arXiv subjects

Ray Li

Publications and source records attributed to Ray Li.

44 records · Page 3Linked to original sources

arXiv2019

Enumeration of Preferred Extensions in Almost Oriented Digraphs

In this paper, we present enumeration algorithms to list all preferred extensions of an argumentation framework. This task is equivalent to enumerating all maximal semikernels of a directed graph. For directed graphs on $n$ vertices, all preferred extensions can be enumerated in $O^*(3^{n/3})$ time and there are directed graphs with $Ω(3^{n/3})$ preferred extensions. We give faster enumeration algorithms for directed graphs with at most $0.8004\cdot n$ vertices occurring in $2$-cycles. In particular, for oriented graphs (digraphs with no 2-cycles) one of our algorithms runs in time $O(1.2321^n)$, and we show that there are oriented graphs with $Ω(3^{n/6}) > Ω(1.2009^n)$ preferred extensions. A combination of three algorithms leads to the fastest enumeration times for various proportions of the number of vertices in $2$-cycles. The most innovative one is a new 2-stage sampling algorithm, combined with a new parameterized enumeration algorithm, analyzed with a combination of the recent monotone local search technique (STOC 2016) and an extension thereof (ICALP 2017).

cs.DS↗

arXiv2019

Polynomial time decodable codes for the binary deletion channel

In the random deletion channel, each bit is deleted independently with probability $p$. For the random deletion channel, the existence of codes of rate $(1-p)/9$, and thus bounded away from $0$ for any $p < 1$, has been known. We give an explicit construction with polynomial time encoding and deletion correction algorithms with rate $c_0 (1-p)$ for an absolute constant $c_0 > 0$.

cs.IT↗

arXiv2019

On Ramsey numbers of hedgehogs

The hedgehog $H_t$ is a 3-uniform hypergraph on vertices $1,\dots,t+\binom{t}{2}$ such that, for any pair $(i,j)$ with $1\le i t$ such that $\{i,j,k\}$ is an edge. Conlon, Fox, and Rödl proved that the two-color Ramsey number of the hedgehog grows polynomially in the number of its vertices, while the four-color Ramsey number grows exponentially in the number of its vertices. They asked whether the two-color Ramsey number of the hedgehog $H_t$ is nearly linear in the number of its vertices. We answer this question affirmatively, proving that $r(H_t) = O(t^2\ln t)$.

math.CO↗

arXiv2017

Coding against deletions in oblivious and online models

We consider binary error correcting codes when errors are deletions. A basic challenge concerning deletion codes is determining $p_0^{(adv)}$, the zero-rate threshold of adversarial deletions, defined to be the supremum of all $p$ for which there exists a code family with rate bounded away from 0 capable of correcting a fraction $p$ of adversarial deletions. A recent construction of deletion-correcting codes [Bukh et al 17] shows that $p_0^{(adv)} \ge \sqrt{2}-1$, and the trivial upper bound, $p_0^{(adv)}\le\frac{1}{2}$, is the best known. Perhaps surprisingly, we do not know whether or not $p_0^{(adv)} = 1/2$. In this work, to gain further insight into deletion codes, we explore two related error models: oblivious deletions and online deletions, which are in between random and adversarial deletions in power. In the oblivious model, the channel can inflict an arbitrary pattern of $pn$ deletions, picked without knowledge of the codeword. We prove the existence of binary codes of positive rate that can correct any fraction $p < 1$ of oblivious deletions, establishing that the associated zero-rate threshold $p_0^{(obliv)}$ equals $1$. For online deletions, where the channel decides whether to delete bit $x_i$ based only on knowledge of bits $x_1x_2\dots x_i$, define the deterministic zero-rate threshold for online deletions $p_0^{(on,d)}$ to be the supremum of $p$ for which there exist deterministic codes against an online channel causing $pn$ deletions with low average probability of error. That is, the probability that a randomly chosen codeword is decoded incorrectly is small. We prove $p_0^{(adv)}=\frac{1}{2}$ if and only if $p_0^{(on,d)}=\frac{1}{2}$.

cs.IT↗

arXiv2016

Central Limit Theorems for Gaps of Generalized Zeckendorf Decompositions

Zeckendorf proved that every integer can be written uniquely as a sum of non-adjacent Fibonacci numbers $\{1,2,3,5,\dots\}$. This has been extended to many other recurrence relations $\{G_n\}$ (with their own notion of a legal decomposition) and to proving that the distribution of the number of summands of an $M \in [G_n, G_{n+1})$ converges to a Gaussian as $n\to\infty$. We prove that for any non-negative integer $g$ the average number of gaps of size $g$ in many generalized Zeckendorf decompositions is $C_μn+d_μ+o(1)$ for constants $C_μ> 0$ and $d_μ$ depending on $g$ and the recurrence, the variance of the number of gaps of size $g$ is similarly $C_σn + d_σ+ o(1)$ with $C_σ> 0$, and the number of gaps of size $g$ of an $M\in[G_n,G_{n+1})$ converges to a Gaussian as $n\to\infty$. The proof is by analysis of an associated two-dimensional recurrence; we prove a general result on when such behavior converges to a Gaussian, and additionally re-derive other results in the literature.

math.NT↗

arXiv2016

Efficiently decodable insertion/deletion codes for high-noise and high-rate regimes

This work constructs codes that are efficiently decodable from a constant fraction of \emph{worst-case} insertion and deletion errors in three parameter settings: (i) Binary codes with rate approaching 1; (ii) Codes with constant rate for error fraction approaching 1 over fixed alphabet size; and (iii) Constant rate codes over an alphabet of size $k$ for error fraction approaching $(k-1)/(k+1)$. When errors are constrained to deletions alone, efficiently decodable codes in each of these regimes were constructed recently. We complete the picture by constructing similar codes that are efficiently decodable in the insertion/deletion regime.

cs.IT↗

arXiv2014

An Elementary Proof of the Cayley Formula Using Random Maps

Cayley's formula states that the number of labelled trees on $n$ vertices is $n^{n-2}$, and many of the current proofs involve complex structures or rigorous computation. We present a bijective proof of the formula by providing an elementary calculation of the probability that a cycle occurs in a random map from an $n$-element set to an $n+1$-element set.

math.CO↗

arXiv2013

A Simple Proof of the Cayley Formula using Random Graphs

We present a nice result on the probability of a cycle occurring in a randomly generated graph. We then provide some extensions and applications, including the proof of the famous Cayley formula, which states that the number of labeled trees on $n$ vertices is $n^{n-2}.$

math.CO↗