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N. V. Vinodchandran

Publications and source records attributed to N. V. Vinodchandran.

At least 19 recordsLinked to original sources
arXiv2026

Counting Triangles in Graph Streams with Repeatable and Forgettable Edges

Most existing graph streaming algorithms assume the ideal scenario where each edge arrives only once. Real-world graph streams, such as communication or transaction logs, often contain many repeated occurrences of the same edge. In general, the algorithms developed for the single-edge arrival case can fail when edges can arrive multiple times. Motivated by this, we study the {\em repeated-edge arrival graph streaming model} where an edge is allowed to arrive multiple times. In this work, we study the triangle counting problem in the repeated-edge arrival model: approximate the number of triangles in the underlying {\em simple graph} despite arbitrary edge repetitions. We design the first algorithms for triangle counting with optimal space complexity. In particular, we present a single-pass algorithm that computes an $(\varepsilon,δ)$-approximation of the number of triangles with optimal space complexity. We introduce {\em right-to-be-forgotten graph streaming} (RFGS) model, where a forget operation can cause all previous occurrences of an edge to disappear. We show that our single-pass algorithm can be extended to the RFGS model with optimal space complexity. Finally, we present optimal constant-pass algorithms that compute an $(\varepsilon,δ)$-approximation of the number of triangles and cliques for the repeated-edge arrival graph streams.

cs.DS↗
arXiv2026

Estimating Size of the Union of Sets in Streaming Model

We study estimating the size of the union of sets $S_1,\dots,S_M$, where each $S_i\subseteqΩ$ is presented implicitly and arrives in a stream. We introduce Delphic sets, a class of streaming problems in which membership, sampling, and counting queries to each set are efficient, and show that this notion captures three well-known problems: Klee's measure problem (discrete version), test coverage estimation in combinatorial testing, and model counting of DNF formulas. Our primary contribution is a simple and efficient sampling-based algorithm that outputs an $(\varepsilon,δ)$-approximation of the cardinality of the union of Delphic sets in the streaming setting. It has space complexity $O(R\log|Ω|)$ and update time $O(R\log R\cdot\log(M/δ)\cdot\log|Ω|)$, where $R=O(\log(M/δ)\cdot\varepsilon^{-2})$. For the streaming Klee's measure problem, this gives the first algorithm whose update time depends linearly on the dimension $d$ for $d>1$, settling an open problem of Tirthapura and Woodruff (PODS 2012), and it directly yields efficient streaming algorithms for coverage estimation and DNF model counting. We further show that the space for coverage estimation can be made near-optimal at the cost of an update procedure in $\mathrm{P}^{\mathrm{NP}}$, revealing a time-space trade-off. A key strength of our approach is the simplicity of both the algorithm and its analysis, which makes it amenable to practical implementation. In this revised version, the algorithm and its correctness analysis have additionally been formalized and machine-checked in Lean 4. (Shortened for Arxiv)

cs.DS↗
arXiv2026

Computing over Data Streams using Catalytic Space

We introduce a streaming model with \emph{catalytic memory}, an auxiliary workspace that must be returned to its initial state at the end of the computation. We show that catalytic space yields dramatic space savings for data stream algorithms. We first study the exact computation of frequency moments in insertion-only data streams. For every $k\ge1$, we give an exact four-pass algorithm for computing $\mathbb{F}_{k}$ using $O(k\log m)$ clean space, where $m$ is the stream length. We also present a $(k+1)$-pass algorithm with the same clean-space complexity that uses a factor of $k$ less catalytic space than the four-pass algorithm. For small moments, we obtain stronger results. In particular, we show that $\mathbb{F}_{2}$ and $\mathbb{F}_{3}$ can be computed exactly in two and three passes, respectively, using only $O(\log m)$ clean space. Additionally, we show that exact $\mathbb{F}_{0}$ computation reduces to computing $\mathbb{F}_{k}$ for a suitably chosen large value of $k$, resulting in an exact four-pass algorithm for $\mathbb{F}_{0}$ using only $O(\log m)$ clean space. We further show how our frequency-moment algorithms can be used to exactly count induced occurrences of any fixed graph $H$ in a graph stream, yielding a four-pass algorithm that uses $O_H(\log n)$ clean space, where $n$ is the number of vertices in the graph. As a special case, we obtain an exact three-pass algorithm for triangle counting using $O(\log n)$ clean space. All of our algorithms are multi-pass. We complement these algorithmic results with a matching limitation showing that catalytic memory does not provide additional power in the single-pass setting. Specifically, we prove that every randomized or deterministic single-pass streaming algorithm using $s$ bits of clean memory and catalytic space can be simulated in the standard streaming model, without catalytic memory, using $O(s)$ space.

