arXiv subjects

Omri Ben-Eliezer

Publications and source records attributed to Omri Ben-Eliezer.

34 records · Page 2Linked to original sources

arXiv2020

Very fast construction of bounded-degree spanning graphs via the semi-random graph process

Semi-random processes involve an adaptive decision-maker, whose goal is to achieve some predetermined objective in an online randomized environment. They have algorithmic implications in various areas of computer science, as well as connections to biological processes involving decision making. In this paper, we consider a recently proposed semi-random graph process, described as follows: we start with an empty graph on $n$ vertices, and in each round, the decision-maker, called Builder, receives a uniformly random vertex $v$, and must immediately (in an online manner) choose another vertex $u$, adding the edge $\{u,v\}$ to the graph. Builder's end goal is to make the constructed graph satisfy some predetermined monotone graph property. We consider the property of containing a spanning graph $H$ as a subgraph. It was asked by N. Alon whether for any bounded-degree $H$, Builder can construct a copy of $H$ w.h.p. in $O(n)$ rounds. We answer this question positively in a strong sense, showing that any graph with maximum degree $Δ$ can be constructed w.h.p. in $(3Δ/2 + o(Δ)) n$ rounds. This is tight (even for the offline case) up to a multiplicative factor of $3 + o_Δ(1)$. Furthermore, for the special case where $H$ is a spanning forest of maximum degree $Δ$, we show that $H$ can be constructed w.h.p. in $O(n \log Δ)$ rounds. This is tight up to a multiplicative constant, even for the offline setting. Finally, we show a separation between adaptive and non-adaptive strategies, proving a lower bound of $Ω(n\sqrt{\log n})$ on the number of rounds necessary to eliminate all isolated vertices w.h.p. using a non-adaptive strategy. This bound is tight, and in fact there are non-adaptive strategies for constructing a Hamilton cycle or a $K_r$-factor, which are successful w.h.p. within $O(n\sqrt{\log n})$ rounds.

math.CO↗

arXiv2020

READ: Recursive Autoencoders for Document Layout Generation

Layout is a fundamental component of any graphic design. Creating large varieties of plausible document layouts can be a tedious task, requiring numerous constraints to be satisfied, including local ones relating different semantic elements and global constraints on the general appearance and spacing. In this paper, we present a novel framework, coined READ, for REcursive Autoencoders for Document layout generation, to generate plausible 2D layouts of documents in large quantities and varieties. First, we devise an exploratory recursive method to extract a structural decomposition of a single document. Leveraging a dataset of documents annotated with labeled bounding boxes, our recursive neural network learns to map the structural representation, given in the form of a simple hierarchy, to a compact code, the space of which is approximated by a Gaussian distribution. Novel hierarchies can be sampled from this space, obtaining new document layouts. Moreover, we introduce a combinatorial metric to measure structural similarity among document layouts. We deploy it to show that our method is able to generate highly variable and realistic layouts. We further demonstrate the utility of our generated layouts in the context of standard detection tasks on documents, showing that detection performance improves when the training data is augmented with generated documents whose layouts are produced by READ.

cs.CV↗

arXiv2020

The hat guessing number of graphs

Consider the following hat guessing game: $n$ players are placed on $n$ vertices of a graph, each wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. Given a graph $G$, its hat guessing number ${\rm{HG}}(G)$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors. In 2008, Butler et al. asked whether the hat guessing number of the complete bipartite graph $K_{n,n}$ is at least some fixed positive (fractional) power of $n$. We answer this question affirmatively, showing that for sufficiently large $n$, the complete $r$-partite graph $K_{n,\ldots,n}$ satisfies ${\rm{HG}}(K_{n,\ldots,n})=Ω(n^{\frac{r-1}{r}-o(1)})$. Our guessing strategy is based on a probabilistic construction and other combinatorial ideas, and can be extended to show that ${\rm{HG}}(\vec{C}_{n,\ldots,n})=Ω(n^{\frac{1}{r}-o(1)})$, where $\vec{C}_{n,\ldots,n}$ is the blow-up of a directed $r$-cycle, and where for directed graphs each player sees only the hat colors of his outneighbors.

