A Simple Dynamization of Trapezoidal Point Location in Planar Subdivisions

Brankovic, Milutin; Grujic, Nikola; Renssen, André van; Seybold, Martin P.

doi:10.4230/lipics.icalp.2020.18

Cited by 2 publications

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“…Early algorithms generalize common search tree rotations to abstract, complex structures in order to reuse the point location search and leaf insertion algorithms [17]. Simpler search and recursive top-down algorithms were described recently [6]. e bounds on expected insertion and deletion time of both methods however require that the update request entails a random object.…”

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

Gudmundsson¹,

Seybold²

2021

Preprint

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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 O(n) expected size and O(n ln n) expected construction time.Our main result is an improved tail bound, with exponential decay, for the size of the TSD: ere 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 . is yields improved bounds on the TSD construction and their maintenance. I.e. TSD construction takes with high probability O(n ln n) time and TSD size can be made worst case O(n) with an expected rebuild cost of O( 1).e proposed analysis technique also shows that the expected depth is O(ln n), which partially solves a recent conjecture by Hemmer et al. that is used in the CGAL implementation of the TSD.

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confidence: 99%