最短路径图等价分解及应用
Shortest Path Map Equivalence Decompositions and Applications
AI总结：
研究平面多边形域\(P\)的最短路径图等价分解，证明不同条件下其组合复杂度上界，提出计算分解的新算法，还解决了包括两点最短路径查询等其他问题。
摘要：
给定平面多边形域\(P\)，关于点\(s\)的最短路径图\(SPM(s)\)是将\(P\)分解为单元，使得同一单元内从\(s\)到所有点\(t\)的最短路径有相同顶点序列。\(P\)的最短路径图等价分解是使同一单元内所有点\(s\)的\(SPM(s)\)拓扑等价的分解。本文证明了不同设置下\(SPM\)等价分解组合复杂度的新上界，提出计算这些分解的新算法，还得出其他问题的新解。
英文摘要：
Given a polygonal domain $P$ in the plane, the shortest path map with respect to a point $s$, denoted by $SPM(s)$, is the decomposition of $P$ into cells such that shortest paths from $s$ to all points $t$ in the same cell have the same vertex sequence. The shortest path map equivalence decomposition of $P$ is the decomposition of $P$ into cells so that $SPM(s)$ is topologically equivalent for all points $s$ in the same cell. In this paper, we prove new upper bounds on the combinatorial complexities of the $SPM$-equivalence decompositions under various settings, depending on whether $s$ and/or $t$ are restricted to the boundary of $P$. We also propose new algorithms to compute these decompositions. Further, our results lead to new solutions to several other problems, including answering two-point shortest path queries in $P$, and computing geodesic diameter and center of $P$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
