A Near-Linear Time Approximation Scheme for Geometric Transportation with Arbitrary Supplies and Spread

K. Fox, Jiashuai Lu
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引用次数: 10

Abstract

The geometric transportation problem takes as input a set of points $P$ in $d$-dimensional Euclidean space and a supply function $\mu : P \to \mathbb{R}$. The goal is to find a transportation map, a non-negative assignment $\tau : P \times P \to \mathbb{R}_{\geq 0}$ to pairs of points, so the total assignment leaving each point is equal to its supply, i.e., $\sum_{r \in P} \tau(q, r) - \sum_{p \in P} \tau(p, q) = \mu(q)$ for all points $q \in P$. The goal is to minimize the weighted sum of Euclidean distances for the pairs, $\sum_{(p, q) \in P \times P} \tau(p, q) \cdot ||q - p||_2$. We describe the first algorithm for this problem that returns, with high probability, a $(1 + \epsilon)$-approximation to the optimal transportation map in $O(n\:\text{poly}(1 / \epsilon)\:\text{polylog}{n})$ time. In contrast to the previous best algorithms for this problem, our near-linear running time bound is independent of the spread of $P$ and the magnitude of its real-valued supplies.
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具有任意供给和扩展的几何运输的近线性时间逼近格式
几何运输问题以$d$维欧氏空间中的一组点$P$和一个供给函数$\mu : P \to \mathbb{R}$作为输入。目标是找到一个交通地图,对点对的非负分配$\tau : P \times P \to \mathbb{R}_{\geq 0}$,所以每个点的总分配等于它的供给,即对所有点$q \in P$的总分配为$\sum_{r \in P} \tau(q, r) - \sum_{p \in P} \tau(p, q) = \mu(q)$。目标是最小化对欧几里得距离的加权和,$\sum_{(p, q) \in P \times P} \tau(p, q) \cdot ||q - p||_2$。我们描述了该问题的第一种算法,它以高概率返回$O(n\:\text{poly}(1 / \epsilon)\:\text{polylog}{n})$时间内最优交通地图的$(1 + \epsilon)$ -近似值。与此问题的先前最佳算法相比,我们的近线性运行时间界限独立于$P$的传播及其实值供应的大小。
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来源期刊
CiteScore
0.80
自引率
0.00%
发文量
4
审稿时长
>12 weeks
期刊介绍: The International Journal of Computational Geometry & Applications (IJCGA) is a quarterly journal devoted to the field of computational geometry within the framework of design and analysis of algorithms. Emphasis is placed on the computational aspects of geometric problems that arise in various fields of science and engineering including computer-aided geometry design (CAGD), computer graphics, constructive solid geometry (CSG), operations research, pattern recognition, robotics, solid modelling, VLSI routing/layout, and others. Research contributions ranging from theoretical results in algorithm design — sequential or parallel, probabilistic or randomized algorithms — to applications in the above-mentioned areas are welcome. Research findings or experiences in the implementations of geometric algorithms, such as numerical stability, and papers with a geometric flavour related to algorithms or the application areas of computational geometry are also welcome.
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