arXiv subjects

Kevin Zhao

Publications and source records attributed to Kevin Zhao.

28 records · Page 2Linked to original sources

arXiv2021

SWP: Microsecond Network SLOs Without Priorities

The increasing use of cloud computing for latency-sensitive applications has sparked renewed interest in providing tight bounds on network tail latency. Achieving this in practice at reasonable network utilization has proved elusive, due to a combination of highly bursty application demand, faster link speeds, and heavy-tailed message sizes. While priority scheduling can be used to reduce tail latency for some traffic, this comes at a cost of much worse delay behavior for all other traffic on the network. Most operators choose to run their networks at very low average utilization, despite the added cost, and yet still suffer poor tail behavior. This paper takes a different approach. We build a system, swp, to help operators (and network designers) to understand and control tail latency without relying on priority scheduling. As network workload changes, swp is designed to give real-time advice on the network switch configurations needed to maintain tail latency objectives for each traffic class. The core of swp is an efficient model for simulating the combined effect of traffic characteristics, end-to-end congestion control, and switch scheduling on service-level objectives (SLOs), along with an optimizer that adjusts switch-level scheduling weights assigned to each class. Using simulation across a diverse set of workloads with different SLOs, we show that to meet the same SLOs as swp provides, FIFO would require 65% greater link capacity, and 79% more for scenarios with tight SLOs on bursty traffic classes.

cs.NI↗

arXiv2020

Chip-Firing and Fractional Bases

We study a particular chip-firing process on an infinite path graph. At any time when there are at least $a+b$ chips at a vertex, $a$ chips fire to the left and $b$ chips fire to the right. We describe the final state of this process when we start with $n$ chips at the origin.

math.CO↗

arXiv2020

The coset factorization of finite cyclic group

Let $G$ be a finite cyclic group, written additively, and let $A,\ B$ be nonempty subsets of $G$. We will say that $G= A+B$ is a \textit{factorization} if for each $g$ in $G$ there are unique elements $a,\ b$ of $G$ such that $g=a+b, \ a\in A, b\in B$. In particular, if $A$ is a complete set of residues $modulo$ $|A|$, then we call the factorization a \textit{coset factorization} of $G$. In this paper, we mainly study a factorization $G= A+B$, where $G$ is a finite cyclic group and $A=[0,n-k-1]\cup\{i_0,i_1,\ldots i_{k-1}\}$ with $|A|=n$ and $n\geq 2k+1$. We obtain the following conclusion: If $(i)$ $k\leq 2$ or $(ii)$ The number of distinct prime divisors of $gcd(|A|,|B|)$ is at most $1$ or $(iii)$ $gcd(|A|,|B|)=pq$ with $gcd(pq,\frac{|B|}{gcd(|A|,|B|)})=1$, then $A$ is a complete set of residues $modulo$ $n$.

math.CO↗

arXiv2020

$1$-product problems with congruence conditions in nonabelian groups

Let $G$ be a finite group and $D_{2n}$ be the dihedral group of $2n$ elements. For a positive integer $d$, let $\mathsf{s}_{d\mathbb{N}}(G)$ denote the smallest integer $\ell\in \mathbb{N}_0\cup \{+\infty\}$ such that every sequence $S$ over $G$ of length $|S|\geq \ell$ has a nonempty $1$-product subsequence $T$ with $|T|\equiv 0$ (mod $d$). In this paper, we mainly study the problem for dihedral groups $D_{2n}$ and determine their exact values: $\mathsf{s}_{d\mathbb{N}}(D_{2n})=2d+\lfloor log_2n\rfloor$, if $d$ is odd with $n|d$; $\mathsf{s}_{d\mathbb{N}}(D_{2n})=nd+1$, if $gcd(n,d)=1$. Furthermore, we also analysis the problem for metacyclic groups $C_p\ltimes_s C_q$ and obtain a result: $\mathsf{s}_{kp\mathbb{N}}(C_p\ltimes_s C_q)=lcm(kp,q)+p-2+gcd(kp,q)$, where $p\geq 3$ and $p|q-1$.

