| Summary: | 碩士 === 國立交通大學 === 資訊工程系所 === 92 === We study the water jug problem and obtain new lower and upper
bounds on the minimum number of measuring steps. These bounds are
tight and significantly improve previous results. We prove that to
compute the crucial number $\mu_{\mathbf{c}}(x)$ (i.e.,
$\min\limits_{x=\mathbf{x\cdot c}}||\mathbf{x}||_1$, where
$\mathbf{c} \in \mathbf{N}^n, x \in \mathbf{N}, ||\mathbf{x}||_1
=\sum_{i=1}^n |x_i|$) for estimating the minimum measuring steps
is indeed a problem in $P^{NP}$. Moreover, we prove that testing
whether $\mu_{\mathbf{c}}(x)$ is bounded by a fixed number is
indeed NP-complete and thus the optimal jug measuring problem is
NP-hard, which was also proved independently by~\cite{Havas}.
Finally, we give a pseudo-polynomial time algorithm for computing
$\mu_{\mathbf{c}}(x)$ and a polynomial time algorithm, which is
based on {\it LLL} basis reduction algorithm, for approximating
the minimum number of jug measuring steps efficiently.
|
|---|
