On Finer Separations Between Subclasses of Read-Once Oblivious ABPs

Read-once Oblivious Algebraic Branching Programs (ROABPs) compute polynomials as products of univariate polynomials that have matrices as coefficients. In an attempt to understand the landscape of algebraic complexity classes surrounding ROABPs, we study classes of ROABPs based on the algebraic structure of these coefficient matrices. We study connections between polynomials computed by these structured variants of ROABPs and other well-known classes of polynomials (such as depth-three powering circuits, tensor-rank and Waring rank of polynomials). Our main result concerns commutative ROABPs, where all coefficient matrices commute with each other, and diagonal ROABPs, where all the coefficient matrices are just diagonal matrices. In particular, we show a somewhat surprising connection between these models and the model of depth-three powering circuits that is related to the Waring rank of polynomials. We show that if the dimension of partial derivatives captures Waring rank up to polynomial factors, then the model of diagonal ROABPs efficiently simulates the seemingly more expressive model of commutative ROABPs. Further, a commutative ROABP that cannot be efficiently simulated by a diagonal ROABP will give an explicit polynomial that gives a super-polynomial separation between dimension of partial derivatives and Waring rank. Our proof of the above result builds on the results of Marinari, M\"oller and Mora (1993), and M\"oller and Stetter (1995), that characterise rings of commuting matrices in terms of polynomials that have small dimension of partial derivatives. The algebraic structure of the coefficient matrices of these ROABPs plays a crucial role in our proofs.Comment: Accepted to STACS 202

Analysis of basic Architectures used for Lifecycle Management and Orchestration of Network Service in Network Function Virtualization Environment

The Network Function Virtualization (NFV), Software Defined Networking are technologies, so which are in combination inorder to provide a high flexibility for network and dynamical continuum of resources for the deployment of services in the environment of high network programmability. A Network Function Virtualization Orchestration (NFVO) is an important topic played a major role in above scenario and in high availability of Virtual Network Functions (VNF), lifecycle and configuration management of network elements. However, the hardware usage is one of the obstacle towards network programmability and is generally considered as a contrast with respect to NFV concepts. In this paper shows many architectures, workflow in virtualization environment, compatibility, flexibility is discussed. These architectures involve in great enhancement of network infrastructure in virtualized environment. Each architecture is needed to gain better results in network function virtualization environment