Estimating the second frequency moment F₂ of a data stream up to a (1 ± ε) factor is a central problem in the streaming literature. For errors ε > Ω(1/√n), the tight bound Θ(log(ε² n)/ε²) was recently established by Braverman and Zamir. In this work, we complete the picture by resolving the remaining regime of small error, ε < 1/√n, showing that the optimal space complexity is Θ(min(n, 1/ε²)⋅(1 + |log(ε² n)|)) bits for all ε ≥ 1/n², assuming a sufficiently large universe. This closes the gap between the best known Ω(n) lower bound and the straightforward O(n log n) upper bound in that range, and shows that essentially storing the entire stream is necessary for high-precision estimation. To derive this bound, we fully characterize the two-party communication complexity of estimating the size of a set intersection up to an arbitrary additive error ε n. In particular, we prove a tight Ω(n log n) lower bound for one-way communication protocols when ε < n^{-1/2-Ω(1)}, in contrast to classical O(n)-bit protocols that use two-way communication. Motivated by this separation, we present a two-pass streaming algorithm that computes the exact histogram of a stream with high probability using only O(n log log n) bits of space, in contrast to the Θ(n log n) bits required in one pass even to approximate F₂ with small error. This yields the first asymptotic separation between one-pass and O(1)-passes space complexity for small frequency moment estimation.

Traditional error-correcting codes (ECCs) assume a fixed message length, but many scenarios involve ongoing or indefinite transmissions where the message length is not known in advance. For example, when streaming a video, the user should be able to fix a fraction of errors that occurred before any point in time. We introduce unbounded error-correcting codes (unbounded codes), a natural generalization of ECCs that supports arbitrarily long messages without a predetermined length. An unbounded code with rate \(R\) and distance \(\varepsilon\) ensures that for every sufficiently large \(k\), the message prefix of length \(R_k\) can be recovered from the code prefix of length \(k\) even if an adversary corrupts up to an \(\varepsilon\) fraction of the symbols in this code prefix.

Cryptographic primitives have been used for various non-cryptographic objectives, such as eliminating or reducing randomness and interaction. We show how to use cryptography to improve the time complexity of solving computational problems. Specifically, we show that under standard cryptographic assumptions, we can design algorithms that are asymptotically faster than existing ones while maintaining correctness.

A subset S of the Boolean hypercube 𝔽₂ⁿ is a sumset if S = {a + b : a, b ∈ A} for some A ⊆ 𝔽₂ⁿ. Sumsets are central objects of study in additive combinatorics, where they play a role in several of the field’s most important results. We prove a lower bound of Ω(2^{n/2}) for the number of queries needed to test whether a Boolean function f:𝔽₂ⁿ → {0,1} is the indicator function of a sumset, ruling out an efficient testing algorithm for sumsets. Our lower bound for testing sumsets follows from sharp bounds on the related problem of shift testing, which may be of independent interest. We also give a near-optimal {2^{n/2} ⋅ poly(n)}-query algorithm for a smoothed analysis formulation of the sumset refutation problem. Finally, we include a simple proof that the number of different sumsets in 𝔽₂ⁿ is 2^{(1±o(1))2^{n-1}}.

Abstract. A subset [Formula: see text] of the Boolean hypercube [Formula: see text] is a sumset if [Formula: see text] for some [Formula: see text]. We prove that the number of sumsets in [Formula: see text] is asymptotically [Formula: see text]. Furthermore, we show that the family of sumsets in [Formula: see text] is almost identical to the family of all subsets of [Formula: see text] that contain a complete linear subspace of codimension 1.

Recent advances in the capabilities of large language models such as GPT-4 have spurred increasing concern about our ability to detect AI-generated text. Prior works have suggested methods of embedding watermarks in model outputs, by *noticeably* altering the output distribution. We ask: Is it possible to introduce a watermark without incurring *any detectable* change to the output distribution? To this end, we introduce a cryptographically-inspired notion of undetectable watermarks for language models. That is, watermarks can be detected only with the knowledge of a secret key; without the secret key, it is computationally intractable to distinguish watermarked outputs from those of the original model. In particular, it is impossible for a user to observe any degradation in the quality of the text. Crucially, watermarks remain undetectable even when the user is allowed to adaptively query the model with arbitrarily chosen prompts. We construct undetectable watermarks based on the existence of one-way functions, a standard assumption in cryptography.

