Noga Alon

Noga Alon
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Noga Alon

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Pub Categories

Mathematics - Combinatorics (36)
Computer Science - Data Structures and Algorithms (9)
Computer Science - Discrete Mathematics (6)
Computer Science - Information Theory (5)
Mathematics - Information Theory (5)
Computer Science - Computational Complexity (5)
Computer Science - Learning (4)
Computer Science - Computer Science and Game Theory (3)
Computer Science - Distributed; Parallel; and Cluster Computing (3)
Statistics - Machine Learning (2)
Mathematics - Number Theory (2)
Mathematics - Probability (2)
Mathematics - Metric Geometry (2)
Computer Science - Multiagent Systems (1)
Quantitative Biology - Genomics (1)
Quantitative Biology - Populations and Evolution (1)

Publications Authored By Noga Alon

In this work we derive a variant of the classic Glivenko-Cantelli Theorem, which asserts uniform convergence of the empirical Cumulative Distribution Function (CDF) to the CDF of the underlying distribution. Our variant allows for tighter convergence bounds for extreme values of the CDF. We apply our bound in the context of revenue learning, which is a well-studied problem in economics and algorithmic game theory. Read More

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. Read More

For two fixed graphs $T$ and $H$ let $ex(G(n,p),T,H)$ be the random variable counting the maximum number of copies of $T$ in an $H$-free subgraph of the random graph $G(n,p)$. We show that for the case $T=K_m$ and $\chi(H)> m$ the behavior of $ex(G(n,p),K_m,H)$ depends strongly on the relation between $p$ and $m_2(H)=\max_{H'\subset H, |V(H')|'\geq 3}\left\{ \frac{e(H')-1}{v(H')-2} \right\}$. When $m_2(H)> m_2(K_m)$ we prove that with high probability, depending on the value of $p$, either one can maintain almost all copies of $K_m$, or it is asymptotically best to take a $\chi(H)-1$ partite subgraph of $G(n,p)$. Read More

The clique chromatic number of a graph G=(V,E) is the minimum number of colors in a vertex coloring so that no maximal (with respect to containment) clique is monochromatic. We prove that the clique chromatic number of the binomial random graph G=G(n,1/2) is, with high probability, \Omega(log n). This settles a problem of McDiarmid, Mitsche and Pralat who proved that it is O(log n) with high probability. Read More

We study the tandem duplication distance between binary sequences and their roots. In other words, the quantity of interest is the number of tandem duplication operations of the form $\seq x = \seq a \seq b \seq c \to \seq y = \seq a \seq b \seq b \seq c$, where $\seq x$ and $\seq y$ are sequences and $\seq a$, $\seq b$, and $\seq c$ are their substrings, needed to generate a binary sequence of length $n$ starting from a square-free sequence from the set $\{0,1,01,10,010,101\}$. This problem is a restricted case of finding the duplication/deduplication distance between two sequences, defined as the minimum number of duplication and deduplication operations required to transform one sequence to the other. Read More

For a hypergraph $H$ let $\beta(H)$ denote the minimal number of edges from $H$ covering $V(H)$. An edge $S$ of $H$ is said to represent {\em fairly} (resp. {\em almost fairly}) a partition $(V_1,V_2, \ldots, V_m)$ of $V(H)$ if $|S\cap V_i|\ge \lfloor\frac{|V_i|}{\beta(H)}\rfloor$ (resp. Read More

Let $X$ be a set of $n$ points of norm at most $1$ in the Euclidean space $R^k$, and suppose $\varepsilon>0$. An $\varepsilon$-distance sketch for $X$ is a data structure that, given any two points of $X$ enables one to recover the square of the (Euclidean) distance between them up to an {\em additive} error of $\varepsilon$. Let $f(n,k,\varepsilon)$ denote the minimum possible number of bits of such a sketch. Read More

The graph removal lemma is an important structural result in graph theory. It has many different variants and is closely related to property testing and other areas. Our main aim is to develop removal lemmas of the same spirit for two dimensional matrices. Read More

A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. In 1980 Hahn conjectured that every properly edge-coloured complete graph $K_n$ has a rainbow Hamiltonian path. Although this conjecture turned out to be false, it was widely believed that such a colouring always contains a rainbow cycle of length almost $n$. Read More

We show that for any constant $\Delta \ge 2$, there exists a graph $G$ with $O(n^{\Delta / 2})$ vertices which contains every $n$-vertex graph with maximum degree $\Delta$ as an induced subgraph. For odd $\Delta$ this significantly improves the best-known earlier bound of Esperet et al. and is optimal up to a constant factor, as it is known that any such graph must have at least $\Omega(n^{\Delta/2})$ vertices. Read More