math.CO↗

arXiv2019

Hard properties with (very) short PCPPs and their applications

We show that there exist properties that are maximally hard for testing, while still admitting PCPPs with a proof size very close to linear. Specifically, for every fixed $\ell$, we construct a property $\mathcal{P}^{(\ell)}\subseteq\{0,1\}^n$ satisfying the following: Any testing algorithm for $\mathcal{P}^{(\ell)}$ requires $Ω(n)$ many queries, and yet $\mathcal{P}^{(\ell)}$ has a constant query PCPP whose proof size is $O(n\cdot \log^{(\ell)}n)$, where $\log^{(\ell)}$ denotes the $\ell$ times iterated log function (e.g., $\log^{(2)}n = \log \log n$). The best previously known upper bound on the PCPP proof size for a maximally hard to test property was $O(n \cdot \mathrm{poly}\log{n})$. As an immediate application, we obtain stronger separations between the standard testing model and both the tolerant testing model and the erasure-resilient testing model: for every fixed $\ell$, we construct a property that has a constant-query tester, but requires $Ω(n/\log^{(\ell)}(n))$ queries for every tolerant or erasure-resilient tester.

cs.CC↗

arXiv2019

Optimal Adaptive Detection of Monotone Patterns

We investigate adaptive sublinear algorithms for detecting monotone patterns in an array. Given fixed $2 \leq k \in \mathbb{N}$ and $\varepsilon > 0$, consider the problem of finding a length-$k$ increasing subsequence in an array $f \colon [n] \to \mathbb{R}$, provided that $f$ is $\varepsilon$-far from free of such subsequences. Recently, it was shown that the non-adaptive query complexity of the above task is $Θ((\log n)^{\lfloor \log_2 k \rfloor})$. In this work, we break the non-adaptive lower bound, presenting an adaptive algorithm for this problem which makes $O(\log n)$ queries. This is optimal, matching the classical $Ω(\log n)$ adaptive lower bound by Fischer [2004] for monotonicity testing (which corresponds to the case $k=2$), and implying in particular that the query complexity of testing whether the longest increasing subsequence (LIS) has constant length is $Θ(\log n)$.

cs.DS↗

arXiv2019

Finding monotone patterns in sublinear time

We study the problem of finding monotone subsequences in an array from the viewpoint of sublinear algorithms. For fixed $k \in \mathbb{N}$ and $\varepsilon > 0$, we show that the non-adaptive query complexity of finding a length-$k$ monotone subsequence of $f \colon [n] \to \mathbb{R}$, assuming that $f$ is $\varepsilon$-far from free of such subsequences, is $Θ((\log n)^{\lfloor \log_2 k \rfloor})$. Prior to our work, the best algorithm for this problem, due to Newman, Rabinovich, Rajendraprasad, and Sohler (2017), made $(\log n)^{O(k^2)}$ non-adaptive queries; and the only lower bound known, of $Ω(\log n)$ queries for the case $k = 2$, followed from that on testing monotonicity due to Ergün, Kannan, Kumar, Rubinfeld, and Viswanathan (2000) and Fischer (2004).

cs.DS↗

arXiv2019

Semi-random graph process

We introduce and study a novel semi-random multigraph process, described as follows. The process starts with an empty graph on $n$ vertices. In every round of the process, one vertex $v$ of the graph is picked uniformly at random and independently of all previous rounds. We then choose an additional vertex (according to a strategy of our choice) and connect it by an edge to $v$. For various natural monotone increasing graph properties $P$, we prove tight upper and lower bounds on the minimum (extended over the set of all possible strategies) number of rounds required by the process to obtain, with high probability, a graph that satisfies $P$. Along the way, we show that the process is general enough to approximate (using suitable strategies) several well-studied random graph models.

