arXiv subjects

Huacheng Yu

Publications and source records attributed to Huacheng Yu.

At least 19 recordsLinked to original sources

arXiv2026

Adaptive Data Analysis and Adversarial Robustness on Insertion-Deletion Streams

Classical streaming algorithms and statistical mechanisms frequently fail when future inputs or queries depend adaptively on prior outputs. In this work, we resolve central open questions in both adaptive data analysis (ADA) and adversarially robust turnstile streaming by introducing a unified hierarchical estimator-corrector-learner framework. For adaptive data analysis over an unrestricted domain, we design a randomized mechanism that answers $k$ adaptively chosen statistical queries to within additive error $α$ using $\tilde{O}(\frac{\sqrt{k}}α + \frac{1}{α^2})$ samples, nearly matching the lower bounds of Hardt and Ullman [FOCS 2014] and Steinke and Ullman [COLT 2015] while improving upon the classic $\tilde{O}(\frac{\sqrt{k}}{α^2})$ bound of Dwork et al. [STOC 2015] and Bassily et al. [STOC 2016]. For known finite domains $U$, we combine our mechanism with private multiplicative weights to improve the sample complexity to $\tilde{O}\left(\frac{1}{α^2}+ \min\left(\frac{\sqrt{k}}α, \frac{\sqrt{\log |U|}}{α^2}\right)\right)$. Building on our ADA mechanism, we construct a fixed linear sketch with a stateful decoder that answers $\tilde{O}(\frac{1}{ε^2})$ adaptive $F_2$ moment estimation queries to within a $(1\pmε)$-multiplicative factor using $\tilde{O}(\frac{1}{ε^2})$ bits of space. Finally, by embedding this sketch into a binary-tree recursion, we obtain the first nearly space-optimal adversarially robust algorithm for $F_2$ estimation on insertion-deletion streams. Our algorithm uses $\tilde O(\frac{1}{ε^2})$ bits of space, improving upon the $\tilde{O}(\frac{1}{ε^4})$ bound of Gribelyuk et al. [STOC 2026] and matching the insertion-only bound of Woodruff and Zhou [FOCS 2021]. This demonstrates that no asymptotic space overhead in $ε$ is required to achieve adversarial robustness in the turnstile model.

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arXiv2026

Turnstile Streaming Algorithms Might (Still) as Well Be Linear Sketches, for Polynomial-Length Streams

A fundamental question in streaming complexity is whether every space-efficient turnstile algorithm is implicitly a linear sketch. The work of Li, Nguyen, and Woodruff (STOC~2014) established an equivalence between the two, but their reduction requires a stream length that is at least doubly exponential in the dimension $n$. In the opposite direction, Kallaugher and Price (STOC~2020) showed a separation for streams of linear length, so the equivalence does not hold in general. The most natural and practically relevant regime, that of polynomial-length streams, has therefore remained open. We show that polynomial-length turnstile algorithms admit linear-sketch simulations. More precisely, if a turnstile algorithm uses $S$ bits of space and succeeds on all streams of length $\textrm{poly}(D, n)$, then on final vectors $x$ with $\|x\|_2 \le D$, its output can be recovered from $O(S)$ linear measurements of $x$, using $O(S)$ bits overall. For smooth problems under appropriate input distributions, a mollified version of the reduction yields a bounded-entry sketch with $O(S / \log D)$ measurements and optimal $O(S)$ total space. Our results extend to strict turnstile streams and non-uniform Read-Once Branching Programs (ROBPs). Instead of the transition-graph arguments of prior work, our proof uses Fourier analysis and tools from additive combinatorics to extract discrete linear measurements. We show that an $S$-bit algorithm can only be sensitive to a small set of heavy Fourier frequencies, which lie close to a finite group of order $2^{O(S)}$, and we use generators of this group as the rows of the sketching matrix. Combined with existing lower bounds for real sketches and for communication, this gives new lower bounds for polynomial-length turnstile streams.

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arXiv2026

Quadratic Probing Insertions Are $ε^{-(1+o(1))}$ Time

First proposed in 1968, quadratic probing has stood for more than half a century as one of the simplest and most widely used hash-table designs in computer science. It is conjectured that, at load factor $1 - ε$, the hash table achieves $O(ε^{-1})$ expected insertion time. But even proving a bound of the form $f(ε^{-1})$ for any function $f$ has remained open. In this paper, we prove that the expected insertion time is $ε^{-(1 + o(1))}$. This settles the complexity of the data structure up to sub-polynomial factors in $ε^{-1}$.

