We revisit the randomized incremental construction of the Trapezoidal Search DAG (TSD) for a set of{\cal   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\emph{ [  Hemmer et al. that is used in the CGAL implementation of the TSD [ .
The Randomized Incremental Construction (RIC) of search DAGs for point location in planar subdivisions, nearest-neighbor search in 2D points, and extreme point search in 3D convex hulls, are well known to take{\cal O ( n \log n) expected time for structures of {\cal O}(n) expected size. Moreover, searching takes w.h.p. {\cal O} (\log n) comparisons in the first and w.h.p. {\cal O}( \log^2 n) comparisons in the latter two DAGs. However, the expected depth of the DAGs and high probability bounds for their size are unknown. Using a novel analysis technique, we show that the three DAGs have w.h.p. i) a size of {\cal O}(n) , ii) a depth of {\cal O}( \log n), and iii) a construction time of {\cal O}( n \log n). One application of these new and improved results are\emph{remarkably simple Las Vegas verifiers to obtain search DAGs with optimal worst-case bounds. This positively answers the conjectured logarithmic search cost in the DAG of Delaunay triangulations[Guibas et al.; ICALP 1990 and a conjecture on the depth of the DAG of Trapezoidal subdivisions Hemmer et al. ; ESA 2012 . It also shows that history-based RIC circumvents a lower bound on runtime tail estimates of conflict-graph RICs[Sen; STACS 2019 .
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We revisit the randomized incremental construction of the Trapezoidal Search DAG (TSD) for a set of
The Randomized Incremental Construction (RIC) of search DAGs for point location in planar subdivisions, nearest-neighbor search in 2D points, and extreme point search in 3D convex hulls, are well known to take
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O
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(
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non-crossing segments, e.g. edges from planar subdivisions. It is well known that this point location structure has
\log n) expected time for structures of
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size and
size. Moreover, searching takes w.h.p.
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(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
(\log n) comparisons in the first and w.h.p.
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n \ln n) time and TSD size can be made worst case
\log^2 n) comparisons in the latter two DAGs. However, the expected depth of the DAGs and high probability bounds for their size are unknown. Using a novel analysis technique, we show that the three DAGs have w.h.p. i) a size of
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with an expected rebuild cost
, ii) a depth
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1). The proposed analysis technique also shows that the expected depth is
\log n), and iii) a construction time of
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\ln n ), which partially solves a recent conjecture by
n \log n). One application of these new and improved results are
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remarkably simple
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Las Vegas verifiers to obtain search DAGs with optimal worst-case bounds. This positively answers the conjectured logarithmic search cost in the DAG of Delaunay triangulations
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Guibas et al.; ICALP 1990
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and a conjecture on the depth of the DAG of Trapezoidal subdivisions
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that is used in the CGAL implementation of the TSD
; ESA 2012
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. It also shows that history-based RIC circumvents a lower bound on runtime tail estimates of conflict-graph RICs
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Sen; STACS 2019