math.CO↗

arXiv2019

The Adversarial Robustness of Sampling

Random sampling is a fundamental primitive in modern algorithms, statistics, and machine learning, used as a generic method to obtain a small yet "representative" subset of the data. In this work, we investigate the robustness of sampling against adaptive adversarial attacks in a streaming setting: An adversary sends a stream of elements from a universe $U$ to a sampling algorithm (e.g., Bernoulli sampling or reservoir sampling), with the goal of making the sample "very unrepresentative" of the underlying data stream. The adversary is fully adaptive in the sense that it knows the exact content of the sample at any given point along the stream, and can choose which element to send next accordingly, in an online manner. Well-known results in the static setting indicate that if the full stream is chosen in advance (non-adaptively), then a random sample of size $Ω(d / \varepsilon^2)$ is an $\varepsilon$-approximation of the full data with good probability, where $d$ is the VC-dimension of the underlying set system $(U,R)$. Does this sample size suffice for robustness against an adaptive adversary? The simplistic answer is \emph{negative}: We demonstrate a set system where a constant sample size (corresponding to VC-dimension $1$) suffices in the static setting, yet an adaptive adversary can make the sample very unrepresentative, as long as the sample size is (strongly) sublinear in the stream length, using a simple and easy-to-implement attack. However, this attack is "theoretical only", requiring the set system size to (essentially) be exponential in the stream length. This is not a coincidence: We show that to make Bernoulli or reservoir sampling robust against adaptive adversaries, the modification required is solely to replace the VC-dimension term $d$ in the sample size with the cardinality term $\log |R|$. This nearly matches the bound imposed by the attack.

cs.DS↗

arXiv2019

On the separation conjecture in Avoider-Enforcer games

Given a fixed graph $H$ with at least two edges and positive integers $n$ and $b$, the strict $(1 \colon b)$ Avoider-Enforcer $H$-game, played on the edge set of $K_n$, has the following rules: In each turn Avoider picks exactly one edge, and then Enforcer picks exactly $b$ edges. Avoider wins if and only if the subgraph containing her/his edges is $H$-free after all edges of $K_n$ are taken. The lower threshold of a graph $H$ with respect to $n$ is the largest $b_0$ for which Enforcer has a winning strategy for the $(1\colon b)$ $H$-game played on $K_n$ for any $b \leq b_0$, and the upper threshold is the largest $b$ for which Enforcer wins the $(1 \colon b)$ game. The separation conjecture of Hefetz, Krivelevich, Stojaković and Szabó states that for any connected $H$, the lower threshold and the upper threshold of the Avoider-Enforcer $H$-game played on $K_n$ are not of the same order in $n$. Until now, the conjecture has been verified only for stars, by Grzesik, Mikalački, Nagy, Naor, Patkos and Skerman. We show that the conjecture holds for every connected graph $H$ with at most one cycle (and at least two edges), with a polynomial separation between the lower and upper thresholds. We also prove an upper bound for the lower threshold of any graph $H$ with at least two edges, and show that this bound is tight for all graphs in which each connected component contains at most one cycle. Along the way, we establish number-theoretic tools that might be useful for other problems of this type.

math.CO↗

arXiv2018

Testing local properties of arrays

We study testing of local properties in one-dimensional and multi-dimensional arrays. A property of $d$-dimensional arrays $f:[n]^d \to Σ$ is $k$-local if it can be defined by a family of $k \times \ldots \times k$ forbidden consecutive patterns. This definition captures numerous interesting properties. For example, monotonicity, Lipschitz continuity and submodularity are $2$-local; convexity is (usually) $3$-local; and many typical problems in computational biology and computer vision involve $o(n)$-local properties. In this work, we present a generic approach to test all local properties of arrays over any finite (and not necessarily bounded size) alphabet. We show that any $k$-local property of $d$-dimensional arrays is testable by a simple canonical one-sided error non-adaptive $\varepsilon$-test, whose query complexity is $O(ε^{-1}k \log{\frac{εn}{k}})$ for $d = 1$ and $O(c_d ε^{-1/d} k \cdot n^{d-1})$ for $d > 1$. The queries made by the canonical test constitute sphere-like structures of varying sizes, and are completely independent of the property and the alphabet $Σ$. The query complexity is optimal for a wide range of parameters: For $d=1$, this matches the query complexity of many previously investigated local properties, while for $d > 1$ we design and analyze new constructions of $k$-local properties whose one-sided non-adaptive query complexity matches our upper bounds. For some previously studied properties, our method provides the first known sublinear upper bound on the query complexity.

