arXiv subjects

Ramesh Krishnan S. Pallavoor

Publications and source records attributed to Ramesh Krishnan S. Pallavoor.

6 recordsLinked to original sources

arXiv2026

Testing Connectedness and Edge Connectedness

We revisit the classical problem of testing connectedness of graphs represented in the adjacency lists model and obtain optimal bounds. Specifically, we show that $\varepsilon$-testing connectedness of graphs with maximum degree at most $d$ requires $Ω(\frac{1}{\varepsilon} \log \frac{1}{\varepsilon d})$ queries (when $\varepsilon \le \frac{1}{4d^5}$). Our result implies that the connectedness tester of Levi, Pallavoor, Raskhodnikova, and Varma ({\em ACM Trans.\ Comput.\ Theory, 2022}) is optimal. We also study the problem of testing $k$-edge-connectedness of both directed and undirected graphs of average degree $\overline{d}$ in the adjacency lists model and design a tester with query complexity $O(\frac{k}{\varepsilon} \log \frac{1}{\varepsilon \overline{d}})$, improving upon the bounds by Forster, Nanongkai, Saranurak, Yang, and Yingchareonthawornchai (SODA 2020). Furthermore, we complement our upper bound by showing that every one-sided error $k$-edge-connectedness tester must make $Ω(k/\varepsilon)$ queries.

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arXiv2021

Approximating the Distance to Monotonicity of Boolean Functions

We design a nonadaptive algorithm that, given oracle access to a function $f: \{0,1\}^n \to \{0,1\}$ which is $α$-far from monotone, makes poly$(n, 1/α)$ queries and returns an estimate that, with high probability, is an $\widetilde{O}(\sqrt{n})$-approximation to the distance of $f$ to monotonicity. The analysis of our algorithm relies on an improvement to the directed isoperimetric inequality of Khot, Minzer, and Safra (SIAM J. Comput., 2018). Furthermore, we rule out a poly$(n, 1/α)$-query nonadaptive algorithm that approximates the distance to monotonicity significantly better by showing that, for all constant $κ> 0,$ every nonadaptive $n^{1/2 - κ}$-approximation algorithm for this problem requires $2^{n^κ}$ queries. This answers a question of Seshadhri (Property Testing Review, 2014) for the case of nonadaptive algorithms. We obtain our lower bound by proving an analogous bound for erasure-resilient (and tolerant) testers. Our method also yields the same lower bounds for unateness and being a $k$-junta.

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arXiv2020

Erasure-Resilient Sublinear-Time Graph Algorithms

We investigate sublinear-time algorithms that take partially erased graphs represented by adjacency lists as input. Our algorithms make degree and neighbor queries to the input graph and work with a specified fraction of adversarial erasures in adjacency entries. We focus on two computational tasks: testing if a graph is connected or $\varepsilon$-far from connected and estimating the average degree. For testing connectedness, we discover a threshold phenomenon: when the fraction of erasures is less than $\varepsilon$, this property can be tested efficiently (in time independent of the size of the graph); when the fraction of erasures is at least $\varepsilon,$ then a number of queries linear in the size of the graph representation is required. Our erasure-resilient algorithm (for the special case with no erasures) is an improvement over the previously known algorithm for connectedness in the standard property testing model and has optimal dependence on the proximity parameter $\varepsilon$. For estimating the average degree, our results provide an "interpolation" between the query complexity for this computational task in the model with no erasures in two different settings: with only degree queries, investigated by Feige (SIAM J. Comput. `06), and with degree queries and neighbor queries, investigated by Goldreich and Ron (Random Struct. Algorithms `08) and Eden et al. (ICALP `17). We conclude with a discussion of our model and open questions raised by our work.

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arXiv2017

A Lower Bound for Nonadaptive, One-Sided Error Testing of Unateness of Boolean Functions over the Hypercube

A Boolean function $f:\{0,1\}^d \mapsto \{0,1\}$ is unate if, along each coordinate, the function is either nondecreasing or nonincreasing. In this note, we prove that any nonadaptive, one-sided error unateness tester must make $Ω(\frac{d}{\log d})$ queries. This result improves upon the $Ω(\frac{d}{\log^2 d})$ lower bound for the same class of testers due to Chen et al. (STOC, 2017).

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arXiv2017

Optimal Unateness Testers for Real-Valued Functions: Adaptivity Helps

We study the problem of testing unateness of functions $f:\{0,1\}^d \to \mathbb{R}.$ We give a $O(\frac{d}ε \cdot \log\frac{d}ε)$-query nonadaptive tester and a $O(\frac{d}ε)$-query adaptive tester and show that both testers are optimal for a fixed distance parameter $ε$. Previously known unateness testers worked only for Boolean functions, and their query complexity had worse dependence on the dimension both for the adaptive and the nonadaptive case. Moreover, no lower bounds for testing unateness were known. We also generalize our results to obtain optimal unateness testers for functions $f:[n]^d \to \mathbb{R}$. Our results establish that adaptivity helps with testing unateness of real-valued functions on domains of the form $\{0,1\}^d$ and, more generally, $[n]^d$. This stands in contrast to the situation for monotonicity testing where there is no adaptivity gap for functions $f:[n]^d \to \mathbb{R}$.

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arXiv2016

Testing Unateness of Real-Valued Functions

We give a unateness tester for functions of the form $f:[n]^d\rightarrow R$, where $n,d\in \mathbb{N}$ and $R\subseteq \mathbb{R}$ with query complexity $O(\frac{d\log (\max(d,n))}ε)$. Previously known unateness testers work only for Boolean functions over the domain $\{0,1\}^d$. We show that every unateness tester for real-valued functions over hypergrid has query complexity $Ω(\min\{d, |R|^2\})$. Consequently, our tester is nearly optimal for real-valued functions over $\{0,1\}^d$. We also prove that every nonadaptive, 1-sided error unateness tester for Boolean functions needs $Ω(\sqrt{d}/ε)$ queries. Previously, no lower bounds for testing unateness were known.

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