math.CO↗

arXiv2020

The complete splittings of finite abelian groups

Let $G$ be a finite group. We will say that $M$ and $S$ form a \textsl{complete splitting} (\textsl{splitting}) of $G$ if every element (nonzero element) $g$ of $G$ has a unique representation of the form $g=ms$ with $m\in M$ and $s\in S$, and $0$ has a such representation (while $0$ has no such representation). In this paper, we determine the structures of complete splittings of finite abelian groups. In particular, for complete splittings of cyclic groups our description is more specific. Furthermore, we show some results for existence and nonexistence of complete splittings of cyclic groups and find a relationship between complete splittings and splittings for finite groups.

math.CO↗

arXiv2020

Purely singular splittings of cyclic groups

Let $G$ be a finite abelian group. We say that $M$ and $S$ form a \textsl{splitting} of $G$ if every nonzero element $g$ of $G$ has a unique representation of the form $g=ms$ with $m\in M$ and $s\in S$, while $0$ has no such representation. The splitting is called \textit{purely singular} if for each prime divisor $p$ of $|G|$, there is at least one element of $M$ is divisible by $p$. In this paper, we mainly study the purely singular splittings of cyclic groups. We first prove that if $k\ge3$ is a positive integer such that $[-k+1, \,k]^*$ splits a cyclic group $\mathbb{Z}_m$, then $m=2k$. Next, we have the following general result. Suppose $M=[-k_1, \,k_2]^*$ splits $\mathbb{Z}_{n(k_1+k_2)+1}$ with $1\leq k_1< k_2$. If $n\geq 2$, then $k_1\leq n-2$ and $k_2\leq 2n-5$. Applying this result, we prove that if $M=[-k_1, \,k_2]^*$ splits $\mathbb{Z}_m$ purely singularly, and either $(i)$ $\gcd(s, \,m)=1$ for all $s\in S$ or $(ii)$ $m=2^αp^β$ or $2^αp_1p_2$ with $α\geq 0$, $β\geq 1$ and $p$, $p_1$, $p_2$ odd primes, then $m=k_1+k_2+1$ or $k_1=0$ and $m=k_2+1$ or $2k_2+1$.

math.NT↗

arXiv2019

It's Common Knowledge

We discuss some old common knowledge puzzles and introduce a lot of new common knowledge puzzles.

math.HO↗

arXiv2019

On the Existence of Perfect Splitter Sets

Given integers $k_1, k_2$ with $0\le k_1<k_2$, the determinations of all positive integers $q$ for which there exists a perfect Splitter $B[-k_1, k_2](q)$ set is a wide open question in general. In this paper, we obtain new necessary and sufficient conditions for an odd prime $p$ such that there exists a nonsingular perfect $B[-1,3](p)$ set. We also give some necessary conditions for the existence of purely singular perfect splitter sets. In particular, we determine all perfect $B[-k_1, k_2](2^n)$ sets for any positive integers $k_1,k_2$ with $k_1+k_2\ge4$. We also prove that there are infinitely many prime $p$ such that there exists a perfect $B[-1,3](p)$ set.

cs.IT↗

arXiv2019

Variants of Base 3 over 2

We discuss two different systems of number representations that both can be called 'base 3/2'. We explain how they are connected. Unlike classical fractional extension, these two systems provide a finite representation for integers. We also discuss a connection between these systems and 3-free sequences.

math.NT↗

arXiv2019

The Harborth Constant of Dihedral Groups

The Harborth constant of a finite group $G$, denoted $\gs(G)$, is the smallest integer $k$ such that the following holds: For $A\subseteq G$ with $|A|=k$, there exists $B\subseteq A$ with $|B|=\exp(G)$ such that the elements of $B$ can be rearranged into a sequence whose product equals $1_G$, the identity element of $G$. The Harborth constant is a well studied combinatorial invariant in the case of abelian groups. In this paper, we consider a generalization $\gs(G)$ of this combinatorial invariant for nonabelian groups and prove that if $G$ is a dihedral group of order $2n$ with $n\ge 3$, then $\gs(G) = n + 2$ if $n$ is even and $\gs(G) = 2n + 1$ otherwise.

math.CO↗