arXiv subjects

Mihir Singhal

Publications and source records attributed to Mihir Singhal.

17 recordsLinked to original sources

arXiv2026

Average-Radius List-Decodability of Random Linear Codes

We prove that for every prime power $q$ and every $p \in (0, 1-1/q)$, a random $\mathbb{F}_q$-linear code of rate $1 - h_q(p) - \epsilon$ is $(p, C_{p,q}/\epsilon)$-average-radius list-decodable with probability at least $1 - q^{-\Omega(n)}$, i.e., for every center $y \in \mathbb{F}_q^n$, the $C_{p,q}/\epsilon$ codewords closest to $y$ have average fractional Hamming distance at least $p$ from $y$. This extends a similar result for (standard) list-decoding due to Guruswami, H\r{a}stad, and Kopparty (2010) to the stronger average-radius guarantee, with the same $O(1/\epsilon)$ list size. For average-radius list-decoding, such a result was previously known only for binary linear codes (Guruswami, Li, Mosheiff, Resch, Silas, and Wootters, 2021) and for general (non-linear) random codes over arbitrary alphabets (Elias, 1991).

cs.IT↗

arXiv2026

Improved Local Computation Algorithms for Greedy Set Cover via Retroactive Updates

In this work, we focus on designing an efficient Local Computation Algorithm (LCA) for the set cover problem, which is a core optimization task. The state-of-the-art LCA for computing $O(\log \Delta)$-approximate set cover, developed by Grunau, Mitrovi\'c, Rubinfeld, and Vakilian [SODA '20], achieves query complexity of $\Delta^{O(\log \Delta)} \cdot f^{O(\log \Delta \cdot (\log \log \Delta + \log \log f))}$, where $\Delta$ is the maximum set size, and $f$ is the maximum frequency of any element in sets. We present a new LCA that solves this problem using $f^{O(\log \Delta)}$ queries. Specifically, for instances where $f = \text{poly} \log \Delta$, our algorithm improves the query complexity from $\Delta^{O(\log \Delta)}$ to $\Delta^{O(\log \log \Delta)}$. Our central technical contribution in designing LCAs is to aggressively sparsify the input instance but to allow for \emph{retroactive updates}. Namely, our main LCA sometimes ``corrects'' decisions it made in the previous recursive LCA calls. It enables us to achieve stronger concentration guarantees, which in turn allows for more efficient and ``sparser'' LCA execution. We believe that this technique will be of independent interest.

cs.DS↗

arXiv2026

Nearly Time-Optimal Pure State Tomography with Pauli Measurements

We give an algorithm for pure state tomography with near-optimal copy and time complexity using only single-qubit measurements. Specifically, given $\widetilde{O}(2^n/\epsilon)$ copies of an unknown $n$-qubit pure state $|\psi\rangle$, the algorithm performs only nonadaptive Pauli measurements, runs in time $\widetilde{O}(2^n/\epsilon)$, and outputs $|\widehat{\psi} \rangle$ with fidelity at least $1-\epsilon$ with $|\psi\rangle$ with high probability. This is the first algorithm for pure state tomography that achieves near-optimal running time.

quant-ph↗

arXiv2025

The communication complexity of distributed estimation

We study an extension of the standard two-party communication model in which Alice and Bob hold probability distributions $p$ and $q$ over domains $X$ and $Y$, respectively. Their goal is to estimate \[ \mathbb{E}_{x \sim p,\, y \sim q}[f(x, y)] \] to within additive error $\varepsilon$ for a bounded function $f$, known to both parties. We refer to this as the distributed estimation problem. Special cases of this problem arise in a variety of areas including sketching, databases and learning. Our goal is to understand how the required communication scales with the communication complexity of $f$ and the error parameter $\varepsilon$. The random sampling approach -- estimating the mean by averaging $f$ over $O(1/\varepsilon^2)$ random samples -- requires $O(R(f)/\varepsilon^2)$ total communication, where $R(f)$ is the randomized communication complexity of $f$. We design a new debiasing protocol which improves the dependence on $1/\varepsilon$ to be linear instead of quadratic. Additionally we show better upper bounds for several special classes of functions, including the Equality and Greater-than functions. We introduce lower bound techniques based on spectral methods and discrepancy, and show the optimality of many of our protocols: the debiasing protocol is tight for general functions, and that our protocols for the equality and greater-than functions are also optimal. Furthermore, we show that among full-rank Boolean functions, Equality is essentially the easiest.