cs.DS↗
arXiv2025

Fast Rational Search via Stern-Brocot Tree

We revisit the problem of rational search: given an unknown rational number $α= \frac{a}{b} \in (0,1)$ with $b \leq n$, the goal is to identify $α$ using comparison queries of the form ``$β\leq α$?''. The problem has been studied several decades ago and optimal query algorithms are known. We present a new algorithm for rational search based on a compressed traversal of the Stern--Brocot tree, which appeared to have been overlooked in the literature. This approach also naturally extends to two related problems that, to the best of our knowledge, have not been previously addressed: (i) unbounded rational search, where the bound $n$ is unknown, and (ii) computing the best (in a precise sense) rational approximation of an unknown real number using only comparison queries.

cs.DS↗
arXiv2025

List Replicable Reinforcement Learning

Replicability is a fundamental challenge in reinforcement learning (RL), as RL algorithms are empirically observed to be unstable and sensitive to variations in training conditions. To formally address this issue, we study \emph{list replicability} in the Probably Approximately Correct (PAC) RL framework, where an algorithm must return a near-optimal policy that lies in a \emph{small list} of policies across different runs, with high probability. The size of this list defines the \emph{list complexity}. We introduce both weak and strong forms of list replicability: the weak form ensures that the final learned policy belongs to a small list, while the strong form further requires that the entire sequence of executed policies remains constrained. These objectives are challenging, as existing RL algorithms exhibit exponential list complexity due to their instability. Our main theoretical contribution is a provably efficient tabular RL algorithm that guarantees list replicability by ensuring the list complexity remains polynomial in the number of states, actions, and the horizon length. We further extend our techniques to achieve strong list replicability, bounding the number of possible policy execution traces polynomially with high probability. Our theoretical result is made possible by key innovations including (i) a novel planning strategy that selects actions based on lexicographic order among near-optimal choices within a randomly chosen tolerance threshold, and (ii) a mechanism for testing state reachability in stochastic environments while preserving replicability. Finally, we demonstrate that our theoretical investigation sheds light on resolving the \emph{instability} issue of RL algorithms used in practice. In particular, we show that empirically, our new planning strategy can be incorporated into practical RL frameworks to enhance their stability.

cs.LG↗
arXiv2025

Distribution Learning Meets Graph Structure Sampling

This work establishes a novel link between the problem of PAC-learning high-dimensional graphical models and the task of (efficient) counting and sampling of graph structures, using an online learning framework. We observe that if we apply the exponentially weighted average (EWA) or randomized weighted majority (RWM) forecasters on a sequence of samples from a distribution P using the log loss function, the average regret incurred by the forecaster's predictions can be used to bound the expected KL divergence between P and the predictions. Known regret bounds for EWA and RWM then yield new sample complexity bounds for learning Bayes nets. Moreover, these algorithms can be made computationally efficient for several interesting classes of Bayes nets. Specifically, we give a new sample-optimal and polynomial time learning algorithm with respect to trees of unknown structure and the first polynomial sample and time algorithm for learning with respect to Bayes nets over a given chordal skeleton.