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arXiv2026

Quadratic Probing Revisited: Smoothed Analysis and the Fall of Robin Hood

Quadratic probing is one of the most widely used open-addressing hash-table schemes in practice, but after more than half a century, even its most basic performance guarantees remain poorly understood. In this paper, we revisit quadratic probing through the lens of a smoothed variant in which each key follows a random probe sequence where its $k$th probe is expected at offset $Θ(k^2)$. This is simultaneously a toy model for better understanding regular quadratic probing and a natural hashing scheme in its own right. We analyse smoothed quadratic probing for both Robin Hood ordering and anti-Robin Hood ordering and reveal a surprising separation: At load factor $1-\varepsilon$, anti-Robin Hood achieves an expected query time of $Θ(\log \varepsilon^{-1})$, which matches the conjectured expected average successful query time for regular quadratic probing, while Robin Hood falls short at $Θ(\varepsilon^{-1/2})$. Our analysis generalises to degree-$d$ probing for any $d \ge 1$ with expected query time $O(\max(\log \varepsilon^{-1}, \varepsilon^{1-2/d}))$ for anti-Robin Hood and $Θ(\varepsilon^{-1/d})$ for Robin Hood. Finally, we go beyond smoothed analysis: using the probabilistic method, we show that for every $d \ge 2$, almost every random fixed-offset degree-$d$ probing sequence achieves expected query time $O(\log \varepsilon^{-1})$ under anti-Robin Hood ordering, simultaneously over all admissible table sizes and load factors. Thus, while quadratic probing itself remains elusive, we prove that essentially all quadratic-probing-like fixed-offset schemes achieve the ideal performance under the anti-Robin Hood ordering.

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arXiv2026

Gap-Majority Lemmas in Communication Complexity

We prove an information-theoretically optimal \emph{gap-majority lemma} in the two-player randomized communication model. For a base function $f: \mathcal{X} \to \{\pm 1\}$, its $n$-fold \emph{gap-majority composition}, denoted $\mathsf{GapMAJ} \circ f^n$, takes $n$ inputs $(X_1, \ldots, X_n)$ and distinguishes whether $f^{+n}(X_1,\ldots,X_n) := f(X_1) + \ldots + f(X_n)$ is at least $0.01\sqrt{n}$ or at most $-0.01\sqrt{n}$. We show that if computing $f$ with success probability $0.501$ requires $I$ bits of information, then computing $\mathsf{GapMAJ} \circ f^n$ with success probability $0.99$ requires $n \cdot (I - O(1))$ bits of information. This result is asymptotically optimal in two aspects: it achieves the correct linear scaling of information cost and the correct constant-constant tradeoff between error rates. This makes $\mathsf{GapMAJ}$, to our knowledge, only the third explicit outer gadget that admits a strong composition theorem in the two-player communication setting, following the identity and XOR gadgets. From an application side, our gap-majority lemma can be viewed as a generic amplification tool that lifts the hardness of deciding $f$ into the hardness of approximating $f^{+n}$. Using this framework, we give a new proof to the communication lower bound of Gap-Hamming and derive a tight streaming lower bound of triangle counting, demonstrating the versatility of the gap-majority lemma.

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arXiv2026

On the Communication Complexity of Maximum Matching and Negative-Weight Shortest Paths

We revisit several fundamental graph problems in the deterministic two-party communication model. Our main contributions include: (1) a new $\widetilde{O}(n^{3/2})$-bit protocol for computing a maximum matching in general graphs. While the same upper bound can be obtained by simulating the classic algorithms of Micali-Vazirani and Gabow, our protocol is conceptually simple and avoids the intricacies of finding a maximal set of shortest augmenting paths; (2) a new $\widetilde{O}(n)$-bit protocol for negative-cycle detection and negative-weight single-source shortest paths. Our protocol simplifies that of Blikstad et al. by replacing a long chain of reductions with a more direct approach based on vertex potentials; (3) a combinatorial $\widetilde{O}(n)$-bit protocol for computing a maximum matching in bipartite graphs, obtained by reinterpreting the near-linear communication protocol of Blikstad et al. through a discretized analysis. Together, these results provide simpler protocols for several basic graph problems. We hope they will inspire further advances on the communication complexity of a wide range of graph problems.

