Near Optimal Parallel Algorithms for Dynamic DFS in Undirected Graphs

Depth first search (DFS) tree is a fundamental data structure for solving graph problems. The classical algorithm [SiComp74] for building a DFS tree requires $O(m+n)$ time for a given graph $G$ having $n$ vertices and $m$ edges. Recently, Baswana et al. [SODA16] presented a simple algorithm for updating DFS tree of an undirected graph after an edge/vertex update in $\tilde{O}(n)$ time. However, their algorithm is strictly sequential. We present an algorithm achieving similar bounds, that can be adopted easily to the parallel environment. In the parallel model, a DFS tree can be computed from scratch using $m$ processors in expected $\tilde{O}(1)$ time [SiComp90] on an EREW PRAM, whereas the best deterministic algorithm takes $\tilde{O}(\sqrt{n})$ time [SiComp90,JAlg93] on a CRCW PRAM. Our algorithm can be used to develop optimal (upto polylog n factors deterministic algorithms for maintaining fully dynamic DFS and fault tolerant DFS, of an undirected graph. 1- Parallel Fully Dynamic DFS: Given an arbitrary online sequence of vertex/edge updates, we can maintain a DFS tree of an undirected graph in $\tilde{O}(1)$ time per update using $m$ processors on an EREW PRAM. 2- Parallel Fault tolerant DFS: An undirected graph can be preprocessed to build a data structure of size O(m) such that for a set of $k$ updates (where $k$ is constant) in the graph, the updated DFS tree can be computed in $\tilde{O}(1)$ time using $n$ processors on an EREW PRAM. Moreover, our fully dynamic DFS algorithm provides, in a seamless manner, nearly optimal (upto polylog n factors) algorithms for maintaining a DFS tree in semi-streaming model and a restricted distributed model. These are the first parallel, semi-streaming and distributed algorithms for maintaining a DFS tree in the dynamic setting.Comment: Accepted to appear in SPAA'17, 32 Pages, 5 Figure

Coherent evolution via reservoir driven holonomy

We show that in the limit of strongly interacting environment a system initially prepared in a Decoherence Free Subspace (DFS) coherently evolves in time, adiabatically following the changes of the DFS. If the reservoir cyclicly evolves in time, the DFS states acquire an holonomy.Comment: 4 page

Diclofenac sodium ion exchange resin complex loaded melt cast films for sustained release ocular delivery

The goal of the present study is to develop polymeric matrix films loaded with a combination of free diclofenac sodium (DFSfree) and DFS:Ion exchange resin complexes (DFS:IR) for immediate and sustained release profiles, respectively. Effect of ratio of DFS and IR on the DFS:IR complexation efficiency was studied using batch processing. DFS:IR complex, DFSfree, or a combination of DFSfree+DFS:IR loaded matrix films were prepared by melt-cast technology. DFS content was 20% w/w in these matrix films. In vitro transcorneal permeability from the film formulations were compared against DFS solution, using a side-by-side diffusion apparatus, over a 6 h period. Ocular disposition of DFS from the solution, films and corresponding suspensions were evaluated in conscious New Zealand albino rabbits, 4 h and 8 h post-topical administration. All in vivo studies were carried out as per the University of Mississippi IACUC approved protocol. Complexation efficiency of DFS:IR was found to be 99% with a 1:1 ratio of DFS:IR. DFS release from DFS:IR suspension and the film were best-fit to a Higuchi model. In vitro transcorneal flux with the DFSfree+DFS:IR(1:1)(1 + 1) was twice that of only DFS:IR(1:1) film. In vivo, DFS solution and DFS:IR(1:1) suspension formulations were not able to maintain therapeutic DFS levels in the aqueous humor (AH). Both DFSfree and DFSfree+DFS:IR(1:1)(3 + 1) loaded matrix films were able to achieve and maintain high DFS concentrations in the AH, but elimination of DFS from the ocular tissues was much faster with the DFSfree formulation. DFSfree+DFS:IR combination loaded matrix films were able to deliver and maintain therapeutic DFS concentrations in the anterior ocular chamber for up to 8 h. Thus, free drug/IR complex loaded matrix films could be a potential topical ocular delivery platform for achieving immediate and sustained release characteristics

Principles of Control for Decoherence-Free Subsystems

Decoherence-Free Subsystems (DFS) are a powerful means of protecting quantum information against noise with known symmetry properties. Although Hamiltonians theoretically exist that can implement a universal set of logic gates on DFS encoded qubits without ever leaving the protected subsystem, the natural Hamiltonians that are available in specific implementations do not necessarily have this property. Here we describe some of the principles that can be used in such cases to operate on encoded qubits without losing the protection offered by the DFS. In particular, we show how dynamical decoupling can be used to control decoherence during the unavoidable excursions outside of the DFS. By means of cumulant expansions, we show how the fidelity of quantum gates implemented by this method on a simple two-physical-qubit DFS depends on the correlation time of the noise responsible for decoherence. We further show by means of numerical simulations how our previously introduced "strongly modulating pulses" for NMR quantum information processing can permit high-fidelity operations on multiple DFS encoded qubits in practice, provided that the rate at which the system can be modulated is fast compared to the correlation time of the noise. The principles thereby illustrated are expected to be broadly applicable to many implementations of quantum information processors based on DFS encoded qubits.Comment: 12 pages, 7 figure