New Classes of Distributed Time Complexity

A number of recent papers -- e.g. Brandt et al. (STOC 2016), Chang et al. (FOCS 2016), Ghaffari & Su (SODA 2017), Brandt et al. (PODC 2017), and Chang & Pettie (FOCS 2017) -- have advanced our understanding of one of the most fundamental questions in theory of distributed computing: what are the possible time complexity classes of LCL problems in the LOCAL model? In essence, we have a graph problem $\Pi$ in which a solution can be verified by checking all radius-$O(1)$ neighbourhoods, and the question is what is the smallest $T$ such that a solution can be computed so that each node chooses its own output based on its radius-$T$ neighbourhood. Here $T$ is the distributed time complexity of $\Pi$. The time complexity classes for deterministic algorithms in bounded-degree graphs that are known to exist by prior work are $\Theta(1)$, $\Theta(\log^* n)$, $\Theta(\log n)$, $\Theta(n^{1/k})$, and $\Theta(n)$. It is also known that there are two gaps: one between $\omega(1)$ and $o(\log \log^* n)$, and another between $\omega(\log^* n)$ and $o(\log n)$. It has been conjectured that many more gaps exist, and that the overall time hierarchy is relatively simple -- indeed, this is known to be the case in restricted graph families such as cycles and grids. We show that the picture is much more diverse than previously expected. We present a general technique for engineering LCL problems with numerous different deterministic time complexities, including $\Theta(\log^{\alpha}n)$ for any $\alpha\ge1$, $2^{\Theta(\log^{\alpha}n)}$ for any $\alpha\le 1$, and $\Theta(n^{\alpha})$ for any $\alpha <1/2$ in the high end of the complexity spectrum, and $\Theta(\log^{\alpha}\log^* n)$ for any $\alpha\ge 1$, $\smash{2^{\Theta(\log^{\alpha}\log^* n)}}$ for any $\alpha\le 1$, and $\Theta((\log^* n)^{\alpha})$ for any $\alpha \le 1$ in the low end; here $\alpha$ is a positive rational number

JohnnyVon: Self-Replicating Automata in Continuous Two-Dimensional Space

JohnnyVon is an implementation of self-replicating automata in continuous two-dimensional space. Two types of particles drift about in a virtual liquid. The particles are automata with discrete internal states but continuous external relationships. Their internal states are governed by finite state machines but their external relationships are governed by a simulated physics that includes brownian motion, viscosity, and spring-like attractive and repulsive forces. The particles can be assembled into patterns that can encode arbitrary strings of bits. We demonstrate that, if an arbitrary “seed” pattern is put in a “soup” of separate individual particles, the pattern will replicate by assembling the individual particles into copies of itself. We also show that, given sufficient time, a soup of separate individual particles will eventually spontaneously form self-replicating patterns. We discuss the implications of JohnnyVon for research in nanotechnology, theoretical biology, and artificial life

Molecular simulation studies of cyanine-based chromonic mesogens: spontaneous symmetry breaking to form chiral aggregates and the formation of a novel lamellar structure

All‐atom molecular dynamics simulations are performed on two chromonic mesogens in aqueous solution: 5,5′‐dimethoxy‐bis‐(3,3′‐di‐sulphopropyl)‐thiacyanine triethylammonium salt (Dye A) and 5,5′‐dichloro‐bis‐(3,3′‐di‐sulphopropyl)‐thiacyanine triethylammonuim salt (Dye B). Simulations demonstrate the formation of self‐assembled chromonic aggregates with an interlayer distance of ≈0.35 nm, with neighboring molecules showing a predominantly head‐to‐tail antiparallel stacking arrangement to minimize electrostatic repulsion between hydrophilic groups. Strong overlap of the aromatic rings occurs within the self‐assembled columns, characteristic of H‐aggregation in aqueous solution. At low concentrations, aggregates of Dye A form chiral columns, despite the presence of strictly achiral species. Chirality arises out of the minimization of steric repulsion between methoxy groups, which would otherwise disrupt the stacking of aromatic molecular cores. At higher concentrations, simulations suggest the interaction of short columns leads to the formation of an achiral‐layered structure in which hydrophobic aromatic regions of the molecule are sandwiched between two layers of hydrophilic groups. This novel lamellar structure is suggested as a likely candidate for the structure of a J‐aggregate. The latter is known to exhibit intense red‐shifted absorption peaks in solution but their structure has not yet been characterized. Self‐organization of such structures provides a route to the formation of “smectic” chromonic mesophases

The Cowl - v.79 - n.6 - Oct 9, 2014

The Cowl - student newspaper of Providence College. Vol 79 - No. 6 - October 9, 2014. 24 pages

The Cowl -v.62 - n.16 - Feb 26, 1998

The Cowl - student newspaper of Providence College. Vol 62 - No. 16 - Feb 26, 1998. 20 pages