cs.CC↗

arXiv2025

Breaking the Quadratic Barrier: Robust Cardinality Sketches for Adaptive Queries

Cardinality sketches are compact data structures that efficiently estimate the number of distinct elements across multiple queries while minimizing storage, communication, and computational costs. However, recent research has shown that these sketches can fail under {\em adaptively chosen queries}, breaking down after approximately $\tilde{O}(k^2)$ queries, where $k$ is the sketch size. In this work, we overcome this \emph{quadratic barrier} by designing robust estimators with fine-grained guarantees. Specifically, our constructions can handle an {\em exponential number of adaptive queries}, provided that each element participates in at most $\tilde{O}(k^2)$ queries. This effectively shifts the quadratic barrier from the total number of queries to the number of queries {\em sharing the same element}, which can be significantly smaller. Beyond cardinality sketches, our approach expands the toolkit for robust algorithm design.

cs.DS↗

arXiv2025

Locally computing edge orientations

We consider the question of orienting the edges in a graph $G$ such that every vertex has bounded out-degree. For graphs of arboricity $\alpha$, there is an orientation in which every vertex has out-degree at most $\alpha$ and, moreover, the best possible maximum out-degree of an orientation is at least $\alpha - 1$. We are thus interested in algorithms that can achieve a maximum out-degree of close to $\alpha$. A widely studied approach for this problem in the distributed algorithms setting is a ``peeling algorithm'' that provides an orientation with maximum out-degree $\alpha(2+\epsilon)$ in a logarithmic number of iterations. We consider this problem in the local computation algorithm (LCA) model, which quickly answers queries of the form ``What is the orientation of edge $(u,v)$?'' by probing the input graph. When the peeling algorithm is executed in the LCA setting by applying standard techniques, e.g., the Parnas-Ron paradigm, it requires $\Omega(n)$ probes per query on an $n$-vertex graph. In the case where $G$ has unbounded degree, we show that any LCA that orients its edges to yield maximum out-degree $r$ must use $\Omega(\sqrt n/r)$ probes to $G$ per query in the worst case, even if $G$ is known to be a forest (that is, $\alpha=1$). We also show several algorithms with sublinear probe complexity when $G$ has unbounded degree. When $G$ is a tree such that the maximum degree $\Delta$ of $G$ is bounded, we demonstrate an algorithm that uses $\Delta n^{1-\log_\Delta r + o(1)}$ probes to $G$ per query. To obtain this result, we develop an edge-coloring approach that ultimately yields a graph-shattering-like result. We also use this shattering-like approach to demonstrate an LCA which $4$-colors any tree using sublinear probes per query.

cs.DS↗

arXiv2024

One Attack to Rule Them All: Tight Quadratic Bounds for Adaptive Queries on Cardinality Sketches

Cardinality sketches are compact data structures for representing sets or vectors. These sketches are space-efficient, typically requiring only logarithmic storage in the input size, and enable approximation of cardinality (or the number of nonzero entries). A crucial property in applications is \emph{composability}, meaning that the sketch of a union of sets can be computed from individual sketches. Existing designs provide strong statistical guarantees, ensuring that a randomly sampled sketching map remains robust for an exponential number of queries in terms of the sketch size $k$. However, these guarantees degrade to quadratic in $k$ when queries are \emph{adaptive}, meaning they depend on previous responses. Prior works on statistical queries (Steinke and Ullman, 2015) and specific MinHash cardinality sketches (Ahmadian and Cohen, 2024) established that this is tight in that they can be compromised using a quadratic number of adaptive queries. In this work, we develop a universal attack framework that applies to broad classes of cardinality sketches. We show that any union-composable sketching map can be compromised with $\tilde{O}(k^4)$ adaptive queries and this improves to a tight bound of $\tilde{O}(k^2)$ for monotone maps (including MinHash, statistical queries, and Boolean linear maps). Similarly, any linear sketching map over the reals $\mathbb{R}$ and finite fields $\mathbb{F}_p$ can be compromised using $\tilde{O}(k^2)$ adaptive queries, which is optimal and strengthens some of the recent results by~\citet{GribelyukLWYZ:FOCS2024}, who established a weaker polynomial bound.

cs.DS↗

arXiv2024

Tight bounds for stream decodable error-correcting codes

In order to communicate a message over a noisy channel, a sender (Alice) uses an error-correcting code to encode her message $x$ into a codeword. The receiver (Bob) decodes it correctly whenever there is at most a small constant fraction of adversarial error in the transmitted codeword. This work investigates the setting where Bob is computationally bounded. Specifically, Bob receives the message as a stream and must process it and write $x$ in order to a write-only tape while using low (say polylogarithmic) space. We show three basic results about this setting, which are informally as follows: (1) There is a stream decodable code of near-quadratic length. (2) There is no stream decodable code of sub-quadratic length. (3) If Bob need only compute a private linear function of the input bits, instead of writing them all to the output tape, there is a stream decodable code of near-linear length.

