Related papers: Tight Bounds for Hashing Block Sources
It is well established that the notion of min-entropy fails to satisfy the \emph{chain rule} of the form $H(X,Y) = H(X|Y)+H(Y)$, known for Shannon Entropy. Such a property would help to analyze how min-entropy is split among smaller blocks.…
Universal hash functions map the output of a source to random strings over a finite alphabet, aiming to approximate the uniform distribution on the set of strings. A classic result on these functions, called the Leftover Hash Lemma, gives…
The monotone minimal perfect hash function (MMPHF) problem is the following indexing problem. Given a set $S= \{s_1,\ldots,s_n\}$ of $n$ distinct keys from a universe $U$ of size $u$, create a data structure $DS$ that answers the following…
Min-entropy sampling gives a bound on the min-entropy of a randomly chosen subset of a string, given a bound on the min-entropy of the whole string. K\"onig and Renner showed a min-entropy sampling theorem that holds relative to quantum…
In large scale systems, approximate nearest neighbour search is a crucial algorithm to enable efficient data retrievals. Recently, deep learning-based hashing algorithms have been proposed as a promising paradigm to enable data dependent…
We show that a randomly chosen linear map over a finite field gives a good hash function in the $\ell_\infty$ sense. More concretely, consider a set $S \subset \mathbb{F}_q^n$ and a randomly chosen linear map $L : \mathbb{F}_q^n \to…
The weak law of large numbers implies that, under mild assumptions on the source, the Renyi entropy per produced symbol converges (in probability) towards the Shannon entropy rate. This paper quantifies the speed of this convergence for…
It is well known that the entropy $H(X)$ of a discrete random variable $X$ is always greater than or equal to the entropy $H(f(X))$ of a function $f$ of $X$, with equality if and only if $f$ is one-to-one. In this paper, we give tight…
Let X_1, ..., X_n be a sequence of n classical random variables and consider a sample of r positions selected at random. Then, except with (exponentially in r) small probability, the min-entropy of the sample is not smaller than, roughly, a…
We put forth a new computational notion of entropy, measuring the (in)feasibility of sampling high-entropy strings that are consistent with a given generator. Specifically, the i'th output block of a generator G has accessible entropy at…
The problem of minimizing convex functionals of probability distributions is solved under the assumption that the density of every distribution is bounded from above and below. A system of sufficient and necessary first-order optimality…
De, Trevisan and Tulsiani [CRYPTO 2010] show that every distribution over $n$-bit strings which has constant statistical distance to uniform (e.g., the output of a pseudorandom generator mapping $n-1$ to $n$ bit strings), can be…
A natural measure for the amount of quantum information that a physical system E holds about another system A = A_1,...,A_n is given by the min-entropy Hmin(A|E). Specifically, the min-entropy measures the amount of entanglement between E…
In this paper we study the problem of computing max-entropy distributions over a discrete set of objects subject to observed marginals. Interest in such distributions arises due to their applicability in areas such as statistical physics,…
We present novel and sharp lower bounds for higher load moments in the classical problem of mapping $M$ balls into $N$ bins by $q$-universal hashing, specialized to the case when $M=N$. As a corollary we prove a tight counterpart for the…
Given an increasing sequence of integers $x_1,\ldots,x_n$ from a universe $\{0,\ldots,u-1\}$, the monotone minimal perfect hash function (MMPHF) for this sequence is a data structure that answers the following rank queries: $rank(x) = i$ if…
In the problem of minimal perfect hashing, we are given a size $k$ subset $\mathcal{A}$ of a universe of keys $[n] = \{1,2, \cdots, n\}$, for which we wish to construct a hash function $h: [n] \to [k]$ such that $h(\cdot)$ maps…
A random hash function $h$ is $\varepsilon$-minwise if for any set $S$, $|S|=n$, and element $x\in S$, $\Pr[h(x)=\min h(S)]=(1\pm\varepsilon)/n$. Minwise hash functions with low bias $\varepsilon$ have widespread applications within…
Dependence among marginally constrained observations can break a finite-sample barrier. To formalize this phenomenon, we introduce the \emph{minimum list entropy coupling} $H(P\|Q_1,\dots,Q_m)$, the minimum conditional entropy…
Given $m \ge 2$ discrete probability distributions over $n$ states each, the minimum-entropy coupling is the minimum-entropy joint distribution whose marginals are the same as the input distributions. Computing the minimum-entropy coupling…