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Abstract. A [Formula: see text]- vertex connectivity oracle for an undirected graph [Formula: see text] is a data structure that, given [Formula: see text], reports [Formula: see text], where [Formula: see text] is the pairwise vertex connectivity between [Formula: see text]. There are three main measures of efficiency: construction time, query time, and space. Prior work of Izsak and Nutov [ Inform. Process. Lett., 112 (2012), pp. 39–43] produced a data structure of [Formula: see text] words, which can even be encoded as a [Formula: see text]-bit labeling scheme, that can answer vertex connectivity queries in [Formula: see text] time. The construction time is polynomial but unspecified. In this paper, we address the top three complexity measures. (1) Space: We prove that any [Formula: see text]-vertex connectivity oracle requires [Formula: see text] bits of space for any [Formula: see text]. This proves that the Iszak–Nutov data structure is optimal up to polylogarithmic factors for every [Formula: see text] and that the sparsifiers of Nagamochi and Ibaraki [ Algorithmica, 7 (1992), pp. 583–596] are optimal compression schemes for [Formula: see text]-vertex connectivity up to a logarithmic factor. In particular, whereas all edge connectivities can be efficiently compressed (as a weighted [Formula: see text]-edge Gomory–Hu tree), vertex connectivity admits no asymptotic compression: [Formula: see text] bits are necessary. We design a variation on Izsak and Nutov’s data structure that uses [Formula: see text] words of space. (2) Query time: We answer queries in [Formula: see text] time, improving on the [Formula: see text] time bound of Izsak and Nutov [ Inform. Process. Lett., 112 (2012), pp. 39–43]. The main idea is to build instances of [Formula: see text] data structures, with additional structure based on affine planes. This structure allows for query time that is linear in the output size, which evades some conditional lower bounds that are polynomial in the query set sizes [ 42 ,  51 ]. (3) Construction time: Our data structure can be constructed in the time of [Formula: see text] max-flow computations, namely, [Formula: see text] time, using the recent near-linear time flow algorithm of [ 13 ]. The main technical contribution here is a fast algorithm to compute a [Formula: see text]-approximate Gomory–Hu tree for element connectivity in the time of [Formula: see text] max-flow computations. Element connectivity is a notion that generalizes edge and vertex connectivity. 

Abstract The theory of forbidden 0–1 matrices generalizes Turán-style (bipartite) subgraph avoidance, Davenport-Schinzel theory, and Zarankiewicz-type problems, and has been influential in many areas, such as discrete and computational geometry, the analysis of self-adjusting data structures, and the development of the graph parametertwin width. The foremost open problem in this area is to resolve thePach-Tardos conjecturefrom 2005, which states that if a forbidden pattern$$P\in \{0,1\}^{k\times l}$$ $P\in {\{0,1\}}^{k\times l}$ isacyclic, meaning it is the bipartite incidence matrix of a forest, then$$\operatorname {Ex}(P,n) = O(n\log ^{C_P} n)$$ $Ex(P,n)=O\left(n{log}^{C_P}n\right)$ , where$$\operatorname {Ex}(P,n)$$ $Ex(P,n)$ is the maximum number of 1s in aP-free$$n\times n$$ $n\times n$ 0–1 matrix and$$C_P$$ $C_P$ is a constant depending only onP. This conjecture has been confirmed on many small patterns, specifically allPwith weight at most 5, and all but two with weight 6. The main result of this paper is a clean refutation of the Pach-Tardos conjecture. Specifically, we prove that$$\operatorname {Ex}(S_0,n),\operatorname {Ex}(S_1,n) \ge n2^{\Omega (\sqrt{\log n})}$$ $Ex(S_0,n),Ex(S_1,n)\ge n{2}^{\Omega \left(\sqrt{logn}\right)}$ , where$$S_0,S_1$$ $S_0,S_1$ are the outstanding weight-6 patterns. We also prove sharp bounds on the entire class ofalternatingpatterns$$(P_t)$$ $\left(P_t\right)$ , specifically that for every$$t\ge 2$$ $t\ge 2$ ,$$\operatorname {Ex}(P_t,n)=\Theta (n(\log n/\log \log n)^t)$$ $Ex(P_t,n)=\Theta \left(n{(logn/loglogn)}^t\right)$ . This is the first proof of an asymptotically sharp bound that is$$\omega (n\log n)$$ $\omega (nlogn)$ . 

In the d-dimensional turnstile streaming model, a frequency vector 𝐱 = (𝐱(1),…,𝐱(n)) ∈ (ℝ^d)ⁿ is updated entry-wisely over a stream. We consider the problem of f-moment estimation for which one wants to estimate f(𝐱)=∑_{v ∈ [n]}f(𝐱(v)) with a small-space sketch. A function f is tractable if the f-moment can be estimated to within a constant factor using polylog(n) space. The f-moment estimation problem has been intensively studied in the d = 1 case. Flajolet and Martin estimate the F₀-moment (f(x) = 1 (x > 0), incremental stream); Alon, Matias, and Szegedy estimate the L₂-moment (f(x) = x²); Indyk estimates the L_α-moment (f(x) = |x|^α), α ∈ (0,2]. For d ≥ 2, Ganguly, Bansal, and Dube estimate the L_{p,q} hybrid moment (f:ℝ^d → ℝ,f(x) = (∑_{j = 1}^d |x_j|^p)^q), p ∈ (0,2],q ∈ (0,1). For tractability, Bar-Yossef, Jayram, Kumar, and Sivakumar show that f(x) = |x|^α is not tractable for α > 2. Braverman, Chestnut, Woodruff, and Yang characterize the class of tractable one-variable functions except for a class of nearly periodic functions. In this work we present a simple and generic scheme to construct sketches with the novel idea of hashing indices to Lévy processes, from which one can estimate the f-moment f(𝐱) where f is the characteristic exponent of the Lévy process. The fundamental Lévy-Khintchine representation theorem completely characterizes the space of all possible characteristic exponents, which in turn characterizes the set of f-moments that can be estimated by this generic scheme. The new scheme has strong explanatory power. It unifies the construction of many existing sketches (F₀, L₀, L₂, L_α, L_{p,q}, etc.) and it implies the tractability of many nearly periodic functions that were previously unclassified. Furthermore, the scheme can be conveniently generalized to multidimensional cases (d ≥ 2) by considering multidimensional Lévy processes and can be further generalized to estimate heterogeneous moments by projecting different indices with different Lévy processes. We conjecture that the set of tractable functions can be characterized using the Lévy-Khintchine representation theorem via what we called the Fourier-Hahn-Lévy method.