R edinerem t qz oidaldecomposit.io

Oversimplified last time

Input n segments page 1792 pn.gs disjoint fornee

Start with trivial decomposition one trap over 1122

wehaven'tfor i 1 to n inrandomorder

described find trapezoid containing pithis yet Add vertical wall through pi

ns.T.i.ae I sina.aeas

n n

his part takes time to

Of new trapezoids o

iii iii ii f.lt ooFH'llO_Ocrossedbypiqi 1011

f fOf new traps with u

Pi9i on floororceiling 1011d T d

Buk is I 41

1,11ithatdemgosfpf.it aphsas I 09

I 1exactly 3itt o

LEE.tn iEtzaoo Pif fEEG.ffrap segment pairs 1 I fSo if we choose a segment5 at random

Not countingpoint locationEf trapswithfloorkeilings e8 Eftime for its insertion OCI

6 C time for all insertions _OG

Howtolocateepi Quioksa

Method 1 conflict listsHtt

Every trap maintains a list of all left endpoints it contains

o Whenever we splItoutof

smallertrapezoid TEPEED t into

t we split KCE KLEA and KL teby brute forcein OC Idt time

Whenever we merge two traps te and te

EFFI i

Total time 011 x endpoints whosetrapezoid changesConsider the ith insertion conflict changes

For eachpointpj with j i

PrftrapIpj changes Prftraplpj is new

Prfoneofthe segmentsdefiningtrapLpj 4was just inserted E

i

So E conflict changes 41i

so Eftoutedtime spent maintaining conflictsOCD OCD EEE Ocn logn

Anth harmonic number Hn

In Intl Hn E Inn 1for allu O

Method Historygraph TreqAs we build the decomposition we also build a data structureto locate arbitrary points in it

Exanp 1pt xWE fight

a

compare x coordinates

a e

oriutation testHI I1177ge

DJ leaf trapezoidabove

outlivesStart at the root andwalk down guided byX comparisons and orientation tests

Insertingasegment pigiUse historydrag to locate piwalk through decomposition as usual

ftplt be E

t

Daye beyonD D

IFai Yes

Talreadyexists

qq.ggits

o

in olddecomp DdEach trapezoid t that intersects pigi

add E3 nodes below t in history day

SpaceTEE nodes added in ith insertion

Ti decampC 3 If traps in Te intersectingpig after an

Olt Ef traps in Te touching piginsertion

0113So total expected nodes in historydag is 04

Timeirpointhierx comps and orientation tests to locate pLet tilp denote trapezoid int containing p

tests p E 3 times tilp changes

Prftilp ti 11psPr pig is one of E4 segs defining tip4I ceiling

so Ef testsCPD E3 If Itf friesE 12 Hn login floor

Overdlconstructiontimei Eftime Calogu

Intesectingsegmats intersecting pairsfChiming time Oculegntk