cs.LG↗
arXiv2025

Algorithms and Hardness for Estimating Statistical Similarity

We introduce and study the computational problem of determining statistical similarity between probability distributions. For distributions $P$ and $Q$ over a finite sample space, their statistical similarity is defined as $S_{\mathrm{stat}}(P, Q) := \sum_x \min(P(x), Q(x))$. Despite its fundamental nature as a measure of similarity between distributions, capturing essential concepts such as Bayes error in prediction and hypothesis testing, this computational problem has not been previously explored. Recent work on computing statistical distance has established that, somewhat surprisingly, even for the simple class of product distributions, exactly computing statistical similarity is $\#\mathsf{P}$-hard. This motivates the question of designing approximation algorithms for statistical similarity. Our first contribution is a Fully Polynomial-Time deterministic Approximation Scheme (FPTAS) for estimating statistical similarity between two product distributions. Furthermore, we also establish a complementary hardness result. In particular, we show that it is $\mathsf{NP}$-hard to estimate statistical similarity when $P$ and $Q$ are Bayes net distributions of in-degree $2$.

cs.DS↗
arXiv2024

Computational Explorations of Total Variation Distance

We investigate some previously unexplored (or underexplored) computational aspects of total variation (TV) distance. First, we give a simple deterministic polynomial-time algorithm for checking equivalence between mixtures of product distributions, over arbitrary alphabets. This corresponds to a special case, whereby the TV distance between the two distributions is zero. Second, we prove that unless $\mathsf{NP} \subseteq \mathsf{RP}$, it is impossible to efficiently estimate the TV distance between arbitrary Ising models, even in a bounded-error randomized setting.

cs.DS↗
arXiv2024

Point Prediction for Streaming Data

We present two new approaches for point prediction with streaming data. One is based on the Count-Min sketch (CMS) and the other is based on Gaussian process priors with a random bias. These methods are intended for the most general predictive problems where no true model can be usefully formulated for the data stream. In statistical contexts, this is often called the $\mathcal{M}$-open problem class. Under the assumption that the data consists of i.i.d samples from a fixed distribution function $F$, we show that the CMS-based estimates of the distribution function are consistent. We compare our new methods with two established predictors in terms of cumulative $L^1$ error. One is based on the Shtarkov solution (often called the normalized maximum likelihood) in the normal experts setting and the other is based on Dirichlet process priors. These comparisons are for two cases. The first is one-pass meaning that the updating of the predictors is done using the fact that the CMS is a sketch. For predictors that are not one-pass, we use streaming $K$-means to give a representative subset of fixed size that can be updated as data accumulate. Preliminary computational work suggests that the one-pass median version of the CMS method is rarely outperformed by the other methods for sufficiently complex data. We also find that predictors based on Gaussian process priors with random biases perform well. The Shtarkov predictors we use here did not perform as well probably because we were only using the simplest example. The other predictors seemed to perform well mainly when the data did not look like they came from an M-open data generator.

stat.ML↗
arXiv2024

Total Variation Distance Meets Probabilistic Inference

In this paper, we establish a novel connection between total variation (TV) distance estimation and probabilistic inference. In particular, we present an efficient, structure-preserving reduction from relative approximation of TV distance to probabilistic inference over directed graphical models. This reduction leads to a fully polynomial randomized approximation scheme (FPRAS) for estimating TV distances between same-structure distributions over any class of Bayes nets for which there is an efficient probabilistic inference algorithm. In particular, it leads to an FPRAS for estimating TV distances between distributions that are defined over a common Bayes net of small treewidth. Prior to this work, such approximation schemes only existed for estimating TV distances between product distributions. Our approach employs a new notion of $partial$ couplings of high-dimensional distributions, which might be of independent interest.

