论文标题
部分可观测时空混沌系统的无模型预测
The Hamiltonian Path Graph is Connected for Simple $s,t$ Paths in Rectangular Grid Graphs
论文作者
论文摘要
矩形网格图中的A \ emph {simple} $ s，t $ pAth $ p $ p $ \ m m缩$ \ mathbb {g} $是从左上角$ s $到右下角$ t $的哈密顿式路径 bends needed to travel from $a$ to $b$ (i.e., $0$, $1$, or $2$ bends, depending on whether $a$ and $b$ are on opposite, adjacent, or the same side of the bounding rectangle).在这里，我们表明$ p $可以重新配置与\ emph {switching $ 2 \ times 2 $ squares}的任何其他简单$ s，t $ path的t $路径，其中最多需要$ {5} | \ mathbb {g} |/{4} $此类操作。此外，每个\ emph {square-switch}均以$ o（1）$时间完成，并将所得路径保留在同一简单$ s，t $ paths的家族中。我们的重新配置结果证明了\ emph {hamiltonian路径图} $ \ cal {g} $对于简单$ s，t $ paths已连接，最多只有$ {5} | \ mathbb {g} |/{4} |/{4} $，这是渐近紧的。
A \emph{simple} $s,t$ path $P$ in a rectangular grid graph $\mathbb{G}$ is a Hamiltonian path from the top-left corner $s$ to the bottom-right corner $t$ such that each \emph{internal} subpath of $P$ with both endpoints $a$ and $b$ on the boundary of $\mathbb{G}$ has the minimum number of bends needed to travel from $a$ to $b$ (i.e., $0$, $1$, or $2$ bends, depending on whether $a$ and $b$ are on opposite, adjacent, or the same side of the bounding rectangle). Here, we show that $P$ can be reconfigured to any other simple $s,t$ path of $\mathbb{G}$ by \emph{switching $2\times 2$ squares}, where at most ${5}|\mathbb{G}|/{4}$ such operations are required. Furthermore, each \emph{square-switch} is done in $O(1)$ time and keeps the resulting path in the same family of simple $s,t$ paths. Our reconfiguration result proves that the \emph{Hamiltonian path graph} $\cal{G}$ for simple $s,t$ paths is connected and has diameter at most ${5}|\mathbb{G}|/{4}$ which is asymptotically tight.
