A sub-quadratic algorithm for bipartite matching of planar points with bounded integer coordinates
A sub-quadratic algorithm for bipartite matching of planar points with bounded integer coordinates
复制标题
有界整数坐标平面点二分匹配的次二次算法
DOI：
10.1145/2462356.2480283
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发表时间：
2013
影响因子：
24
通讯作者：
R. Sharathkumar
中科院分区：
文献类型：
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作者：
R. Sharathkumar
让<i> a，b </i>∈[δ] <sup> 2 </sup>，| <i> a </i> | = | = | </i> | = <i> n> | = <i> n </i>是一个点集，每个点具有一个由δ界定的正整数坐标。 /i> in <i> o </i>（<i> n </i> <sup> {3/2+Δ</sup> log> log（<i> n </i>δ））欧几里得双方匹配的先前的精确算法，即使点集有界定整数坐标，也要以ω（<i> n </i> <sup> 2 </sup>）时间。
首先，我们在<i> o </i>中计算（<i> n </i> <sup> {3/2+Δ</sup> log <i>nΔ）时间，候选人设置ε⊆>a </i> x x b </i>，以至于<i> m </i>*⊆ε和图G（<i> a a </i>∪<i>） B平面是平面； <i> m </i>*从<i> o </i>（<i> n </i> <sup> 3/2 </sup> log> log <i> n <> n </ I </ i>）使用[6]中描述的算法。
Let <i>A, B</i> ∈ [Δ]<sup>2</sup>, |<i>A</i>|=|<i>B</i>|=<i>n</i>, be point sets where each point has a positive integer coordinate bounded by Δ. For an arbitrary small constant δ > 0, we design an algorithm to compute a minimum-cost Euclidean bipartite matching of <i>A,B</i> in <i>O</i>(<i>n</i><sup>{3/2+δ</sup>log (<i>n</i>Δ)) time; all previous exact algorithms for the Euclidean bipartite matching, even when the point sets have bounded integer coordinates take Ω(<i>n</i><sup>2</sup>) time.
First, we compute in <i>O</i>(<i>n</i><sup>{3/2+δ</sup>log <i>n</i>Δ) time, a candidate set Ε ⊆ <i>A</i> x <i>B</i> such that <i>M</i>* ⊆ Ε and the graph G(<i>A</i>∪<i>B</i>, Ε) is planar; here <i>M</i>* is a minimum-cost matching of <i>A</i> and <i>B</i>. Next, we compute <i>M</i>* from this weighted bipartite planar graph in <i>O</i>(<i>n</i><sup>3/2</sup>log <i>n</i>) using the algorithm described in [6].