In this paper, we study the static cell probe complexity of non-adaptive data structures that maintain a subset of $n$ points from a universe consisting of $m=n^{1+Ω(1)}$ points. A data structure is defined to be non-adaptive when the memory locations that are chosen to be accessed during a query depend only on the query inputs and not on the contents of memory. We prove an $Ω(\log m / \log (sw/n\log m))$ static cell probe complexity lower bound for non-adaptive data structures that solve the fundamental dictionary problem where $s$ denotes the space of the data structure in the number of cells and $w$ is the cell size in bits. Our lower bounds hold for all word sizes including the bit probe model ($w = 1$) and are matched by the upper bounds of Boninger et al. [FSTTCS'17]. Our results imply a sharp dichotomy between dictionary data structures with one round of adaptive and at least two rounds of adaptivity. We show that $O(1)$, or $O(\log^{1-ε}(m))$, overhead dictionary constructions are only achievable with at least two rounds of adaptivity. In particular, we show that many $O(1)$ dictionary constructions with two rounds of adaptivity such as cuckoo hashing are optimal in terms of adaptivity. On the other hand, non-adaptive dictionaries must use significantly more overhead. Finally, our results also imply static lower bounds for the non-adaptive predecessor problem. Our static lower bounds peak higher than the previous, best known lower bounds of $Ω(\log m / \log w)$ for the dynamic predecessor problem by Boninger et al. [FSTTCS'17] and Ramamoorthy and Rao [CCC'18] in the natural setting of linear space $s = Θ(n)$ where each point can fit in a single cell $w = Θ(\log m)$. Furthermore, our results are stronger as they apply to the static setting unlike the previous lower bounds that only applied in the dynamic setting.

We present a general method to convert algorithms into faster algorithms for almost-regular input instances. Informally, an almost-regular input is an input in which the maximum degree is larger than the average degree by at most a constant factor. This family of inputs vastly generalizes several families of inputs for which we commonly have improved algorithms, including bounded-degree inputs and random inputs. It also generalizes families of inputs for which we don’t usually have faster algorithms, including regular-inputs of arbitrarily high degree and very dense inputs. We apply our method to achieve breakthroughs in exact algorithms for several central NP-Complete problems including k-SAT, Graph Coloring, and Maximum Independent Set.

Constraint satisfaction problems (CSPs) models many important intractable NP-hard problems such as propositional satisfiability problem (SAT). Algorithms with non-trivial upper bounds on running time for restricted SAT with bounded clause length k (k-SAT) can be classified into three styles: DPLL-like, PPSZ-like and Local Search, with local search algorithms having already been generalized to CSP with bounded constraint arity k (k-CSP). We generalize a DPLL-like algorithm in its simplest form and a PPSZ-like algorithm from k-SAT to k-CSP. As far as we know, this is the first attempt to use PPSZ-like strategy to solve k-CSP, and before little work has been focused on the DPLL-like or PPSZ-like strategies for k-CSP.

Given the computational cost and technical expertise required to train machine learning models, users may delegate the task of learning to a service provider. Delegation of learning has clear benefits, and at the same time raises serious concerns of trust. This work studies possible abuses of power by untrusted learners.We show how a malicious learner can plant an undetectable backdoor into a classifier. On the surface, such a backdoored classifier behaves normally, but in reality, the learner maintains a mechanism for changing the classification of any input, with only a slight perturbation. Importantly, without the appropriate “backdoor key,” the mechanism is hidden and cannot be detected by any computationally-bounded observer. We demonstrate two frameworks for planting undetectable backdoors, with incomparable guarantees.•First, we show how to plant a backdoor in any model, using digital signature schemes. The construction guarantees that given query access to the original model and the backdoored version, it is computationally infeasible to find even a single input where they differ. This property implies that the backdoored model has generalization error comparable with the original model. Moreover, even if the distinguisher can request backdoored inputs of its choice, they cannot backdoor a new input—a property we call non-replicability.•Second, we demonstrate how to insert undetectable backdoors in models trained using the Random Fourier Features (RFF) learning paradigm (Rahimi, Recht; NeurIPS 2007). In this construction, undetectability holds against powerful white-box distinguishers: given a complete description of the network and the training data, no efficient distinguisher can guess whether the model is “clean” or contains a backdoor. The backdooring algorithm executes the RFF algorithm faithfully on the given training data, tampering only with its random coins. We prove this strong guarantee under the hardness of the Continuous Learning With Errors problem (Bruna, Regev, Song, Tang; STOC 2021). We show a similar white-box undetectable backdoor for random ReLU networks based on the hardness of Sparse PCA (Berthet, Rigollet; COLT 2013).Our construction of undetectable backdoors also sheds light on the related issue of robustness to adversarial examples. In particular, by constructing undetectable backdoor for an “adversarially-robust” learning algorithm, we can produce a classifier that is indistinguishable from a robust classifier, but where every input has an adversarial example! In this way, the existence of undetectable backdoors represent a significant theoretical roadblock to certifying adversarial robustness.

