The Next-to-Shortest Path Problem in Graphs

• Main Authors: Kuo-Hua Kao, 高國華
• Other Authors: Yue-Li Wang
• Format: Others
• Language: en_US
• Published: 2010
• Online Access: http://ndltd.ncl.edu.tw/handle/08917498716992523332

博士 === 國立暨南國際大學 === 資訊工程學系 === 98 === Given an edge-weighted graph $G=(V,E,w)$ with two prescribed vertices $u$ and $v$, finding a shortest path from $u$ to $v$ is a classic problem in graph theory. An interesting variation of the shortest path problem is to find a next-to-shortest path from $u$ to $v$. It is a variation of the $k$-shortest paths problem and can be applied in VLSI design and in optimizing compilers for embedded systems. Given an edge-weighted undirected graph $G=(V,E,w)$ and two prescribed vertices $u$ and $v$, a next-to-shortest $(u,v)$-path is a shortest $(u,v)$-path among all $(u,v)$-paths having length strictly greater than the length of a shortest $(u,v)$-path. The next-to-shortest path problem is to find a next-to-shortest $(u,v)$-path. In 2006, Li, Sun and Chen proposed an ${\mathcal O}(n^3)$-time algorithm for the next-to-shortest problem. In this dissertation, we investigate the fundamental properties of a next-to-shortest $(u,v)$-path and present a quadratic time algorithm which significantly improves the bound of a previous one in ${\mathcal O}(n^3)$ time, where $n$ is the number of vertices in $G$.