The Discrete and Semicontinuous Fréchet Distance with Shortcuts via Approximate Distance Counting and Selection
The Discrete and Semicontinuous Fréchet Distance with Shortcuts via Approximate Distance Counting and Selection
复制标题
通过近似距离计数和选择的离散和半连续 Fréchet 距离的快捷方式
DOI：
10.1145/2700222
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
M. Sharir
中科院分区：
文献类型：
--
作者：
Rinat Ben Avraham;O. Filtser;Haim Kaplan;M. J. Katz;M. Sharir
<i>fréchet距离</i>是曲线之间的一个良好的相似性度量。 <i> n </i>点，其中这些点通常是从输入曲线采样的。在异常值的情况下快捷方式仅在一个含噪声的曲线中，我们给出一种以<i> o </i>运行的随机算法（（（（<i> m </i> + <i> n> n </i>））<sup> 6/5 + ϵ </sup>）预期时间，对于任何ϵ> 0。当两条曲线允许捷径时，我们给出一个<i> o </i>（（（<i> m </i> <sup） > 2/3 </sup> <i> n </i> <sup> 2/3 </sup> + <i> m </i> + <i> n </i>）log <sup> 3 </sup>（<i> m </i> + <i> n </i>）） - 时间确定性算法。
我们还考虑具有单方面快捷方式的半连续fréchet距离，在该距离中，我们的序列序列是<i> m </i>的序列和一个边缘的多边形曲线，并且仅允许快捷方式序列。 sup> <i> m </i> <sup> 2/3 </sup> <i> n </i> <sup> 1/3 </sup> log（<i> m </i> +） <i> n </i>））。
我们的技术是新颖的，可能会找到进一步的应用。 i </i>，我们开发了一种算法，该算法决定对（x x </i>，<i> y>）的数量是否</i>在<i> i> y> y> y> y> y> i> </i>中的距离dist（<i> x </i> </i> 。此算法的运行时间会随着<i> l的增加而减小。获取一对对近似距离的对，即我们可以大约“一分子” <i> i </i>）。使用A的最佳单侧Fréchet距离，使用A非常快速的决策过程一般，新技术可以应用于决策过程非常快的优化问题，但是标准技术（例如参数搜索）使优化算法大大慢。
The <i>Fréchet distance</i> is a well-studied similarity measure between curves. The <i>discrete Fréchet distance</i> is an analogous similarity measure, defined for two sequences of <i>m</i> and <i>n</i> points, where the points are usually sampled from input curves. We consider a variant, called the <i>discrete Fréchet distance with shortcuts</i>, which captures the similarity between (sampled) curves in the presence of outliers. When shortcuts are allowed only in one noise-containing curve, we give a randomized algorithm that runs in <i>O</i>((<i>m</i> + <i>n</i>)<sup>6/5 + ϵ</sup>) expected time, for any ϵ > 0. When shortcuts are allowed in both curves, we give an <i>O</i>((<i>m</i><sup>2/3</sup><i>n</i><sup>2/3</sup> + <i>m</i> + <i>n</i>)log <sup>3</sup>(<i>m</i> + <i>n</i>))-time deterministic algorithm.
We also consider the semicontinuous Fréchet distance with one-sided shortcuts, where we have a sequence of <i>m</i> points and a polygonal curve of <i>n</i> edges, and shortcuts are allowed only in the sequence. We show that this problem can be solved in randomized expected time <i>O</i>((<i>m</i> + <i>n</i>)<sup>2/3</sup><i>m</i><sup>2/3</sup><i>n</i><sup>1/3</sup>log (<i>m</i> + <i>n</i>)).
Our techniques are novel and may find further applications. One of the main new technical results is: Given two sets of points <i>A</i> and <i>B</i> in the plane and an interval <i>I</i>, we develop an algorithm that decides whether the number of pairs (<i>x</i>, <i>y</i>) ∈ <i>A</i> × <i>B</i> whose distance dist(<i>x</i>, <i>y</i>) is in <i>I</i> is less than some given threshold <i>L</i>. The running time of this algorithm decreases as <i>L</i> increases. In case there are more than <i>L</i> pairs of points whose distance is in <i>I</i>, we can get a small sample of pairs that contain a pair at approximate median distance (i.e., we can approximately “bisect” <i>I</i>). We combine this procedure with additional ideas to search, with a small overhead, for the optimal one-sided Fréchet distance with shortcuts, using a very fast decision procedure. We also show how to apply this technique for approximating distance selection (with respect to rank), and a somewhat more involved variant of this technique is used in the solution of the semicontinuous Fréchet distance with one-sided shortcuts. In general, the new technique can be applied to optimization problems for which the decision procedure is very fast but standard techniques like parametric search makes the optimization algorithm substantially slower.
使用可伸缩皮带计算 Fréchet 距离
影响因子：
0.8
作者：
Kevin Buchin;Maike Buchin;Rolf van Leusden;Wouter Meulemans;Wolfgang Mulzer
通讯作者：
Wolfgang Mulzer