It is well known that a queue can be simulated by two stacks using a constant number of stack operations per queue operation. In this paper we consider the forgotten converse problem of simulating a stack using several queues. We consider several variants of this problem. For the offline variant, we obtain a tight upper and lower bounds for the worst-case number of queue operations needed to simulate a sequence of n stack operations using k queues. For the online variant, when the number of queues k is constant, and n is the maximum number of items in the stack at any given time, we obtain tight Θ(n1/k) upper and lower bounds on the worst-case and amortized number of queue operations needed to simulate one stack operation. When k is allowed to grow with n, we prove an upper bound of O(n1/k + logk n) and a lower bound of on the amortized number of queue operations per stack operation. We also prove an upper bound of O(kn1/k) and a lower bound of Ω(n1/k + logk n) on the worst-case number of queue operations per stack operation. We also show that the specific but interesting sequence of n pushes followed by n pops can be implemented much faster using a total number of only Θ(n logk n) queue operations, for every k ≥ 2, an amortized number of Θ(logk n) queue operations per stack operation, and this bound is tight. On the other hand, we show that the same sequence requires at least Ω(n1/k) queue operations per stack operation in the worst case.

We present a new technique for efficiently removing almost all short cycles in a graph without unintentionally removing its triangles. Consequently, triangle finding problems do not become easy even in almost k-cycle free graphs, for any constant k≥ 4.

In the coin problem we are asked to distinguish, with probability at least 2/3, between$n\ i.i.d$. coins which are heads with probability$\frac{1}{2}+\beta$from ones which are heads with probability$\frac{1}{2}-\beta$. We are interested in the space complexity of the coin problem, corresponding to the width of a read-once branching program solving the problem. The coin problem becomes more difficult as$\beta$becomes smaller. Statistically, it can be solved whenever$\beta= \Omega(n^{-1/2})$, using counting. It has been previously shown that for$\beta=O(n^{-1/2})$, counting is essentially optimal (equivalently, width$poly (n)$is necessary [Braverman-Garg-Woodruff FOCS'20]). On the other hand, the coin problem only requires$O(\log n)$width for$\beta > n^{-c}$for any constant$c > \log_{2}(\sqrt{5}-1)\approx 0.306$(following low-width simulation of AND-OR tree of [Valiant Journal of Algorithms'84]). In this paper, we close the gap between the bounds, showing a tight threshold between the values of$\beta=n^{-c}$where$O(\log n)$width suffices and the regime where$poly (n)$width is needed, with a transition at$c=1/3$. This gives a complete characterization (up to constant factors) of the memory complexity of solving the coin problem, for all values of bias$\beta$. We introduce new techniques in both bounds. For the upper bound, we give a construction based on recursive majority that does not require a memory stack of size$\log n$bits. For the lower bound, we introduce new combinatorial techniques for analyzing progression of the success probabilities in read-once branching programs.

The coloring problem (i.e., computing the chromatic number of a graph) can be solved in O^*(2ⁿ) time, as shown by Björklund, Husfeldt and Koivisto in 2009. For k = 3,4, better algorithms are known for the k-coloring problem. 3-coloring can be solved in O(1.33ⁿ) time (Beigel and Eppstein, 2005) and 4-coloring can be solved in O(1.73ⁿ) time (Fomin, Gaspers and Saurabh, 2007). Surprisingly, for k > 4 no improvements over the general O^*(2ⁿ) are known. We show that both 5-coloring and 6-coloring can also be solved in O((2-ε) ⁿ) time for some ε > 0. As a crucial step, we obtain an exponential improvement for computing the chromatic number of a very large family of graphs. In particular, for any constants Δ,α > 0, the chromatic number of graphs with at least α⋅ n vertices of degree at most Δ can be computed in O((2-ε) ⁿ) time, for some ε = ε_{Δ,α} > 0. This statement generalizes previous results for bounded-degree graphs (Björklund, Husfeldt, Kaski, and Koivisto, 2010) and graphs with bounded average degree (Golovnev, Kulikov and Mihajlin, 2016). We generalize the aforementioned statement to List Coloring, for which no previous improvements are known even for the case of bounded-degree graphs.