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arXiv2026

Adversarial Robustness for Small Frequency Moments and a Weak Equivalence Theorem for Turnstile Streams

We study adversarially robust algorithms for insertion-deletion (turnstile) streams, where future updates may depend on past algorithm outputs. While recent work achieved a robust $(1+ε)$-approximation for the second moment $F_2$ in polylogarithmic space, achieving high accuracy for other frequency moments remained a major open question; for $p\in[0,2)$, including the fundamental distinct elements problem ($F_0$), only constant-factor approximations were known in sublinear space. We close this gap, showing that $(1+ε)$-approximate robustness can be achieved in polylogarithmic space for all $p\in[0,2]$. Our approach generalizes the estimator-corrector-learner framework to non-Hilbert spaces by dynamically maintaining implicit isometric embeddings into $L_2$ and performing regularized kernel ridge regression over adaptively discovered hard queries, yielding the first insertion-deletion algorithms that approximate: (1) the $p$-th frequency moment $F_p$ up to a $(1+ε)$-factor in poly$(1/ε, \log n)$ space for all $p\in[0,2]$, including the support size $F_0$, (2) metric and information-theoretic quantities, including the Earth Mover Distance (EMD) and $k$-median clustering cost over $[Δ]^d$ up to an $O(d \log Δ)$-factor, and the Shannon entropy up to an $ε$-additive error, and (3) non-normed symmetric losses defined by Bernstein functions up to a $(1+ε)$-factor. For the $F_p$ moments, our algorithm is optimal up to poly$(1/ε, \log n)$ factors. Furthermore, we establish a weak equivalence between classical oblivious sketching and adversarial robustness. We prove that for any sub-multiplicative norm, the existence of an efficient classical linear sketch is equivalent to the existence of an efficient robust turnstile algorithm, up to polynomial factors, formalizing $L_1$ embeddability as the fundamental mechanism governing both models.

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arXiv2026

Adversarial Robustness on Insertion-Deletion Streams

We study adversarially robust algorithms for insertion-deletion (turnstile) streams, where future updates may depend on past algorithm outputs. While robust algorithms exist for insertion-only streams with only a polylogarithmic overhead in memory over non-robust algorithms, it was widely conjectured that turnstile streams of length polynomial in the universe size $n$ require space linear in $n$. We refute this conjecture, showing that robustness can be achieved using space which is significantly sublinear in $n$. Our framework combines multiple linear sketches in a novel estimator-corrector-learner framework, yielding the first insertion-deletion algorithms that approximate: (1) the second moment $F_2$ up to a $(1+\varepsilon)$-factor in polylogarithmic space, (2) any symmetric function $\cal{F}$ with an $\mathcal{O}(1)$-approximate triangle inequality up to a $2^{\mathcal{O}(C)}$ factor in $\tilde{\mathcal{O}}(n^{1/C}) \cdot S(n)$ bits of space, where $S$ is the space required to approximate $\cal{F}$ non-robustly; this includes a broad class of functions such as the $L_1$-norm, the support size $F_0$, and non-normed losses such as the $M$-estimators, and (3) $L_2$ heavy hitters. For the $F_2$ moment, our algorithm is optimal up to $\textrm{poly}((\log n)/\varepsilon)$ factors. Given the recent results of Gribelyuk et al. (STOC, 2025), this shows an exponential separation between linear sketches and non-linear sketches for achieving adversarial robustness in turnstile streams.