We calculate the exact number of contours of size $n$ containing a fixed vertex in $d$-ary trees and provide sharp estimates for this number for more general trees. We also obtain a characterization of the locally finite trees with infinitely many contours of the same size containing a fixed vertex. Read More

Exclusive social groups are ones in which the group members decide whether or not to admit a candidate to the group. Examples of exclusive social groups include academic departments and fraternal organizations. In the present paper we introduce an analytic framework for studying the dynamics of exclusive social groups. Read More

Let $G=(V,E)$ be a connected undirected graph with $k$ vertices. Suppose that on each vertex of the graph there is a player having an $n$-bit string. Each player is allowed to communicate with its neighbors according to an agreed communication protocol, and the players must decide, deterministically, if their inputs are all equal. Read More

We consider the problem of corruption detection on networks. In this model each vertex of a directed graph can be either truthful or corrupt. Each vertex reports about the types (truthful or corrupt) of all his out-neighbors. Read More

A conjecture of Bandelt and Dress states that the maximum quartet distance between any two phylogenetic trees on $n$ leaves is at most $(\frac 23 +o(1))\binom{n}{4}$. Using the machinery of flag algebras we improve the currently known bounds regarding this conjecture, in particular we show that the maximum is at most $(0.69 +o(1))\binom{n}{4}$. Read More

We continue the study of welfare maximization in unit-demand (matching) markets, in a distributed information model where agent's valuations are unknown to the central planner, and therefore communication is required to determine an efficient allocation. Dobzinski, Nisan and Oren (STOC'14) showed that if the market size is $n$, then $r$ rounds of interaction (with logarithmic bandwidth) suffice to obtain an $n^{1/(r+1)}$-approximation to the optimal social welfare. In particular, this implies that such markets converge to a stable state (constant approximation) in time logarithmic in the market size. Read More

This work studies the maximum possible sign rank of $N \times N$ sign matrices with a given VC dimension $d$. For $d=1$, this maximum is {three}. For $d=2$, this maximum is $\tilde{\Theta}(N^{1/2})$. Read More

We study a general class of online learning problems where the feedback is specified by a graph. This class includes online prediction with expert advice and the multi-armed bandit problem, but also several learning problems where the online player does not necessarily observe his own loss. We analyze how the structure of the feedback graph controls the inherent difficulty of the induced $T$-round learning problem. Read More

A (continuous) necklace is simply an interval of the real line colored measurably with some number of colors. A well-known application of the Borsuk-Ulam theorem asserts that every $k$-colored necklace can be fairly split by at most $k$ cuts (from the resulting pieces one can form two collections, each capturing the same measure of every color). Here we prove that for every $k\geq 1$ there is a measurable $(k+3)$-coloring of the real line such that no interval can be fairly split using at most $k$ cuts. Read More

An $r$-augmented tree is a rooted tree plus $r$ edges added from each leaf to ancestors. For $d,g,r\in\mathbb{N}$, we construct a bipartite $r$-augmented complete $d$-ary tree having girth at least $g$. The height of such trees must grow extremely rapidly in terms of the girth. Read More

Given a family $\mathcal{F}$ of subsets of $[n]$, we say two sets $A, B \in \mathcal{F}$ are comparable if $A \subset B$ or $B \subset A$. Sperner's celebrated theorem gives the size of the largest family without any comparable pairs. This result was later generalised by Kleitman, who gave the minimum number of comparable pairs appearing in families of a given size. Read More

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 $\tilde{\Theta}\left(n ^ {\frac{1}{r+1}} \right)$ (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. Read More

We present and study a partial-information model of online learning, where a decision maker repeatedly chooses from a finite set of actions, and observes some subset of the associated losses. This naturally models several situations where the losses of different actions are related, and knowing the loss of one action provides information on the loss of other actions. Moreover, it generalizes and interpolates between the well studied full-information setting (where all losses are revealed) and the bandit setting (where only the loss of the action chosen by the player is revealed). Read More

For a graph $G=(V,E)$, let $bc(G)$ denote the minimum number of pairwise edge disjoint complete bipartite subgraphs of $G$ so that each edge of $G$ belongs to exactly one of them. It is easy to see that for every graph $G$, $bc(G) \leq n -\alpha(G)$, where $\alpha(G)$ is the maximum size of an independent set of $G$. Erd\H{o}s conjectured in the 80s that for almost every graph $G$ equality holds, i. Read More

For two graphs $T$ and $H$ with no isolated vertices and for an integer $n$, let $ex(n,T,H)$ denote the maximum possible number of copies of $T$ in an $H$-free graph on $n$ vertices. The study of this function when $T=K_2$ is a single edge is the main subject of extremal graph theory. In the present paper we investigate the general function, focusing on the cases of triangles, complete graphs, complete bipartite graphs and trees. Read More