Random dimensionality reduction is a versatile tool for speeding up algorithms for high-dimensional problems. We study its application to two clustering problems: the facility location problem, and the single-linkage hierarchical clustering problem, which is equivalent to computing the minimum spanning tree. We show that if we project the input pointset $X$ onto a random $d = O(d_X)$-dimensional subspace (where $d_X$ is the doubling dimension of $X$), then the optimum facility location cost in the projected space approximates the original cost up to a constant factor. We show an analogous statement for minimum spanning tree, but with the dimension $d$ having an extra $\log \log n$ term and the approximation factor being arbitrarily close to $1$. Furthermore, we extend these results to approximating {\em solutions} instead of just their {\em costs}. Lastly, we provide experimental results to validate the quality of solutions and the speedup due to the dimensionality reduction. Unlike several previous papers studying this approach in the context of $k$-means and $k$-medians, our dimension bound does not depend on the number of clusters but only on the intrinsic dimensionality of $X$.

Each vertex of an arbitrary simple graph on n vertices chooses k random incident edges. What is the expected number of edges in the original graph that connect different connected components of the sampled subgraph? We prove that the answer is O(n/k), when k ≥ c log n, for some large enough c. We conjecture that the same holds for smaller values of k, possibly for any k ≥ 2. Such a result is best possible for any k ≥ 2. As an application, we use this sampling result to obtain a one-way communication protocol with private randomness for finding a spanning forest of a graph in which each vertex sends only O (√n log n) bits to a referee.

We consider word RAM data structures for maintaining ordered sets of integers whose select and rank operations are allowed to return approximate results, i.e., ranks, or items whose rank, differ by less than Delta from the exact answer, where Delta=Delta(n) is an error parameter. Related to approximate select and rank is approximate (one-dimensional) nearest-neighbor. A special case of approximate select queries are approximate min queries. Data structures that support approximate min operations are known as approximate heaps (priority queues). Related to approximate heaps are soft heaps, which are approximate heaps with a different notion of approximation. We prove the optimality of all the data structures presented, either through matching cell-probe lower bounds, or through equivalences to well studied static problems. For approximate select, rank, and nearest-neighbor operations we get matching cell-probe lower bounds. We prove an equivalence between approximate min operations, i.e., approximate heaps, and the static partitioning problem. Finally, we prove an equivalence between soft heaps and the classical sorting problem, on a smaller number of items. Our results have many interesting and unexpected consequences. It turns out that approximation greatly speeds up some of these operations, while others are almost unaffected. In particular, while select and rank have identical operation times, both in comparison-based and word RAM implementations, an interesting separation emerges between the approximate versions of these operations in the word RAM model. Approximate select is much faster than approximate rank. It also turns out that approximate min is exponentially faster than the more general approximate select. Next, we show that implementing soft heaps is harder than implementing approximate heaps. The relation between them corresponds to the relation between sorting and partitioning. Finally, as an interesting byproduct, we observe that a combination of known techniques yields a deterministic word RAM algorithm for (exactly) sorting n items in O(n log log_w n) time, where w is the word length. Even for the easier problem of finding duplicates, the best previous deterministic bound was O(min{n log log n,n log_w n}). Our new unifying bound is an improvement when w is sufficiently large compared with n.

We describe an efficient deterministic adversary that forces any comparison-based sorting algorithm to perform at least n log n comparisons. This improves on previous efficient adversaries of Atallah and Kosaraju (1981), Richards and Vaidya (1988), and of Brodal et al. (1996) that force any sorting algorithm to perform at least ψn log n comparisons.

The PPSZ algorithm, due to Paturi, Pudlak, Saks and Zane, is currently the fastest known algorithm for the k-SAT problem, for every k>3. For 3-SAT, a tiny improvement over PPSZ was obtained by Hertli. We introduce a biased version of the PPSZ algorithm using which we obtain an improvement over PPSZ for every k≥ 3. For k=3 we also improve on Herli’s result and get a much more noticeable improvement over PPSZ, though still relatively small. In particular, for Unique 3-SAT, we improve the current bound from 1.308n to 1.307n.

