arXiv subjects

Zi-Long Liu

Publications and source records attributed to Zi-Long Liu.

3 recordsLinked to original sources

arXiv2022

Large induced distance matchings in certain sparse random graphs

For a fixed integer $k\geqslant 2$, let $G\in \mathcal{G}(n,p)$ be a simple connected graph on $n\rightarrow\infty$ vertices with the expected degree $d=np$ satisfying $d\geqslant c$ and $d^{k-1}= o(n)$ for some large enough constant $c$. We show that the asymptotical size of any maximal collection of edges $M$ in $G$ such that no two edges in $M$ are within distance $k$, which is called a distance $k$-matching, is between $ \frac{(k-1)n\log d}{4d^{k-1}}$ and $ \frac{k n \log d}{2d^{k-1}}$. We also design a randomized greedy algorithm to generate one large distance $k$-matching in $G$ with asymptotical size $ \frac{kn\log d}{4d^{k-1}}$. Our results partially generalize the results on the size of the largest distance $k$-matchings from the case $k=2$ or $d=c$ for some large constant $c$.

math.CO↗

arXiv2022

Reversible Data Hiding in Encrypted Images by Lossless Pixel Conversion

Reversible data hiding in encrypted image (RDHEI) becomes a hot topic, and a lot of algorithms have been proposed to optimize this technology. However, these algorithms cannot achieve strong embedding capacity. Thus, in this paper, we propose an advanced RDHEI scheme based on lossless pixel conversion (LPC). Different from the previous RDHEI algorithms, LPC is inspired by the planar map coloring question, and it performs a dynamic image division process to divide the original image into irregular regions instead of regular blocks as in the previous RDHEI algorithms. In the process of LPC, pixel conversion is performed by region; that is, pixels in the same regions are converted to the same conversion values, which will occupy a smaller size, and then the available room can be reserved to accommodate additional data. LPC is a reversible process, so the original image can be losslessly recovered on the receiver side. Experimental results show that the embedding capacity of the proposed scheme outperforms the existing RDHEI algorithms.

cs.CR↗

arXiv2022

Random $K_k$-removal algorithm

One interesting question is how a graph develops from some constrained random graph process, which is a fundamental mechanism in the formation and evolution of dynamic networks. The problem here is referred to the random $K_k$-removal algorithm. For a fixed integer $k\geqslant 3$, it starts with a complete graph on $n\rightarrow\infty$ vertices and iteratively removes the edges of an uniformly chosen $K_k$. This algorithm terminates once no $K_k$s remain and at the same time it generates one linear $k$-uniform hypergraph. For $k=3$, it was shown that the size in the final graph is $n^{3/2+o(1)}$. Less results are on the cases when $k\geqslant 4$. In this paper, we prove that the exact expected trajectories of various key parameters in the algorithm to some iteration such that the final size in the algorithm is at most $n^{2-1/(k(k-1)-2)+o(1)}$ for $k\geqslant 4$. We also show the bound is a natural barrier.

math.CO↗