cs.DS↗

arXiv2018

Earthmover Resilience and Testing in Ordered Structures

One of the main challenges in property testing is to characterize those properties that are testable with a constant number of queries. For unordered structures such as graphs and hypergraphs this task has been mostly settled. However, for ordered structures such as strings, images, and ordered graphs, the characterization problem seems very difficult in general. In this paper, we identify a wide class of properties of ordered structures - the earthmover resilient (ER) properties - and show that the "good behavior" of such properties allows us to obtain general testability results that are similar to (and more general than) those of unordered graphs. A property P is ER if, roughly speaking, slight changes in the order of the elements in an object satisfying P cannot make this object far from P. The class of ER properties includes, e.g., all unordered graph properties, many natural visual properties of images, such as convexity, and all hereditary properties of ordered graphs and images. A special case of our results implies, building on a recent result of Alon and the authors, that the distance of a given image or ordered graph from any hereditary property can be estimated (with good probability) up to a constant additive error, using a constant number of queries.

cs.DS↗

arXiv2017

Improved Bounds for Testing Forbidden Order Patterns

A sequence $f\colon\{1,\dots,n\}\to\mathbb{R}$ contains a permutation $π$ of length $k$ if there exist $i_1<\dots<i_k$ such that, for all $x,y$, $f(i_x)<f(i_y)$ if and only if $π(x)<π(y)$; otherwise, $f$ is said to be $π$-free. In this work, we consider the problem of testing for $π$-freeness with one-sided error, continuing the investigation of [Newman et al., SODA'17]. We demonstrate a surprising behavior for non-adaptive tests with one-sided error: While a trivial sampling-based approach yields an $\varepsilon$-test for $π$-freeness making $Θ(\varepsilon^{-1/k} n^{1-1/k})$ queries, our lower bounds imply that this is almost optimal for most permutations! Specifically, for most permutations $π$ of length $k$, any non-adaptive one-sided $\varepsilon$-test requires $\varepsilon^{-1/(k-Θ(1))}n^{1-1/(k-Θ(1))}$ queries; furthermore, the permutations that are hardest to test require $Θ(\varepsilon^{-1/(k-1)}n^{1-1/(k-1)})$ queries, which is tight in $n$ and $\varepsilon$. Additionally, we show two hierarchical behaviors here. First, for any $k$ and $l\leq k-1$, there exists some $π$ of length $k$ that requires $\tildeΘ_{\varepsilon}(n^{1-1/l})$ non-adaptive queries. Second, we show an adaptivity hierarchy for $π=(1,3,2)$ by proving upper and lower bounds for (one- and two-sided) testing of $π$-freeness with $r$ rounds of adaptivity. The results answer open questions of Newman et al. and [Canonne and Gur, CCC'17].

cs.DS↗

arXiv2017

Efficient Removal Lemmas for Matrices

The authors and Fischer recently proved that any hereditary property of two-dimensional matrices (where the row and column order is not ignored) over a finite alphabet is testable with a constant number of queries, by establishing the following (ordered) matrix removal lemma: For any finite alphabet $Σ$, any hereditary property $\mathcal{P}$ of matrices over $Σ$, and any $ε> 0$, there exists $f_{\mathcal{P}}(ε)$ such that for any matrix $M$ over $Σ$ that is $ε$-far from satisfying $\mathcal{P}$, most of the $f_{\mathcal{P}}(ε) \times f_{\mathcal{P}}(ε)$ submatrices of $M$ do not satisfy $\mathcal{P}$. Here being $ε$-far from $\mathcal{P}$ means that one needs to modify at least an $ε$-fraction of the entries of $M$ to make it satisfy $\mathcal{P}$. However, in the above general removal lemma, $f_{\mathcal{P}}(ε)$ grows very fast as a function of $ε^{-1}$, even when $\mathcal{P}$ is characterized by a single forbidden submatrix. In this work we establish much more efficient removal lemmas for several special cases of the above problem. In particular, we show the following: For any fixed $s \times t$ binary matrix $A$ and any $ε> 0$ there exists $δ> 0$ polynomial in $ε$, such that for any binary matrix $M$ in which less than a $δ$-fraction of the $s \times t$ submatrices are equal to $A$, there exists a set of less than an $ε$-fraction of the entries of $M$ that intersects every $A$-copy in $M$. We generalize the work of Alon, Fischer and Newman [SICOMP'07] and make progress towards proving one of their conjectures. The proofs combine their efficient conditional regularity lemma for matrices with additional combinatorial and probabilistic ideas.