cs.IT↗

arXiv2024

Optimal quantile estimation: beyond the comparison model

Estimating quantiles is one of the foundational problems of data sketching. Given $n$ elements $x_1, x_2, \dots, x_n$ from some universe of size $U$ arriving in a data stream, a quantile sketch estimates the rank of any element with additive error at most $\varepsilon n$. A low-space algorithm solving this task has applications in database systems, network measurement, load balancing, and many other practical scenarios. Current quantile estimation algorithms described as optimal include the GK sketch (Greenwald and Khanna 2001) using $O(\varepsilon^{-1} \log n)$ words (deterministic) and the KLL sketch (Karnin, Lang, and Liberty 2016) using $O(\varepsilon^{-1} \log\log(1/δ))$ words (randomized, with failure probability $δ$). However, both algorithms are only optimal in the comparison-based model, whereas most typical applications involve streams of integers that the sketch can use aside from making comparisons. If we go beyond the comparison-based model, the deterministic q-digest sketch (Shrivastava, Buragohain, Agrawal, and Suri 2004) achieves a space complexity of $O(\varepsilon^{-1}\log U)$ words, which is incomparable to the previously-mentioned sketches. It has long been asked whether there is a quantile sketch using $O(\varepsilon^{-1})$ words of space (which is optimal as long as $n \leq \mathrm{poly}(U)$). In this work, we present a deterministic algorithm using $O(\varepsilon^{-1})$ words, resolving this line of work.

cs.DS↗

arXiv2024

Omnipredictors for Regression and the Approximate Rank of Convex Functions

Consider the supervised learning setting where the goal is to learn to predict labels $\mathbf y$ given points $\mathbf x$ from a distribution. An \textit{omnipredictor} for a class $\mathcal L$ of loss functions and a class $\mathcal C$ of hypotheses is a predictor whose predictions incur less expected loss than the best hypothesis in $\mathcal C$ for every loss in $\mathcal L$. Since the work of [GKR+21] that introduced the notion, there has been a large body of work in the setting of binary labels where $\mathbf y \in \{0, 1\}$, but much less is known about the regression setting where $\mathbf y \in [0,1]$ can be continuous. Our main conceptual contribution is the notion of \textit{sufficient statistics} for loss minimization over a family of loss functions: these are a set of statistics about a distribution such that knowing them allows one to take actions that minimize the expected loss for any loss in the family. The notion of sufficient statistics relates directly to the approximate rank of the family of loss functions. Our key technical contribution is a bound of $O(1/\varepsilon^{2/3})$ on the $ε$-approximate rank of convex, Lipschitz functions on the interval $[0,1]$, which we show is tight up to a factor of $\mathrm{polylog} (1/ε)$. This yields improved runtimes for learning omnipredictors for the class of all convex, Lipschitz loss functions under weak learnability assumptions about the class $\mathcal C$. We also give efficient omnipredictors when the loss families have low-degree polynomial approximations, or arise from generalized linear models (GLMs). This translation from sufficient statistics to faster omnipredictors is made possible by lifting the technique of loss outcome indistinguishability introduced by [GKH+23] for Boolean labels to the regression setting.

cs.LG↗

arXiv2022

Low-Degree Multicalibration

Introduced as a notion of algorithmic fairness, multicalibration has proved to be a powerful and versatile concept with implications far beyond its original intent. This stringent notion -- that predictions be well-calibrated across a rich class of intersecting subpopulations -- provides its strong guarantees at a cost: the computational and sample complexity of learning multicalibrated predictors are high, and grow exponentially with the number of class labels. In contrast, the relaxed notion of multiaccuracy can be achieved more efficiently, yet many of the most desirable properties of multicalibration cannot be guaranteed assuming multiaccuracy alone. This tension raises a key question: Can we learn predictors with multicalibration-style guarantees at a cost commensurate with multiaccuracy? In this work, we define and initiate the study of Low-Degree Multicalibration. Low-Degree Multicalibration defines a hierarchy of increasingly-powerful multi-group fairness notions that spans multiaccuracy and the original formulation of multicalibration at the extremes. Our main technical contribution demonstrates that key properties of multicalibration, related to fairness and accuracy, actually manifest as low-degree properties. Importantly, we show that low-degree multicalibration can be significantly more efficient than full multicalibration. In the multi-class setting, the sample complexity to achieve low-degree multicalibration improves exponentially (in the number of classes) over full multicalibration. Our work presents compelling evidence that low-degree multicalibration represents a sweet spot, pairing computational and sample efficiency with strong fairness and accuracy guarantees.