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arXiv2026

Optimally detecting uniformly-distributed $\ell_2$ heavy hitters in data streams

Given a stream $x_1,x_2,\dots,x_n$ of items from a Universe $U$ of size poly$(n)$, and a parameter $ε>0$, an item $i\in U$ is said to be an $\ell_2$ heavy hitter if its frequency $f_i$ in the stream is at least $\sqrt{εF_2}$, where $F_2={\sum_{i\in U} f_i^2}$. Efficiently detecting such heavy hitters is a fundamental problem in data streams and has several applications in both theory and in practice. The classical $\mathsf{CountSketch}$ algorithm due to Charikar, Chen, and Farach-Colton [2004], was the first algorithm to detect $\ell_2$ heavy hitters using $O\left(\frac{\log^2 n}ε\right)$ bits of space, and their algorithm is optimal for streams with deletions. A work due to Braverman, Chestnut, Ivkin, Nelson, Wang, and Woodruff [2017] gave the $\mathsf{BPTree}$ algorithm which detects $\ell_2$ heavy hitters in insertion-only streams using only $O\left(\frac{\log(1/ε)}ε\log n \right)$ space. Note that any algorithm requires at least $Ω\left(\frac{1}ε \log n\right)$ space to output $O(1/ε)$ heavy hitters in the worst case. While $\mathsf{BPTree}$ achieves optimal space bound for constant $ε$, their bound could be sub-optimal for $ε=o(1)$. For $\textit{random order}$ streams, where the stream elements can be adversarial but their order of arrival is uniformly random, Braverman, Garg, and Woodruff [2020] showed that it is possible to achieve the optimal space bound of $O\left(\frac{1}ε \log n\right)$ for every $ε= Ω\left(\frac{1}{2^{\sqrt{\log n}}}\right)$. In this work, we generalize their result to $\textit{partially random order}$ streams where only the heavy hitters are required to be uniformly distributed in the stream. We show that it is possible to achieve the same space bound, but with an additional assumption that the algorithm is given a constant approximation to $F_2$ in advance.

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arXiv2025

Static Retrieval Revisited: To Optimality and Beyond

In the static retrieval problem, a data structure must answer retrieval queries mapping a set of $n$ keys in a universe $[U]$ to $v$-bit values. Information-theoretically, retrieval data structures can use as little as $nv$ bits of space. For small value sizes $v$, it is possible to achieve $O(1)$ query time while using space $nv + o(n)$ bits -- whether or not such a result is possible for larger values of $v$ (e.g., $v = Θ(\log n)$) has remained open. In this paper, we obtain a tight lower bound (as well as matching upper bounds) for the static retrieval problem. In the case where values are large, we show that there is actually a significant tension between time and space. It is not possible, for example, to get $O(1)$ query time using $nv + o(n)$ bits of space, when $v = Θ(\log n)$ (and assuming the word RAM model with $O(\log n)$-bit words). At first glance, our lower bound would seem to render retrieval unusable in many settings that aim to achieve very low redundancy. However, our second result offers a way around this: We show that, whenever a retrieval data structure $D_1$ is stored along with another data structure $D_2$ (whose size is similar to or larger than the size of $D_1$), it is possible to implement the combined data structure $D_1 \cup D_2$ so that queries to $D_1$ take $O(1)$ time, operations on $D_2$ take the same asymptotic time as if $D_2$ were stored on its own, and the total space is $nv + \mathrm{Space}(D_2) + n^{0.67}$ bits.

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arXiv2025

A (Very) Nearly Optimal Sketch for $k$-Edge Connectivity Certificates

In this note, we present a simple algorithm for computing a \emph{$k$-connectivity certificate} in dynamic graph streams. Our algorithm uses $O(n \log^2 n \cdot \max\{k, \log n \log k\})$ bits of space which improves upon the $O(kn \log^3 n)$-space algorithm of Ahn, Guha, and McGregor (SODA'12). For the values of $k$ that are truly sublinear, our space usage \emph{very nearly} matches the known lower bound $Ω(n \log^2 n \cdot \max\{k, \log n\})$ established by Nelson and Yu (SODA'19; implicit) and Robinson (DISC'24). In particular, our algorithm fully settles the space complexity at $Θ(kn \log^2{n})$ for $k = Ω(\log n \log \log n)$, and bridges the gap down to only a doubly-logarithmic factor of $O(\log \log n)$ for a smaller range of $k = o(\log n \log \log n)$.

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arXiv2025

Strong XOR Lemma for Information Complexity

For any $\{0,1\}$-valued function $f$, its \emph{$n$-folded XOR} is the function $f^{\oplus n}$ where $f^{\oplus n}(X_1, \ldots, X_n) = f(X_1) \oplus \cdots \oplus f(X_n)$. Given a procedure for computing the function $f$, one can apply a ``naive" approach to compute $f^{\oplus n}$ by computing each $f(X_i)$ independently, followed by XORing the outputs. This approach uses $n$ times the resources required for computing $f$. In this paper, we prove a strong XOR lemma for \emph{information complexity} in the two-player randomized communication model: if computing $f$ with an error probability of $O(n^{-1})$ requires revealing $I$ bits of information about the players' inputs, then computing $f^{\oplus n}$ with a constant error requires revealing $Ω(n) \cdot (I - 1 - o_n(1))$ bits of information about the players' inputs. Our result demonstrates that the naive protocol for computing $f^{\oplus n}$ is both information-theoretically optimal and asymptotically tight in error trade-offs.