The 'separation dimension' of a graph $G$ is the smallest natural number $k$ for which the vertices of $G$ can be embedded in $\mathbb{R}^k$ such that any pair of disjoint edges in $G$ can be separated by a hyperplane normal to one of the axes. Equivalently, it is the smallest possible cardinality of a family $\mathcal{F}$ of total orders of the vertices of $G$ such that for any two disjoint edges of $G$, there exists at least one total order in $\mathcal{F}$ in which all the vertices in one edge precede those in the other. In general, the maximum separation dimension of a graph on $n$ vertices is $\Theta(\log n)$. Read More

For a graph $H$, the {\em extremal number} $ex(n,H)$ is the maximum number of edges in a graph of order $n$ not containing a subgraph isomorphic to $H$. Let $\delta(H)>0$ and $\Delta(H)$ denote the minimum degree and maximum degree of $H$, respectively. We prove that for all $n$ sufficiently large, if $H$ is any graph of order $n$ with $\Delta(H) \le \sqrt{n}/200$, then $ex(n,H)={{n-1} \choose 2}+\delta(H)-1$. Read More

For a graph $G=(V,E)$, let $\tau(G)$ denote the minimum number of pairwise edge disjoint complete bipartite subgraphs of $G$ so that each edge of $G$ belongs to exactly one of them. It is easy to see that for every graph $G$, $\tau(G) \leq n -\alpha(G)$, where $\alpha(G)$ is the maximum size of an independent set of $G$. Erd\H{o}s conjectured in the 80s that for almost every graph $G$ equality holds, i. Read More

Let $f:\mathbb{R}^k\to \mathbb{R}$ be a measurable function, and let $\{U_i\}_{i\in\mathbb{N}}$ be a sequence of i.i.d. Read More

The problem of constructing a minimal rank matrix over GF(2) whose kernel does not intersect a given set S is considered. In the case where S is a Hamming ball centered at 0, this is equivalent to finding linear codes of largest dimension. For a general set, this is an instance of "the critical problem" posed by Crapo and Rota in 1970. Read More

We study the vertex pursuit game of \emph{Cops and Robbers}, in which cops try to capture a robber on the vertices of the graph. The minimum number of cops required to win on a given graph $G$ is called the cop number of $G$. We focus on $G_{d}(n,r)$, a random geometric graph in which $n$ vertices are chosen uniformly at random and independently from $[0,1]^d$, and two vertices are adjacent if the Euclidean distance between them is at most $r$. Read More

Given a set of $n$ real numbers, if the sum of elements of every subset of size larger than $k$ is negative, what is the maximum number of subsets of nonnegative sum? In this note we show that the answer is $\binom{n-1}{k-1} + \binom{n-1}{k-2} + \cdots + \binom{n-1}{0}+1$, settling a problem of Tsukerman. We provide two proofs, the first establishes and applies a weighted version of Hall's Theorem and the second is based on an extension of the nonuniform Erd\H{o}s-Ko-Rado Theorem. Read More

Given a property of Boolean functions, what is the minimum number of queries required to determine with high probability if an input function satisfies this property or is "far" from satisfying it? This is a fundamental question in Property Testing, where traditionally the testing algorithm is allowed to pick its queries among the entire set of inputs. Balcan, Blais, Blum and Yang have recently suggested to restrict the tester to take its queries from a smaller random subset of polynomial size of the inputs. This model is called active testing, and in the extreme case when the size of the set we can query from is exactly the number of queries performed it is known as passive testing. Read More

We consider the partial observability model for multi-armed bandits, introduced by Mannor and Shamir. Our main result is a characterization of regret in the directed observability model in terms of the dominating and independence numbers of the observability graph. We also show that in the undirected case, the learner can achieve optimal regret without even accessing the observability graph before selecting an action. Read More

A {\em faithful (unit) distance graph} in $\mathbb{R}^d$ is a graph whose set of vertices is a finite subset of the $d$-dimensional Euclidean space, where two vertices are adjacent if and only if the Euclidean distance between them is exactly $1$. A {\em (unit) distance graph} in $\mathbb{R}^d$ is any subgraph of such a graph. In the first part of the paper we focus on the differences between these two classes of graphs. Read More

A Boolean function f of n variables is said to be q-locally correctable if, given a black-box access to a function g which is "close" to an isomorphism f_sigma(x)=f_sigma(x_1, ... Read More

When numbers are added in base $b$ in the usual way, carries occur. If two random, independent 1-digit numbers are added, then the probability of a carry is $\frac{b-1}{2b}$. Other choices of digits lead to less carries. Read More

