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FPTAS for Minimizing the Earth Mover's Distance Under Rigid Transformations and Related Problems

机译:FPTAS用于在刚性变换和相关问题下最小化土方的距离

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In this paper, we consider the problem (denoted as EMDRT) of minimizing the earth mover's distance between two sets of weighted points A and B in R-d under rigid transformation. EMDRT is an important problem in both theory and applications and has received considerable attentions in recent years. Previous research on this problem has resulted in only constant factor approximations and it has been an open problem for a long time to achieve PTAS solution. In this paper, we present the first FPTAS algorithm for EMDRT. Our algorithm runs roughly in O((nm)(d+2)(log nm)(2d))time (which is close to a lower bound on any PTAS for this problem), where n and m are the sizes of A and B, respectively. Our result is based on several new techniques, such as the Sequential Orthogonal Decomposition and Optimum Guided Base, and can be extended to several related problems, such as the problem of earth mover's distance under similarity transformation and the alignment problem, to achieve FPTAS for each of them.
机译:在本文中,我们考虑了在刚性变换下使R-d中两组加权点A和B之间的推土机距离最小化的问题(称为EMDRT)。 EMDRT在理论和应用上都是一个重要的问题,近年来受到了相当大的关注。以前对此问题的研究仅得出恒定因子近似值,并且在很长一段时间内一直无法解决PTAS问题。在本文中,我们提出了用于EMDRT的第一个FPTAS算法。我们的算法大致以O((nm)(d + 2)(log nm)(2d))时间运行(此问题接近任何PTAS的下限),其中n和m是A和B分别。我们的结果基于诸如顺序正交分解和最佳导引基础之类的几种新技术,并且可以扩展到若干相关问题,例如相似性变换下的土方距离问题和对齐问题,以实现每种的FPTAS其中。

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