arXiv subjects

Xue Chen

Publications and source records attributed to Xue Chen.

41 records · Page 3Linked to original sources

arXiv2018

Derandomized Balanced Allocation

In this paper, we study the maximum loads of explicit hash families in the $d$-choice schemes when allocating sequentially $n$ balls into $n$ bins. We consider the \emph{Uniform-Greedy} scheme, which provides $d$ independent bins for each ball and places the ball into the bin with the least load, and its non-uniform variant --- the \emph{Always-Go-Left} scheme introduced by Vöcking. We construct a hash family with $O(\log n \log \log n)$ random bits based on the previous work of Celis et al. and show the following results. 1. With high probability, this hash family has a maximum load of $\frac{\log \log n}{\log d} + O(1)$ in the \emph{Uniform-Greedy} scheme. 2. With high probability, it has a maximum load of $\frac{\log \log n}{d \log ϕ_d} + O(1)$ in the \emph{Always-Go-Left} scheme for a constant $ϕ_d>1.61$. The maximum loads of our hash family match the maximum loads of a perfectly random hash function in the \emph{Uniform-Greedy} and \emph{Always-Go-Left} scheme separately, up to the low order term of constants. Previously, the best known hash families matching the same maximum loads of a perfectly random hash function in $d$-choice schemes were $O(\log n)$-wise independent functions, which needs $Θ(\log^2 n)$ random bits.

cs.DS↗

arXiv2016

Parameterized Algorithms for Constraint Satisfaction Problems Above Average with Global Cardinality Constraints

Given a constraint satisfaction problem (CSP) on $n$ variables, $x_1, x_2, \dots, x_n \in \{\pm 1\}$, and $m$ constraints, a global cardinality constraint has the form of $\sum_{i = 1}^{n} x_i = (1-2p)n$, where $p \in (Ω(1), 1 - Ω(1))$ and $pn$ is an integer. Let $AVG$ be the expected number of constraints satisfied by randomly choosing an assignment to $x_1, x_2, \dots, x_n$, complying with the global cardinality constraint. The CSP above average with the global cardinality constraint problem asks whether there is an assignment (complying with the cardinality constraint) that satisfies more than $(AVG+t)$ constraints, where $t$ is an input parameter. In this paper, we present an algorithm that finds a valid assignment satisfying more than $(AVG+t)$ constraints (if there exists one) in time $(2^{O(t^2)} + n^{O(d)})$. Therefore, the CSP above average with the global cardinality constraint problem is fixed-parameter tractable.

cs.DS↗

arXiv2016

Fourier-sparse interpolation without a frequency gap

We consider the problem of estimating a Fourier-sparse signal from noisy samples, where the sampling is done over some interval $[0, T]$ and the frequencies can be "off-grid". Previous methods for this problem required the gap between frequencies to be above 1/T, the threshold required to robustly identify individual frequencies. We show the frequency gap is not necessary to estimate the signal as a whole: for arbitrary $k$-Fourier-sparse signals under $\ell_2$ bounded noise, we show how to estimate the signal with a constant factor growth of the noise and sample complexity polynomial in $k$ and logarithmic in the bandwidth and signal-to-noise ratio. As a special case, we get an algorithm to interpolate degree $d$ polynomials from noisy measurements, using $O(d)$ samples and increasing the noise by a constant factor in $\ell_2$.

cs.DS↗

arXiv2015

Integrality Gaps and Approximation Algorithms for Dispersers and Bipartite Expanders

We study the problem of approximating the quality of a disperser. A bipartite graph $G$ on $([N],[M])$ is a $(ρN,(1-δ)M)$-disperser if for any subset $S\subseteq [N]$ of size $ρN$, the neighbor set $Γ(S)$ contains at least $(1-δ)M$ distinct vertices. Our main results are strong integrality gaps in the Lasserre hierarchy and an approximation algorithm for dispersers. \begin{enumerate} \item For any $α>0$, $δ>0$, and a random bipartite graph $G$ with left degree $D=O(\log N)$, we prove that the Lasserre hierarchy cannot distinguish whether $G$ is an $(N^α,(1-δ)M)$-disperser or not an $(N^{1-α},δM)$-disperser. \item For any $ρ>0$, we prove that there exist infinitely many constants $d$ such that the Lasserre hierarchy cannot distinguish whether a random bipartite graph $G$ with right degree $d$ is a $(ρN, (1-(1-ρ)^d)M)$-disperser or not a $(ρN, (1-Ω(\frac{1-ρ}{ρd + 1-ρ}))M)$-disperser. We also provide an efficient algorithm to find a subset of size exact $ρN$ that has an approximation ratio matching the integrality gap within an extra loss of $\frac{\min\{\fracρ{1-ρ},\frac{1-ρ}ρ\}}{\log d}$. \end{enumerate} Our method gives an integrality gap in the Lasserre hierarchy for bipartite expanders with left degree~$D$. $G$ on $([N],[M])$ is a $(ρN,a)$-expander if for any subset $S\subseteq [N]$ of size $ρN$, the neighbor set $Γ(S)$ contains at least $a \cdot ρN$ distinct vertices. We prove that for any constant $ε>0$, there exist constants $ε'<ε,ρ,$ and $D$ such that the Lasserre hierarchy cannot distinguish whether a bipartite graph on $([N],[M])$ with left degree $D$ is a $(ρN, (1-ε')D)$-expander or not a $(ρN, (1-ε)D)$-expander.

cs.CC↗

arXiv2012

A performance Analysis of the Game of Life based on parallel algorithm

In this article, Conway's Game of Life using OpenMP parallel processing to simulate several different parallel methods, experimental performance results and compare to find the optimal solution of the parallelization of the Game of Life. Finally pointed out the importance of the design of parallel algorithms in solving the parallel problem.

cs.DC↗