Related papers: Sampling Permutations with Cell Probes is Hard

We construct an explicit distribution $\mathbf{D}$ over $\{0,1\}^N$ that exhibits an essentially optimal separation between adaptive and non-adaptive cell-probe sampling. The distribution can be sampled exactly when each output bit is…

We show tight bounds for both online integer multiplication and convolution in the cell-probe model with word size w. For the multiplication problem, one pair of digits, each from one of two n digit numbers that are to be multiplied, is…

Since 1989, the best known lower bound on static data structures was Siegel's classical cell sampling lower bound. Siegel showed an explicit problem with $n$ inputs and $m$ possible queries such that every data structure that answers…

This paper develops an analogy between the cycle structure of, on the one hand, random permutations with cycle lengths restricted to lie in an infinite set $S$ with asymptotic density $\sigma$ and, on the other hand, permutations selected…

In this paper, we study the static cell probe complexity of non-adaptive data structures that maintain a subset of $n$ points from a universe consisting of $m=n^{1+\Omega(1)}$ points. A data structure is defined to be non-adaptive when the…

We show tight bounds for online Hamming distance computation in the cell-probe model with word size w. The task is to output the Hamming distance between a fixed string of length n and the last n symbols of a stream. We give a lower bound…

We consider uniform random permutations of length $n$ conditioned to have no cycle longer than $n^\beta$ with $0<\beta<1$, in the limit of large $n$. Since in unconstrained uniform random permutations most of the indices are in cycles of…

We prove that to store n bits x so that each prefix-sum query Sum(i) := sum_{k < i} x_k can be answered by non-adaptively probing q cells of log n bits, one needs memory > n + n/log^{O(q)} n. Our bound matches a recent upper bound of n +…

In this paper we develop a new technique for proving lower bounds on the update time and query time of dynamic data structures in the cell probe model. With this technique, we prove the highest lower bound to date for any explicit problem,…

The order $O_n(\sigma)$ of a permutation $\sigma$ of $n$ objects is the smallest integer $k \geq 1$ such that the $k$-th iterate of $\sigma$ gives the identity. A remarkable result about the order of a uniformly chosen permutation is due to…

We study the problem of generalized uniformity testing \cite{BC17} of a discrete probability distribution: Given samples from a probability distribution $p$ over an {\em unknown} discrete domain $\mathbf{\Omega}$, we want to distinguish,…

Boolean formulae compactly encode huge, constrained search spaces. Thus, variability-intensive systems are often encoded with Boolean formulae. The search space of a variability-intensive system is usually too large to explore without…

We compute the limiting distribution, as n approaches infinity, of the number of cycles of length between gamma n and delta n in a permutation of [n] chosen uniformly at random, for constants gamma, delta such that 1/(k+1) <= gamma < delta…

An involution is a bijection that is its own inverse. Given a permutation $\sigma$ of $[n],$ let $\mathsf{invol}(\sigma)$ denote the number of ways $\sigma$ can be expressed as a composition of two involutions of $[n].$ We prove that the…

A permutation array $A$ is a set of permutations on a finite set $\Omega$, say of size $n$. Given distinct permutations $\pi, \sigma\in \Omega$, we let $hd(\pi, \sigma) = |\{ x\in \Omega: \pi(x) \ne \sigma(x) \}|$, called the Hamming…

This paper develops a new technique for proving amortized, randomized cell-probe lower bounds on dynamic data structure problems. We introduce a new randomized nondeterministic four-party communication model that enables "accelerated",…

A permutation sequence $(\sigma_n)_{n \in \mathbb{N}}$ is said to be convergent if, for every fixed permutation $\tau$, the density of occurrences of $\tau$ in the elements of the sequence converges. We prove that such a convergent sequence…

A superpermutation is a sequence that contains every permutation of $n$ distinct symbols as a contiguous substring. For instance, a valid example for three symbols is a sequence that contains all six permutations. This paper introduces a…

A circuit $\mathcal{C}$ samples a distribution $\mathbf{X}$ with an error $\epsilon$ if the statistical distance between the output of $\mathcal{C}$ on the uniform input and $\mathbf{X}$ is $\epsilon$. We study the hardness of sampling a…

Efficiently counting or detecting defective items is a crucial task in various fields ranging from biological testing to quality control to streaming algorithms. The \emph{group testing estimation problem} concerns estimating the number of…