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arXiv2025

Optimal Static Dictionary with Worst-Case Constant Query Time

In this paper, we design a new succinct static dictionary with worst-case constant query time. A dictionary data structure stores a set of key-value pairs with distinct keys in $[U]$ and values in $[σ]$, such that given a query $x\in [U]$, it quickly returns if $x$ is one of the input keys, and if so, also returns its associated value. The textbook solution to dictionaries is hash tables. On the other hand, the (information-theoretical) optimal space to encode such a set of key-value pairs is only $\text{OPT} := \log\binom{U}{n}+n\log σ$. We construct a dictionary that uses $\text{OPT} + n^ε$ bits of space, and answers queries in constant time in worst case. Previously, constant-time dictionaries are only known with $\text{OPT} + n/\text{poly}\log n$ space [Pǎtraşcu 2008], or with $\text{OPT}+n^ε$ space but expected constant query time [Yu 2020]. We emphasize that most of the extra $n^ε$ bits are used to store a lookup table that does not depend on the input, and random bits for hash functions. The "main" data structure only occupies $\text{OPT}+\text{poly}\log n$ bits.

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arXiv2025

Lifting Linear Sketches: Optimal Bounds and Adversarial Robustness

We introduce a novel technique for ``lifting'' dimension lower bounds for linear sketches in the real-valued setting to dimension lower bounds for linear sketches with polynomially-bounded integer entries when the input is a polynomially-bounded integer vector. Using this technique, we obtain the first optimal sketching lower bounds for discrete inputs in a data stream, for classical problems such as approximating the frequency moments, estimating the operator norm, and compressed sensing. Additionally, we lift the adaptive attack of Hardt and Woodruff (STOC, 2013) for breaking any real-valued linear sketch via a sequence of real-valued queries, and show how to obtain an attack on any integer-valued linear sketch using integer-valued queries. This shows that there is no linear sketch in a data stream with insertions and deletions that is adversarially robust for approximating any $L_p$ norm of the input, resolving a central open question for adversarially robust streaming algorithms. To do so, we introduce a new pre-processing technique of independent interest which, given an integer-valued linear sketch, increases the dimension of the sketch by only a constant factor in order to make the orthogonal lattice to its row span smooth. This pre-processing then enables us to leverage results in lattice theory on discrete Gaussian distributions and reason that efficient discrete sketches imply efficient continuous sketches. Our work resolves open questions from the Banff '14 and '17 workshops on Communication Complexity and Applications, as well as the STOC '21 and FOCS '23 workshops on adaptivity and robustness.

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arXiv2024

Near-Optimal Relative Error Streaming Quantile Estimation via Elastic Compactors

Computing the approximate quantiles or ranks of a stream is a fundamental task in data monitoring. Given a stream of elements $x_1, x_2, \dots, x_n$ and a query $x$, a relative-error quantile estimation algorithm can estimate the rank of $x$ with respect to the stream, up to a multiplicative $\pm ε\cdot \mathrm{rank}(x)$ error. Notably, this requires the sketch to obtain more precise estimates for the ranks of elements on the tails of the distribution, as compared to the additive $\pm εn$ error regime. Previously, the best-known algorithms for relative error achieved space $\tilde O(ε^{-1}\log^{1.5}(εn))$ (Cormode, Karnin, Liberty, Thaler, Vesel{y}, 2021) and $\tilde O(ε^{-2}\log(εn))$ (Zhang, Lin, Xu, Korn, Wang, 2006). In this work, we present a nearly-optimal streaming algorithm for the relative-error quantile estimation problem using $\tilde O(ε^{-1}\log(εn))$ space, which almost matches the trivial $Ω(ε^{-1} \log (εn))$ lower bound. To surpass the $Ω(ε^{-1}\log^{1.5}(εn))$ barrier of the previous approach, our algorithm crucially relies on a new data structure, called an elastic compactor, which can be dynamically resized over the course of the stream. Interestingly, we design a space allocation scheme which adaptively allocates space to each compactor based on the "hardness" of the input stream. This approach allows us to avoid using the maximal space simultaneously for every compactor and facilitates the improvement in the total space complexity. Along the way, we also propose and study a new problem called the Top Quantiles Problem, which only requires the sketch to provide estimates for a fixed-length tail of the distribution. This problem serves as an important subproblem in our algorithm, though it is also an interesting problem of its own right.

