Rani Hod

3/2 firefighters are not enough

The firefighter problem is a monotone dynamic process in graphs that can be viewed as modeling the use of a limited supply of vaccinations to stop the spread of an epidemic. In more detail, a fire spreads through a graph, from burning vertices to their unprotected neighbors. In every round, a small amount of unburnt vertices can be protected by firefighters. How many firefighters per turn, on average, are needed to stop the fire from advancing? We prove tight lower and upper bounds on the amount of firefighters needed to control a fire in the Cartesian planar grid and in the strong planar grid, resolving two conjectures of Ng and Raff.

Optimal Monotone Encodings

Moran, Naor, and Segev have asked what is the minimal r=r(n, k) for which there exists an (n,k)-monotone encoding of length r, i.e., a monotone injective function from subsets of size up to k of {1, 2,..., n} to r bits. Monotone encodings are relevant to the study of tamper-proof data structures and arise also in the design of broadcast schemes in certain communication networks. To answer this question, we develop a relaxation of k-superimposed families, which we call alpha-fraction k -multiuser tracing ((k, alpha)-FUT (fraction user-tracing) families). We show that r(n, k) = Theta(k log(n/k)) by proving tight asymptotic lower and upper bounds on the size of (k, alpha)-FUT families and by constructing an (n,k)-monotone encoding of length O(k log(n/k)). We also present an explicit construction of an (n, 2)-monotone encoding of length 2 log n+O(1), which is optimal up to an additive constant.

Random Low Degree Polynomials are Hard to Approximate

We study the problem of how well a typical multivariate polynomial can be approximated by lower degree polynomials over

On Active and Passive Testing

{\mathbb{F}_{2}}

Random low-degree polynomials are hard to approximate

. We prove that, with very high probability, a random degree d + 1 polynomial has only an exponentially small correlation with all polynomials of degree d , for all degrees d up to

Component Games on Regular Graphs

\Theta\left(n\right)

A Construction of Almost Steiner Systems

. That is, a random degree d + 1 polynomial does not admit a good approximation of lower degree. In order to prove this, we prove far tail estimates on the distribution of the bias of a random low degree polynomial. Recently, several results regarding the weight distribution of Reed---Muller codes were obtained. Our results can be interpreted as a new large deviation bound on the weight distribution of Reed---Muller codes.

Voronoi Choice Games

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? 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 et al. have recently suggested to restrict the tester to take its queries from a smaller, typically random, subset of the inputs. This model is called active testing, in resemblance of active learning. Active testing gets more dicult as the size of the set we can query from decreases, and the extreme case is when it is exactly the number of queries we perform (so the algorithm actually has no choice). This is known as passive testing, or testing from random examples. In their paper, Balcan et al. have shown that active and passive testing of dictator functions is as hard as learning them, and requires (log n) queries (unlike the classic model, in which it can be done in a constant number of queries). We extend this result to k-linear functions, proving that passive and active testing of them requires ( k logn) queries, assuming k is not too large. Other classes of functions we consider are juntas, partially symmetric functions, linear functions, and low degree polynomials. For linear functions we provide tight bounds on the query complexity in both active and passive models (which asymptotically dier). The analysis for low degree polynomials is less complete and the exact query complexity is given only for passive testing. In both these cases, the query complexity for passive testing is essentially equivalent to that of learning. For juntas and partially symmetric functions, that is, functions that depend on a small number of variables and potentially also on the Hamming weight of the input, we provide some lower and upper bounds for the dierent models. When the functions depend on a constant number of variables, our analysis for both families is asymptotically tight. Moreover, the family of partially symmetric functions is the first example for which the query complexities in all these models are asymptotically dierent. Our methods combine algebraic, combinatorial, and probabilistic techniques, including the Talagrand concentration inequality and the Erdfis‐Rado results on -systems.

CS1001.py: a topic-based introduction to computer science

We study the problem of how well a typical multivariate polynomial can be approximated by lower-degree polynomials over