math.CO↗

arXiv2017

Testing hereditary properties of ordered graphs and matrices

We consider properties of edge-colored vertex-ordered graphs, i.e., graphs with a totally ordered vertex set and a finite set of possible edge colors. We show that any hereditary property of such graphs is strongly testable, i.e., testable with a constant number of queries. We also explain how the proof can be adapted to show that any hereditary property of $2$-dimensional matrices over a finite alphabet (where row and column order is not ignored) is strongly testable. The first result generalizes the result of Alon and Shapira [FOCS'05, SICOMP'08], who showed that any hereditary graph property (without vertex order) is strongly testable. The second result answers and generalizes a conjecture of Alon, Fischer and Newman [SICOMP'07] concerning testing of matrix properties. The testability is proved by establishing a removal lemma for vertex-ordered graphs. It states that for any finite or infinite family $\mathcal{F}$ of forbidden vertex-ordered graphs, and any $ε> 0$, there exist $δ> 0$ and $k$ so that any vertex-ordered graph which is $ε$-far from being $\mathcal{F}$-free contains at least $δn^{|F|}$ copies of some $F\in\mathcal{F}$ (with the correct vertex order) where $|F|\leq k$. The proof bridges the gap between techniques related to the regularity lemma, used in the long chain of papers investigating graph testing, and string testing techniques. Along the way we develop a Ramsey-type lemma for $k$-partite graphs with "undesirable" edges, stating that one can find a Ramsey-type structure in such a graph, in which the density of the undesirable edges is not much higher than the density of those edges in the graph.

cs.DS↗

arXiv2017

Deleting and Testing Forbidden Patterns in Multi-Dimensional Arrays

Understanding the local behaviour of structured multi-dimensional data is a fundamental problem in various areas of computer science. As the amount of data is often huge, it is desirable to obtain sublinear time algorithms, and specifically property testers, to understand local properties of the data. We focus on the natural local problem of testing pattern freeness: given a large $d$-dimensional array $A$ and a fixed $d$-dimensional pattern $P$ over a finite alphabet, we say that $A$ is $P$-free if it does not contain a copy of the forbidden pattern $P$ as a consecutive subarray. The distance of $A$ to $P$-freeness is the fraction of entries of $A$ that need to be modified to make it $P$-free. For any $ε\in [0,1]$ and any large enough pattern $P$ over any alphabet, other than a very small set of exceptional patterns, we design a tolerant tester that distinguishes between the case that the distance is at least $ε$ and the case that it is at most $a_d ε$, with query complexity and running time $c_d ε^{-1}$, where $a_d < 1$ and $c_d$ depend only on $d$. To analyze the testers we establish several combinatorial results, including the following $d$-dimensional modification lemma, which might be of independent interest: for any large enough pattern $P$ over any alphabet (excluding a small set of exceptional patterns for the binary case), and any array $A$ containing a copy of $P$, one can delete this copy by modifying one of its locations without creating new $P$-copies in $A$. Our results address an open question of Fischer and Newman, who asked whether there exist efficient testers for properties related to tight substructures in multi-dimensional structured data. They serve as a first step towards a general understanding of local properties of multi-dimensional arrays, as any such property can be characterized by a fixed family of forbidden patterns.

cs.DS↗

arXiv2015

Local And Global Colorability of Graphs

It is shown that for any fixed $c \geq 3$ and $r$, the maximum possible chromatic number of a graph on $n$ vertices in which every subgraph of radius at most $r$ is $c$ colorable is $\tildeΘ\left(n ^ {\frac{1}{r+1}} \right)$ (that is, $n^\frac{1}{r+1}$ up to a factor poly-logarithmic in $n$). The proof is based on a careful analysis of the local and global colorability of random graphs and implies, in particular, that a random $n$-vertex graph with the right edge probability has typically a chromatic number as above and yet most balls of radius $r$ in it are $2$-degenerate.

math.CO↗