The Subtree Isomorphism problem asks whether a given tree is contained in another given tree. The problem is of fundamental importance and has been studied since the 1960s. For some variants, e.g., ordered trees , near-linear time algorithms are known, but for the general case truly subquadratic algorithms remain elusive. Our first result is a reduction from the Orthogonal Vectors problem to Subtree Isomorphism, showing that a truly subquadratic algorithm for the latter refutes the Strong Exponential Time Hypothesis (SETH). In light of this conditional lower bound, we focus on natural special cases for which no truly subquadratic algorithms are known. We classify these cases against the quadratic barrier, showing in particular that: • Even for binary, rooted trees, a truly subquadratic algorithm refutes SETH. • Even for rooted trees of depth O (log log n ), where n is the total number of vertices, a truly subquadratic algorithm refutes SETH. • For every constant d , there is a constant ε d > 0 and a randomized, truly subquadratic algorithm for degree- d rooted trees of depth at most (1+ ε d ) log d n . In particular, there is an O (min { 2.85 h , n 2 }) algorithm for binary trees of depth h . Our reductions utilize new “tree gadgets” that are likely useful for future SETH-based lower bounds for problems on trees. Our upper bounds apply a folklore result from randomized decision tree complexity.

The Subtree Isomorphism problem asks whether a given tree is contained in another given tree. The problem is of fundamental importance and has been studied since the 1960s. For some variants, e.g., ordered trees, near-linear time algorithms are known, but for the general case truly subquadratic algorithms remain elusive. Our first result is a reduction from the Orthogonal Vectors problem to Subtree Isomorphism, showing that a truly subquadratic algorithm for the latter refutes the Strong Exponential Time Hypothesis (SETH). In light of this conditional lower bound, we focus on natural special cases for which no truly subquadratic algorithms are known. We classify these cases against the quadratic barrier, showing in particular that: Even for binary, rooted trees, a truly subquadratic algorithm refutes SETH. Even for rooted trees of depth O (log log n), where n is the total number of vertices, a truly subquadratic algorithm refutes SETH. For every constant d, there is a constant ∊d > 0 and a randomized, truly subquadratic algorithm for degree-d rooted trees of depth at most (1 + ∊d)logdn. In particular, there is an O(min{2.85h, n2}) algorithm for binary trees of depth h. Our reductions utilize new “tree gadgets” that are likely useful for future SETH-based lower bounds for problems on trees. Our upper bounds apply a folklore result from randomized decision tree complexity.

Gabow and Tarjan showed that the Bottleneck Path (BP) problem, i.e., finding a path between a given source and a given target in a weighted directed graph whose largest edge weight is minimized, as well as the Bottleneck spanning tree (BST) problem, i.e., finding a directed spanning tree rooted at a given vertex whose largest edge weight is minimized, can both be solved deterministically in O(m * log^*(n)) time, where m is the number of edges and n is the number of vertices in the graph. We present a slightly improved randomized algorithm for these problems with an expected running time of O(m * beta(m,n)), where beta(m,n) = min{k >= 1 | log^{(k)}n <= m/n } <= log^*(n) - log^*(m/n)+1. This is the first improvement for these problems in over 25 years. In particular, if m >= n * log^{(k)} * n, for some constant k, the expected running time of the new algorithm is O(m). Our algorithm, as that of Gabow and Tarjan, work in the comparison model. We also observe that in the word-RAM model, both problems can be solved deterministically in O(m) time. Finally, we solve an open problem of Andersson et al., giving a deterministic O(m)-time comparison-based algorithm for solving deterministic 2-player turn-based zero-sum terminal payoff games, also known as Deterministic Graphical Games (DGG).

We obtain an essentially optimal tradeoff between the amortized cost of the three basic priority queue operations insert, delete and find-min in the comparison model. More specifically, we show that for any fixed ε > 0, where n is the number of items in the priority queue and A(insert), A(delete) and A(find-min) are the amortized costs of the insert, delete and find-min operations, respectively. In particular, if A(insert) + A(delete) = O(1), then A(find-min) = Ω(n), and A(find-min) = O(nα), for some α < 1, only if A(insert) + A(delete) = Ω(log n). (We can, of course, have A(insert) = O(1), A(delete) = O(log n), or vice versa, and A(find-min) = O(1).) Our lower bound holds even if randomization is allowed. Surprisingly, such fundamental bounds on the amortized cost of the operations were not known before. Brodal, Chaudhuri and Rad-hakrishnan, obtained similar bounds for the worst-case complexity of find-min.