cs.DS↗
arXiv2023

On Approximating Total Variation Distance

Total variation distance (TV distance) is a fundamental notion of distance between probability distributions. In this work, we introduce and study the problem of computing the TV distance of two product distributions over the domain $\{0,1\}^n$. In particular, we establish the following results. 1. The problem of exactly computing the TV distance of two product distributions is $\#\mathsf{P}$-complete. This is in stark contrast with other distance measures such as KL, Chi-square, and Hellinger which tensorize over the marginals leading to efficient algorithms. 2. There is a fully polynomial-time deterministic approximation scheme (FPTAS) for computing the TV distance of two product distributions $P$ and $Q$ where $Q$ is the uniform distribution. This result is extended to the case where $Q$ has a constant number of distinct marginals. In contrast, we show that when $P$ and $Q$ are Bayes net distributions, the relative approximation of their TV distance is $\mathsf{NP}$-hard.

cs.DS↗
arXiv2023

Neighborhood Variants of the KKM Lemma, Lebesgue Covering Theorem, and Sperner's Lemma on the Cube

We establish a "neighborhood" variant of the cubical KKM lemma and the Lebesgue covering theorem and deduce a discretized version which is a "neighborhood" variant of Sperner's lemma on the cube. The main result is the following: for any coloring of the unit $d$-cube $[0,1]^d$ in which points on opposite faces must be given different colors, and for any $\varepsilon>0$, there is an $\ell_\infty$ $\varepsilon$-ball which contains points of at least $(1+\frac{\varepsilon}{1+\varepsilon})^d$ different colors, (so in particular, at least $(1+\frac{2}{3}\varepsilon)^d$ different colors for all sensible $\varepsilon\in(0,\frac12]$).

math.CO↗
arXiv2023

Distinct Elements in Streams: An Algorithm for the (Text) Book

Given a data stream $\mathcal{A} = \langle a_1, a_2, \ldots, a_m \rangle$ of $m$ elements where each $a_i \in [n]$, the Distinct Elements problem is to estimate the number of distinct elements in $\mathcal{A}$.Distinct Elements has been a subject of theoretical and empirical investigations over the past four decades resulting in space optimal algorithms for it.All the current state-of-the-art algorithms are, however, beyond the reach of an undergraduate textbook owing to their reliance on the usage of notions such as pairwise independence and universal hash functions. We present a simple, intuitive, sampling-based space-efficient algorithm whose description and the proof are accessible to undergraduates with the knowledge of basic probability theory.

cs.DS↗
arXiv2023

Geometry of Rounding: Near Optimal Bounds and a New Neighborhood Sperner's Lemma

A partition $\mathcal{P}$ of $\mathbb{R}^d$ is called a $(k,\varepsilon)$-secluded partition if, for every $\vec{p} \in \mathbb{R}^d$, the ball $\overline{B}_{\infty}(\varepsilon, \vec{p})$ intersects at most $k$ members of $\mathcal{P}$. A goal in designing such secluded partitions is to minimize $k$ while making $\varepsilon$ as large as possible. This partition problem has connections to a diverse range of topics, including deterministic rounding schemes, pseudodeterminism, replicability, as well as Sperner/KKM-type results. In this work, we establish near-optimal relationships between $k$ and $\varepsilon$. We show that, for any bounded measure partitions and for any $d\geq 1$, it must be that $k\geq(1+2\varepsilon)^d$. Thus, when $k=k(d)$ is restricted to ${\rm poly}(d)$, it follows that $\varepsilon=\varepsilon(d)\in O\left(\frac{\ln d}{d}\right)$. This bound is tight up to log factors, as it is known that there exist secluded partitions with $k(d)=d+1$ and $\varepsilon(d)=\frac{1}{2d}$. We also provide new constructions of secluded partitions that work for a broad spectrum of $k(d)$ and $\varepsilon(d)$ parameters. Specifically, we prove that, for any $f:\mathbb{N}\rightarrow\mathbb{N}$, there is a secluded partition with $k(d)=(f(d)+1)^{\lceil\frac{d}{f(d)}\rceil}$ and $\varepsilon(d)=\frac{1}{2f(d)}$. These new partitions are optimal up to $O(\log d)$ factors for various choices of $k(d)$ and $\varepsilon(d)$. Based on the lower bound result, we establish a new neighborhood version of Sperner's lemma over hypercubes, which is of independent interest. In addition, we prove a no-free-lunch theorem about the limitations of rounding schemes in the context of pseudodeterministic/replicable algorithms.