cs.LG↗

arXiv2021

Computations associated with the resonance arrangement

The resonance arrangement $\mathcal{A}_n$ is the arrangement of hyperplanes in $\mathbb{R}^n$ given by all hyperplanes of the form $\sum_{i \in I} x_i = 0$, where $I$ is a nonempty subset of $\{1,\dots,n\}$. We consider the characteristic polynomial $χ(\mathcal{A}_n; t)$ of the resonance arrangement, whose value $R_n$ at $-1$ is of particular interest, and corresponds to counts of generalized retarded functions in quantum field theory, among other things. No formula is known for either the characteristic polynomial or $R_n$, though $R_n$ has been computed up to $n=8$. By exploiting symmetry and using computational methods, we compute the characteristic polynomial of $\mathcal{A}_9$, and thus obtain $R_9$. The coefficients of the characteristic polynomial are also equal to the so-called Betti numbers of the complexified hyperplane arrangement; that is, the coefficient of $t^{n-i}$ is denoted by the Betti number $b_i(\mathcal{A}_n)$. Explicit formulas are known for the Betti numbers up to $b_3(\mathcal{A}_n)$. Using computational methods, we also obtain an explicit formula for $b_4(\mathcal{A}_n)$, which gives the $t^{n-4}$ coefficient of the characteristic polynomial.

math.CO↗

arXiv2021

Lower bounds for superpatterns and universal sequences

A permutation $σ\in S_n$ is said to be $k$-universal or a $k$-superpattern if for every $π\in S_k$, there is a subsequence of $σ$ that is order-isomorphic to $π$. A simple counting argument shows that $σ$ can be a $k$-superpattern only if $n\ge (1/e^2+o(1))k^2$, and Arratia conjectured that this lower bound is best-possible. Disproving Arratia's conjecture, we improve the trivial bound by a small constant factor. We accomplish this by designing an efficient encoding scheme for the patterns that appear in $σ$. This approach is quite flexible and is applicable to other universality-type problems; for example, we also improve a bound by Engen and Vatter on a problem concerning $(k+1)$-ary sequences which contain all $k$-permutations.

math.CO↗

arXiv2020

Tetris is NP-hard even with $O(1)$ rows or columns

We prove that the classic falling-block video game Tetris (both survival and board clearing) remains NP-complete even when restricted to 8 columns, or to 4 rows, settling open problems posed over 15 years ago [BDH+04]. Our reduction is from 3-Partition, similar to the previous reduction for unrestricted board sizes, but with a better packing of buckets. On the positive side, we prove that 2-column Tetris (and 1-row Tetris) is polynomial. We also prove that the generalization of Tetris to larger $k$-omino pieces is NP-complete even when the board starts empty, even when restricted to 3 columns or 2 rows or constant-size pieces. Finally, we present an animated Tetris font.

cs.CC↗

arXiv2020

Families with no perfect matchings

We consider families of $k$-subsets of $\{1, \dots, n\}$, where $n$ is a multiple of $k$, which have no perfect matching. An equivalent condition for a family $\mathcal{F}$ to have no perfect matching is for there to be a blocking set, which is a set of $b$ elements of $\{1, \dots, n\}$ that cannot be covered by $b$ disjoint sets in $\mathcal{F}$. We are specifically interested in the largest possible size of a family $\mathcal{F}$ with no perfect matching and no blocking set of size less than $b$. Frankl resolved the case of families with no singleton blocking set (in other words, the $b=2$ case) for sufficiently large $n$ and conjectured an optimal construction for general $b$. Though Frankl's construction fails to be optimal for $k = 2, 3$, we show that the construction is optimal whenever $k \ge 100$ and $n$ is sufficiently large.

math.CO↗

arXiv2020

Unimodality of a refinement of Lassalle's sequence

Defant, Engen, and Miller defined a refinement of Lassalle's sequence $A_{k+1}$ by considering uniquely sorted permutations of length $2k+1$ whose first element is $\ell$. They showed that each such sequence is symmetric in $\ell$ and conjectured that these sequences are unimodal. We prove that the sequences are unimodal.

math.CO↗

arXiv2019

Erdos-Littlewood-Offord problem with arbitrary probabilities

The classical Erdős-Littlewood-Offord problem concerns the random variable $X = a_1 ξ_1 + \dots + a_n ξ_n$, where $a_i \in \mathbb{R} \setminus \{0\}$ are fixed and $ξ_i \sim \text{Ber}(1/2)$ are independent. The Erdős-Littlewood-Offord theorem states that the maximum possible concentration probability $\max_{x \in \mathbb{R}} \Pr(X = x)$ is $\binom{n}{\lfloor n/2\rfloor} / 2^n$, achieved when the $a_i$ are all $1$. As proposed by Fox, Kwan, and Sauermann, we investigate the general case where $ξ_i \sim \text{Ber}(p)$ instead. Using purely combinatorial techniques, we show that the exact maximum concentration probability is achieved when $a_i \in \{-1, 1\}$ for each $i$. Then, using Fourier-analytic techniques, we investigate the optimal ratio of $1$s to $-1$s. Surprisingly, we find that in some cases, the numbers of $1$s and $-1$s can be far from equal.

math.CO↗