arXiv subjects

Nandakishore Santhi

Publications and source records attributed to Nandakishore Santhi.

23 records · Page 2Linked to original sources

arXiv2010

Understanding Cascading Failures in Power Grids

In the past, we have observed several large blackouts, i.e. loss of power to large areas. It has been noted by several researchers that these large blackouts are a result of a cascade of failures of various components. As a power grid is made up of several thousands or even millions of components (relays, breakers, transformers, etc.), it is quite plausible that a few of these components do not perform their function as desired. Their failure/misbehavior puts additional burden on the working components causing them to misbehave, and thus leading to a cascade of failures. The complexity of the entire power grid makes it difficult to model each and every individual component and study the stability of the entire system. For this reason, it is often the case that abstract models of the working of the power grid are constructed and then analyzed. These models need to be computationally tractable while serving as a reasonable model for the entire system. In this work, we construct one such model for the power grid, and analyze it.

cs.SI↗

arXiv2010

A Quadratic Time-Space Tradeoff for Unrestricted Deterministic Decision Branching Programs

For a decision problem from coding theory, we prove a quadratic expected time-space tradeoff of the form $\eT\eS=Ω(\tfrac{n^2}{q})$ for $q$-way deterministic decision branching programs, where $q\geq 2$. Here $\eT$ is the expected computation time and $\eS$ is the expected space, when all inputs are equally likely. This bound is to our knowledge, the first such to show an exponential size requirement whenever $\eT = O(n^2)$. Previous exponential size tradeoffs for Boolean decision branching programs were valid for time-restricted models with $T=o(n\log_2{n})$. Proving quadratic time-space tradeoffs for unrestricted time decision branching programs has been a major goal of recent research -- this goal has already been achieved for multiple-output branching programs two decades ago. We also show the first quadratic time-space tradeoffs for Boolean decision branching programs verifying circular convolution, matrix-vector multiplication and discrete Fourier transform. Furthermore, we demonstrate a constructive Boolean decision function which has a quadratic expected time-space tradeoff in the Boolean deterministic decision branching program model. When $q$ is a constant the tradeoff results derived here for decision functions verifying various functions are order-comparable to previously known tradeoff bounds for calculating the corresponding multiple-output functions.

cs.CC↗

arXiv2007

On Algebraic Decoding of $q$-ary Reed-Muller and Product-Reed-Solomon Codes

We consider a list decoding algorithm recently proposed by Pellikaan-Wu \cite{PW2005} for $q$-ary Reed-Muller codes $\mathcal{RM}_q(\ell, m, n)$ of length $n \leq q^m$ when $\ell \leq q$. A simple and easily accessible correctness proof is given which shows that this algorithm achieves a relative error-correction radius of $τ\leq (1 - \sqrt{\ell q^{m-1}/{n}})$. This is an improvement over the proof using one-point Algebraic-Geometric codes given in \cite{PW2005}. The described algorithm can be adapted to decode Product-Reed-Solomon codes. We then propose a new low complexity recursive algebraic decoding algorithm for Reed-Muller and Product-Reed-Solomon codes. Our algorithm achieves a relative error correction radius of $τ\leq \prod_{i=1}^m (1 - \sqrt{k_i/q})$. This technique is then proved to outperform the Pellikaan-Wu method in both complexity and error correction radius over a wide range of code rates.

cs.IT↗

arXiv2006

Analog Codes on Graphs

We consider the problem of transmission of a sequence of real data produced by a Nyquist sampled band-limited analog source over a band-limited analog channel, which introduces an additive white Gaussian noise. An analog coding scheme is described, which can achieve a mean-squared error distortion proportional to $(1+SNR)^{-B}$ for a bandwidth expansion factor of $B/R$, where $0 < R < 1$ is the rate of individual component binary codes used in the construction and $B \geq 1$ is an integer. Thus, over a wide range of SNR values, the proposed code performs much better than any single previously known analog coding system.

cs.IT↗

arXiv2006

On an Improvement over Rényi's Equivocation Bound

We consider the problem of estimating the probability of error in multi-hypothesis testing when MAP criterion is used. This probability, which is also known as the Bayes risk is an important measure in many communication and information theory problems. In general, the exact Bayes risk can be difficult to obtain. Many upper and lower bounds are known in literature. One such upper bound is the equivocation bound due to Rényi which is of great philosophical interest because it connects the Bayes risk to conditional entropy. Here we give a simple derivation for an improved equivocation bound. We then give some typical examples of problems where these bounds can be of use. We first consider a binary hypothesis testing problem for which the exact Bayes risk is difficult to derive. In such problems bounds are of interest. Furthermore using the bounds on Bayes risk derived in the paper and a random coding argument, we prove a lower bound on equivocation valid for most random codes over memoryless channels.

cs.IT↗