We revisit the randomized incremental construction of the Trapezoidal Search DAG (TSD) for a set of $n$ non-crossing segments, e.g. edges from planar subdivisions. It is well known that this point location structure has ${\cal O}(n)$ expected size and ${\cal O}(n \ln n)$ expected construction time. Our main result is an improved tail bound, with exponential decay, for the size of the TSD: There is a constant such that the probability for a TSD to exceed its expected size by more than this factor is at most $1/e^n$. This yields improved bounds on the TSD construction and their maintenance. I.e. TSD construction takes with high probability ${\cal O}(n \ln n)$ time and TSD size can be made worst case ${\cal O}(n)$ with an expected rebuild cost of ${\cal O}(1)$. The proposed analysis technique also shows that the expected depth is ${\cal O}(\ln n)$, which partially solves a recent conjecture by Hemmer et al. that is used in the CGAL implementation of the TSD.

中文翻译：

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搜索结构随机增量构造的指数衰减尾估计

我们重新审视了梯形搜索DAG（TSD）的随机增量构造，该构造用于一组$ n $个非交叉线段，例如，平面细分的边缘。众所周知，该点位置结构具有预期的大小{{cal O}（n）$和预期的构建时间$ {\ cal O}（n \ ln n）$。我们的主要结果是针对TSD的大小改进了尾部约束，并且具有指数衰减：存在一个常数，使得TSD超出其预期大小的可能性大于此因子的概率最大为$ 1 / e ^ n $ 。这样可以改善TSD构造及其维护的界限。即，TSD的构建很有可能花费$ {\ cal O}（n \ ln n）$的时间，并且可以使TSD的大小成为最坏的情况$ {\ cal O}（n）$，预期的重建成本为$ {\ cal O }（1）$。所提出的分析技术还表明，预期深度为$ {\ cal O}（\ ln n）$，这部分解决了Hemmer等人最近的猜想。在TSD的CGAL实现中使用的代码。