithms

We introduce a new algorithmic framework for designing dynamic graph algorithms in minor-free graphs, by exploiting the structure of such graphs and a tool called vertex sparsification, which is a way to compress large graphs into small ones that well preserve relevant properties among a subset of vertices and has previously mainly been used in the design of approximation algorithms. Using this framework, we obtain a Monte Carlo randomized fully dynamic algorithm for (1 + epsilon)-approximating the energy of electrical flows in n-vertex planar graphs with tilde{O}(r epsilon^{-2}) worst-case update time and tilde{O}((r + n / sqrt{r}) epsilon^{-2}) worst-case query time, for any r larger than some constant. For r=n^{2/3}, this gives tilde{O}(n^{2/3} epsilon^{-2}) update time and tilde{O}(n^{2/3} epsilon^{-2}) query time. We also extend this algorithm to work for minor-free graphs with similar approximation and running time guarantees. Furthermore, we illustrate our framework on the all-pairs max flow and shortest path problems by giving corresponding dynamic algorithms in minor-free graphs with both sublinear update and query times. To the best of our knowledge, our results are the first to systematically establish such a connection between dynamic graph algorithms and vertex sparsification. We also present both upper bound and lower bound for maintaining the energy of electrical flows in the incremental subgraph model, where updates consist of only vertex activations, which might be of independent interest

Improved guarantees for Vertex Sparsification in planar graphs

Graph Sparsification aims at compressing large graphs into smaller ones while (approximately) preserving important characteristics of the input graph. In this work we study Vertex Sparsifiers, i.e., sparsifiers whose goal is to reduce the number of vertices. Given a weighted graph G=(V,E), and a terminal set K with |K|=k, a quality-q vertex cut sparsifier of G is a graph H with K contained in V_H that preserves the value of minimum cuts separating any bipartition of K, up to a factor of q. We show that planar graphs with all the k terminals lying on the same face admit quality-1 vertex cut sparsifier of size O(k^2) that are also planar. Our result extends to vertex flow and distance sparsifiers. It improves the previous best known bound of O(k^2 2^(2k)) for cut and flow sparsifiers by an exponential factor, and matches an Omega(k^2) lower-bound for this class of graphs. We also study vertex reachability sparsifiers for directed graphs. Given a digraph G=(V,E) and a terminal set K, a vertex reachability sparsifier of G is a digraph H=(V_H,E_H), K contained in V_H that preserves all reachability information among terminal pairs. We introduce the notion of reachability-preserving minors, i.e., we require H to be a minor of G. Among others, for general planar digraphs, we construct reachability-preserving minors of size O(k^2 log^2 k). We complement our upper-bound by showing that there exists an infinite family of acyclic planar digraphs such that any reachability-preserving minor must have Omega(k^2) vertices

Dynamic Effective Resistances and Approximate Schur Complement on Separable Graphs

We consider the problem of dynamically maintaining (approximate) all-pairs effective resistances in separable graphs, which are those that admit an n^{c}-separator theorem for some c0, unless the Online Matrix Vector Multiplication (OMv) conjecture is false. We further show that for general graphs, no incremental or decremental algorithm can maintain the s-t effective resistance problem with worst-case update time O(n^{1-delta}) and query-time O(n^{2-delta}) for any delta >0, unless the OMv conjecture is false

Local algorithms for estimating effective resistance

Effective resistance is an important metric that measures the similarity of two vertices in a graph. It has found applications in graph clustering, recommendation systems and network reliability, among others. In spite of the importance of the effective resistances, we still lack efficient algorithms to exactly compute or approximate them on massive graphs. In this work, we design several local algorithms for estimating effective resistances, which are algorithms that only read a small portion of the input while still having provable performance guarantees. To illustrate, our main algorithm approximates the effective resistance between any vertex pair s,t with an arbitrarily small additive error ε in time O(poly (log n/ε)), whenever the underlying graph has bounded mixing time. We perform an extensive empirical study on several benchmark datasets, validating the performance of our algorithms

Rapid and Efficient Invasion Assay of Glioblastoma in Human Brain Organoids

Glioblastoma (GBM) possesses glioma stem cells (GSCs) that exhibit aggressive invasion behavior in the brain. Current preclinical GBM invasion assays using mouse brain xenografts are time consuming and less efficient. Here, we demonstrate an array of methods that allow rapid and efficient assaying of GSCs invasion in human brain organoids. The assays are versatile to characterize various aspects of GSCs, such as invasion, integration, and interaction with mature neurons of brain organoids. Tissue clearing and quantitative 3D imaging of GSCs in host organoids reveal that invasiveness is inversely correlated with the organoids' age. Importantly, the described invasion assays can distinguish the invasive behaviors of primary and recurrent GSCs. The assays are also amenable to test pharmacological agents. As an example, we show that GI254023X, an inhibitor of ADAM10, could prevent the integration of GSCs into the organoids

E-cadherin integrates mechanotransduction and EGFR signaling to control junctional tissue polarization and tight junction positioning

Generation of a barrier in multi-layered epithelia like the epidermis requires restricted positioning of functional tight junctions (TJ) to the most suprabasal viable layer. This positioning necessitates tissue-level polarization of junctions and the cytoskeleton through unknown mechanisms. Using quantitative whole-mount imaging, genetic ablation, and traction force microscopy and atomic force microscopy, we find that ubiquitously localized E-cadherin coordinates tissue polarization of tension-bearing adherens junction (AJ) and F-actin organization to allow formation of an apical TJ network only in the uppermost viable layer. Molecularly, E-cadherin localizes and tunes EGFR activity and junctional tension to inhibit premature TJ complex formation in lower layers while promoting increased tension and TJ stability in the granular layer 2. In conclusion, our data identify an E-cadherin-dependent mechanical circuit that integrates adhesion, contractile forces and biochemical signaling to drive the polarized organization of junctional tension necessary to build an in vivo epithelial barrier