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arXiv2024

A Strong Separation for Adversarially Robust $\ell_0$ Estimation for Linear Sketches

The majority of streaming problems are defined and analyzed in a static setting, where the data stream is any worst-case sequence of insertions and deletions that is fixed in advance. However, many real-world applications require a more flexible model, where an adaptive adversary may select future stream elements after observing the previous outputs of the algorithm. Over the last few years, there has been increased interest in proving lower bounds for natural problems in the adaptive streaming model. In this work, we give the first known adaptive attack against linear sketches for the well-studied $\ell_0$-estimation problem over turnstile, integer streams. For any linear streaming algorithm $\mathcal{A}$ that uses sketching matrix $\mathbf{A}\in \mathbb{Z}^{r \times n}$ where $n$ is the size of the universe, this attack makes $\tilde{\mathcal{O}}(r^8)$ queries and succeeds with high constant probability in breaking the sketch. We also give an adaptive attack against linear sketches for the $\ell_0$-estimation problem over finite fields $\mathbb{F}_p$, which requires a smaller number of $\tilde{\mathcal{O}}(r^3)$ queries. Finally, we provide an adaptive attack over $\mathbb{R}^n$ against linear sketches $\mathbf{A} \in \mathbb{R}^{r \times n}$ for $\ell_0$-estimation, in the setting where $\mathbf{A}$ has all nonzero subdeterminants at least $\frac{1}{\textrm{poly}(r)}$. Our results provide an exponential improvement over the previous number of queries known to break an $\ell_0$-estimation sketch.

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arXiv2023

Dynamic Dictionary with Subconstant Wasted Bits per Key

Dictionaries have been one of the central questions in data structures. A dictionary data structure maintains a set of key-value pairs under insertions and deletions such that given a query key, the data structure efficiently returns its value. The state-of-the-art dictionaries [Bender, Farach-Colton, Kuszmaul, Kuszmaul, Liu 2022] store $n$ key-value pairs with only $O(n \log^{(k)} n)$ bits of redundancy, and support all operations in $O(k)$ time, for $k \leq \log^* n$. It was recently shown to be optimal [Li, Liang, Yu, Zhou 2023b]. In this paper, we study the regime where the redundant bits is $R=o(n)$, and show that when $R$ is at least $n/\text{poly}\log n$, all operations can be supported in $O(\log^* n + \log (n/R))$ time, matching the lower bound in this regime [Li, Liang, Yu, Zhou 2023b]. We present two data structures based on which range $R$ is in. The data structure for $R<n/\log^{0.1} n$ utilizes a generalization of adapters studied in [Berger, Kuszmaul, Polak, Tidor, Wein 2022] and [Li, Liang, Yu, Zhou 2023a]. The data structure for $R \geq n/\log^{0.1} n$ is based on recursively hashing into buckets with logarithmic sizes.

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arXiv2023

Dynamic "Succincter"

Augmented B-trees (aB-trees) are a broad class of data structures. The seminal work "succincter" by Patrascu showed that any aB-tree can be stored using only two bits of redundancy, while supporting queries to the tree in time proportional to its depth. It has been a versatile building block for constructing succinct data structures, including rank/select data structures, dictionaries, locally decodable arithmetic coding, storing balanced parenthesis, etc. In this paper, we show how to "dynamize" an aB-tree. Our main result is the design of dynamic aB-trees (daB-trees) with branching factor two using only three bits of redundancy (with the help of lookup tables that are of negligible size in applications), while supporting updates and queries in time polynomial in its depth. As an application, we present a dynamic rank/select data structure for $n$-bit arrays, also known as a dynamic fully indexable dictionary (FID). It supports updates and queries in $O(\log n/\log\log n)$ time, and when the array has $m$ ones, the data structure occupies \[ \log\binom{n}{m} + O(n/2^{\log^{0.199}n}) \] bits. Note that the update and query times are optimal even without space constraints due to a lower bound by Fredman and Saks. Prior to our work, no dynamic FID with near-optimal update and query times and redundancy $o(n/\log n)$ was known. We further show that a dynamic sequence supporting insertions, deletions and rank/select queries can be maintained in (optimal) $O(\log n/\log\log n)$ time and with $O(n \cdot \text{poly}\log\log n/\log^2 n)$ bits of redundancy.

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