It is shown that for any outerplanar graph G there is a one to one mapping of the vertices of G to the plane, so that the number of distinct distances between pairs of connected vertices is at most three. This settles a problem of Carmi, Dujmovic, Morin and Wood. The proof combines (elementary) geometric, combinatorial, algebraic and probabilistic arguments. Read More

For two sets $A$ and $M$ of positive integers and for a positive integer $n$, let $p(n,A,M)$ denote the number of partitions of $n$ with parts in $A$ and multiplicities in $M$, that is, the number of representations of $n$ in the form $n=\sum_{a \in A} m_a a$ where $m_a \in M \cup {0}$ for all $a$, and all numbers $m_a$ but finitely many are 0. It is shown that there are infinite sets $A$ and $M$ so that $p(n,A,M)=1$ for every positive integer $n$. This settles (in a strong form) a problem of Canfield and Wilf. Read More

We consider the problem of computing a maximal independent set (MIS) in an extremely harsh broadcast model that relies only on carrier sensing. The model consists of an anonymous broadcast network in which nodes have no knowledge about the topology of the network or even an upper bound on its size. Furthermore, it is assumed that an adversary chooses at which time slot each node wakes up. Read More

The broadcast throughput in a network is defined as the average number of messages that can be transmitted per unit time from a given source to all other nodes when time goes to infinity. Classical broadcast algorithms treat messages as atomic tokens and route them from the source to the receivers by making intermediate nodes store and forward messages. The more recent network coding approach, in contrast, prompts intermediate nodes to mix and code together messages. Read More

In this paper we study sum-free sets of order $m$ in finite Abelian groups. We prove a general theorem on 3-uniform hypergraphs, which allows us to deduce structural results in the sparse setting from stability results in the dense setting. As a consequence, we determine the typical structure and asymptotic number of sum-free sets of order $m$ in Abelian groups $G$ whose order is divisible by a prime $q$ with $q \equiv 2 \pmod 3$, for every $m \ge C(q) \sqrt{n \log n}$, thus extending and refining a theorem of Green and Ruzsa. Read More

We describe two constructions of (very) dense graphs which are edge disjoint unions of large {\em induced} matchings. The first construction exhibits graphs on $N$ vertices with ${N \choose 2}-o(N^2)$ edges, which can be decomposed into pairwise disjoint induced matchings, each of size $N^{1-o(1)}$. The second construction provides a covering of all edges of the complete graph $K_N$ by two graphs, each being the edge disjoint union of at most $N^{2-\delta}$ induced matchings, where $\delta > 0. Read More

A graph property P is strongly testable if for every fixed \epsilon>0 there is a one-sided \epsilon-tester for P whose query complexity is bounded by a function of \epsilon. In classifying the strongly testable graph properties, the first author and Shapira showed that any hereditary graph property (such as P the family of perfect graphs) is strongly testable. A property is easily testable if it is strongly testable with query complexity bounded by a polynomial function of \epsilon^{-1}, and otherwise it is hard. Read More

Recently Rubinfeld et al. (ICS 2011, pp. 223--238) proposed a new model of sublinear algorithms called \emph{local computation algorithms}. Read More

A Boolean function f over n variables is said to be q-locally correctable if, given a black-box access to a function g which is "close" to an isomorphism f_sigma of f, we can compute f_sigma(x) for any x in Z_2^n with good probability using q queries to g. We observe that any k-junta, that is, any function which depends only on k of its input variables, is O(2^k)-locally correctable. Moreover, we show that there are examples where this is essentially best possible, and locally correcting some k-juntas requires a number of queries which is exponential in k. Read More

In this paper we study conditions which guarantee the existence of perfect matchings and perfect fractional matchings in uniform hypergraphs. We reduce this problem to an old conjecture by Erd\H{o}s on estimating the maximum number of edges in a hypergraph when the (fractional) matching number is given, which we are able to solve in some special cases using probabilistic techniques. Based on these results, we obtain some general theorems on the minimum $d$-degree ensuring the existence of perfect (fractional) matchings. Read More

Humans are very good at optimizing solutions for specific problems. Biological processes, on the other hand, have evolved to handle multiple constrained distributed environments and so they are robust and adaptable. Inspired by observations made in a biological system we have recently presented a simple new randomized distributed MIS algorithm \cite{ZScience}. Read More

More than twenty years ago, Manickam, Mikl\'{o}s, and Singhi conjectured that for any integers $n, k$ satisfying $n \geq 4k$, every set of $n$ real numbers with nonnegative sum has at least $\binom{n-1}{k-1}$ $k$-element subsets whose sum is also nonnegative. In this paper we discuss the connection of this problem with matchings and fractional covers of hypergraphs, and with the question of estimating the probability that the sum of nonnegative independent random variables exceeds its expectation by a given amount. Using these connections together with some probabilistic techniques, we verify the conjecture for $n \geq 33k^2$. Read More