arXiv subjects

Meghal Gupta

Publications and source records attributed to Meghal Gupta.

25 records · Page 2Linked to original sources

arXiv2022

Efficient Interactive Coding Achieving Optimal Error Resilience Over the Binary Channel

Given a noiseless protocol $π_0$ computing a function $f(x, y)$ of Alice and Bob's private inputs $x, y$, the goal of interactive coding is to construct an error-resilient protocol $π$ computing $f$ such that even if some fraction of the communication is adversarially corrupted, both parties still learn $f(x, y)$. Ideally, the resulting scheme $π$ should be positive rate, computationally efficient, and achieve optimal error resilience. While interactive coding over large alphabets is well understood, the situation over the binary alphabet has remained evasive. At the present moment, the known schemes over the binary alphabet that achieve a higher error resilience than a trivial adaptation of large alphabet schemes are either still suboptimally error resilient [EKS20], or optimally error resilient with exponential communication complexity [GZ22]. In this work, we construct a scheme achieving optimality in all three parameters: our protocol is positive rate, computationally efficient, and resilient to the optimal $\frac16 - ε$ adversarial errors. Our protocol employs a new type of code that we call a layered code, which may be of independent interest. Like a tree code, a layered code allows the coder to encode a message in an online fashion, but is defined on a graph instead of a tree.

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arXiv2022

An Optimal Algorithm for Certifying Monotone Functions

Given query access to a monotone function $f\colon\{0,1\}^n\to\{0,1\}$ with certificate complexity $C(f)$ and an input $x^{\star}$, we design an algorithm that outputs a size-$C(f)$ subset of $x^{\star}$ certifying the value of $f(x^{\star})$. Our algorithm makes $O(C(f) \cdot \log n)$ queries to $f$, which matches the information-theoretic lower bound for this problem and resolves the concrete open question posed in the STOC '22 paper of Blanc, Koch, Lange, and Tan [BKLT22]. We extend this result to an algorithm that finds a size-$2C(f)$ certificate for a real-valued monotone function with $O(C(f) \cdot \log n)$ queries. We also complement our algorithms with a hardness result, in which we show that finding the shortest possible certificate in $x^{\star}$ may require $Ω\left(\binom{n}{C(f)}\right)$ queries in the worst case.

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arXiv2022

Positive Rate Binary Interactive Error Correcting Codes Resilient to $>\frac12$ Adversarial Erasures

An interactive error correcting code ($\mathsf{iECC}$) is an interactive protocol with the guarantee that the receiver can correctly determine the sender's message, even in the presence of noise. This generalizes the concept of an error correcting code ($\mathsf{ECC}$), which is a non-interactive $\mathsf{iECC}$ that is known to have erasure resilience capped at $\frac12$. The work of \cite{GuptaTZ21} constructed the first $\mathsf{iECC}$ resilient to $> \frac12$ adversarial erasures. However, their $\mathsf{iECC}$ has communication complexity quadratic in the message size. In our work, we construct the first positive rate $\mathsf{iECC}$ resilient to $> \frac12$ adversarial erasures. For any $ε> 0$, our $\mathsf{iECC}$ is resilient to $\frac6{11} - ε$ adversarial erasures and has size $O_ε(n)$.

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arXiv2021

Interactive Error Correcting Codes Over Binary Erasure Channels Resilient to $>\frac12$ Adversarial Corruption

An error correcting code ($\mathsf{ECC}$) allows a sender to send a message to a receiver such that even if a constant fraction of the communicated bits are corrupted, the receiver can still learn the message correctly. Due to their importance and fundamental nature, $\mathsf{ECC}$s have been extensively studied, one of the main goals being to maximize the fraction of errors that the $\mathsf{ECC}$ is resilient to. For adversarial erasure errors (over a binary channel) the maximal error resilience of an $\mathsf{ECC}$ is $\frac12$ of the communicated bits. In this work, we break this $\frac12$ barrier by introducing the notion of an interactive error correcting code ($\mathsf{iECC}$) and constructing an $\mathsf{iECC}$ that is resilient to adversarial erasure of $\frac35$ of the total communicated bits. We emphasize that the adversary can corrupt both the sending party and the receiving party, and that both parties' rounds contribute to the adversary's budget. We also prove an impossibility (upper) bound of $\frac23$ on the maximal resilience of any binary $\mathsf{iECC}$ to adversarial erasures. In the bit flip setting, we prove an impossibility bound of $\frac27$.

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arXiv2021

The Optimal Error Resilience of Interactive Communication Over Binary Channels

In interactive coding, Alice and Bob wish to compute some function $f$ of their individual private inputs $x$ and $y$. They do this by engaging in a non-adaptive (fixed order, fixed length) protocol to jointly compute $f(x,y)$. The goal is to do this in an error-resilient way, such that even given some fraction of adversarial corruptions to the protocol, both parties still learn $f(x,y)$. In this work, we study the optimal error resilience of such a protocol in the face of adversarial bit flip or erasures. While the optimal error resilience of such a protocol over a large alphabet is well understood, the situation over the binary alphabet has remained open. In this work, we resolve this problem of determining the optimal error resilience over binary channels. In particular, we construct protocols achieving $\frac16$ error resilience over the binary bit flip channel and $\frac12$ error resilience over the binary erasure channel, for both of which matching upper bounds are known. We remark that the communication complexity of our binary bit flip protocol is polynomial in the size of the inputs, and the communication complexity of our binary erasure protocol is linear in the size of the minimal noiseless protocol computing $f$.

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arXiv2019

A formula for $F$-Polynomials in terms of $C$-Vectors and Stabilization of $F$-Polynomials

Given a quiver associated to a cluster algebra and a sequence of vertices, iterative mutation leads to $F$-Polynomials which appear in numerous places in the cluster algebraic literature. The coefficients of the monomials in these $F$-Polynomials are difficult to understand and have been an area of study for many years. In this paper, we present a general closed-form formula for these coefficients in terms of elementary manipulations with $C$-matrices. We then demonstrate the effectiveness of the formula by using it to derive simple explicit formulas for $F$-Polynomials of specific classes of quivers and mutation sequences. Work has been done to do these cases in ad-hoc combinatorial ways, but our formula recovers and improves known formulas with a general method. Secondly, we investigate convergence of $F$-polynomials. In themselves, they do not converge, but by changing bases using the $C$-matrix, they conjecturally do. Since our formula relates $C$-matrix entries to coefficients, we are able to apply it to make considerable progress on the conjecture. Specifically, we show stability for green mutation sequences. Finally, we look at exact formulas for these stable deformed $F$-polynomials in instances where they illustrate properties that the $F$-polynomials themselves do not.

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arXiv2016

Bounding extremal functions of forbidden $0-1$ matrices using $(r,s)$-formations

First, we prove tight bounds of $n 2^{\frac{1}{(t-2)!}α(n)^{t-2} \pm O(α(n)^{t-3})}$ on the extremal function of the forbidden pair of ordered sequences $(1 2 3 \ldots k)^t$ and $(k \ldots 3 2 1)^t$ using bounds on a class of sequences called $(r,s)$-formations. Then, we show how an analogous method can be used to derive similar bounds on the extremal functions of forbidden pairs of $0-1$ matrices consisting of horizontal concatenations of identical identity matrices and their horizontal reflections.

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