cs.CC↗
arXiv2023

List and Certificate Complexities in Replicable Learning

We investigate replicable learning algorithms. Ideally, we would like to design algorithms that output the same canonical model over multiple runs, even when different runs observe a different set of samples from the unknown data distribution. In general, such a strong notion of replicability is not achievable. Thus we consider two feasible notions of replicability called list replicability and certificate replicability. Intuitively, these notions capture the degree of (non) replicability. We design algorithms for certain learning problems that are optimal in list and certificate complexity. We establish matching impossibility results.

cs.LG↗
arXiv2023

Constraint Optimization over Semirings

Interpretations of logical formulas over semirings have applications in various areas of computer science including logic, AI, databases, and security. Such interpretations provide richer information beyond the truth or falsity of a statement. Examples of such semirings include Viterbi semiring, min-max or access control semiring, tropical semiring, and fuzzy semiring. The present work investigates the complexity of constraint optimization problems over semirings. The generic optimization problem we study is the following: Given a propositional formula $φ$ over $n$ variable and a semiring $(K,+,\cdot,0,1)$, find the maximum value over all possible interpretations of $φ$ over $K$. This can be seen as a generalization of the well-known satisfiability problem. A related problem is to find an interpretation that achieves the maximum value. In this work, we first focus on these optimization problems over the Viterbi semiring, which we call optConfVal and optConf. We show that for general propositional formulas in negation normal form, optConfVal and optConf are in ${\mathrm{FP}}^{\mathrm{NP}}$. We investigate optConf when the input formula $φ$ is represented as a CNF. For CNF formulae, we first derive an upper bound on optConfVal as a function of the number of maximum satisfiable clauses. In particular, we show that if $r$ is the maximum number of satisfiable clauses in a CNF formula with $m$ clauses, then its optConfVal is at most $1/4^{m-r}$. Building on this we establish that optConfVal for CNF formulae is hard for the complexity class ${\mathrm{FP}}^{\mathrm{NP}[\log]}$. We also design polynomial-time approximation algorithms and establish an inapproximability for optConfVal. We establish similar complexity results for these optimization problems over other semirings including tropical, fuzzy, and access control semirings.

cs.LO↗
arXiv2022

Geometry of Rounding

Rounding has proven to be a fundamental tool in theoretical computer science. By observing that rounding and partitioning of $\mathbb{R}^d$ are equivalent, we introduce the following natural partition problem which we call the {\em secluded hypercube partition problem}: Given $k\in \mathbb{N}$ (ideally small) and $ε>0$ (ideally large), is there a partition of $\mathbb{R}^d$ with unit hypercubes such that for every point $p \in \mathbb{R}^d$, its closed $ε$-neighborhood (in the $\ell_{\infty}$ norm) intersects at most $k$ hypercubes? We undertake a comprehensive study of this partition problem. We prove that for every $d\in \mathbb{N}$, there is an explicit (and efficiently computable) hypercube partition of $\mathbb{R}^d$ with $k = d+1$ and $ε= \frac{1}{2d}$. We complement this construction by proving that the value of $k=d+1$ is the best possible (for any $ε$) for a broad class of ``reasonable'' partitions including hypercube partitions. We also investigate the optimality of the parameter $ε$ and prove that any partition in this broad class that has $k=d+1$, must have $ε\leq\frac{1}{2\sqrt{d}}$. These bounds imply limitations of certain deterministic rounding schemes existing in the literature. Furthermore, this general bound is based on the currently known lower bounds for the dissection number of the cube, and improvements to this bound will yield improvements to our bounds. While our work is motivated by the desire to understand rounding algorithms, one of our main conceptual contributions is the introduction of the {\em secluded hypercube partition problem}, which fits well with a long history of investigations by mathematicians on various hypercube partitions/tilings of Euclidean space.

cs.DM↗