Jug Measuring: Algorithm and Complexity

• Main Authors: Min-Zheng Shieh, 謝旻錚
• Other Authors: Shi-Chun Tsai
• Format: Others
• Language: en_US
• Published: 2004
• Online Access: http://ndltd.ncl.edu.tw/handle/06712731681083419419

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.