A Tail Estimate with Exponential Decay for the Randomized Incremental Construction of Search Structures

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.