tight bounds for dynamic convex hull queries (again)
DESCRIPTION
Tight Bounds for Dynamic Convex Hull Queries (Again). Erik Demaine Mihai P ătraşcu. Dynamic Convex Hull. Set S , |S|≤n points in 2d: insert point delete point. update time t u. linear programming tangents. query time t q. History. π. π. π. So what are you going to improve?. - PowerPoint PPT PresentationTRANSCRIPT
Tight Bounds for Dynamic Convex Hull Queries
(Again)Erik Demaine Mihai Pătraşcu
• linear programming
• tangents
Dynamic Convex Hull
Set S, |S|≤n points in 2d:• insert point• delete point
update time tu
query time tq
History
tu tq
[Overmars, van Leeuwen] STOC’80 O(lg2n) O(lg n)
[Chan] FOCS’99 O(lg1+n) O(lg n)
[Brodal, Jacob] SWAT’99
O(lg n lglg
n) O(lg n)
[Brodal, Jacob] FOCS’99 O(lg n) O(lg n)
[Demaine, Pătraşcu] SoCG’07 So what are you going to improve?
π
π
π
O(lg n) = Optimal?
NO! radix sort, hashing, closest pair in O(n)…
Sorting: O(n√lglg n) n·2O(√lglg n)
Voronoi, segment intersection etc.
Searching: O(min lgwn, lg w) O(min lg n/lglg
n, √w/lg w)
1d 2dPătraşcu FOCS’06Chan FOCS’06Chan, P. STOC’07
predecessor search point location
bounded precisionsay, w bits
Motivation: Information
binary searchin each step, reduce entropy by 1 bit => O(lg n)
fusion trees: a sketch of w bits allows search among √w values
=> each step reduces entropy by ½lg w => O(lgwn)
different information concepts
H(s1,s2)=lg ℓ + lg r
can sketch k segments, if all H(si,si+1)≥H(s1,sk)/k
1d
2d
ℓ r
O(lg
n)
s1
s2
Dynamic Convex Hull
• linear programming=> predecessor search
e.g. O(lg w)<= [Chazelle]
• tangents=> planar point location
e.g. O(√w)
Static
1
2
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5
6
1
2
4 5
3
6
Historytu tq
[Overmars, van Leeuwen] STOC’80 O(lg2n) O(lg n)
• all queries (tangents) NEW O(lg2n)O(lg n/lglg
n)
[Chan] FOCS’99 O(lg1+n) O(lg n)
[Brodal, Jacob] SWAT’99
O(lg n lglg
n) O(lg n)
• some queries (LP) NEW
O(lg n lglg
n) O(lgwn)
• all queries NEW lgO(1)n Ω(lgwn)
[Brodal, Jacob] FOCS’99 O(lg n) O(lg n)
Updating
Review of [Overmars, van Leeuwen]
• split with vertical line• compute 2 hulls recursively => O(lg n) levels• find bridges -- O(lg n) • cut+merge hull trees -- O(lg n)
=> tu=O(lg2n)
• examine bridges• recurse left or right
=> tq=O(lg n)
Proof sketch
• split into lg n subhulls => depth O(lg n/lglg n)
• query:
• remember: “can sketch k segments, if all H(si,si+1)≥w/k”=> superconstant time/level if some H is small
• information efficiency:H only decreases through recursion
• info efficiency => cannot be slow too many timesH acts as potential, bounding running time
• locate among 2lg n bridges• recurse
Summary: Our Contribution
• “dynamic geometry with bounded precision”• lots of geometry =>
[Overmars, van Leeuwen] is informationally efficient
• lower bound
• 1d-like structure for LP
OPEN: [Chan], [Brodal-Jacob] not info efficient…
OPEN: O(lg n/lglg n) vs. Ω(lgwn)
OPEN: Improve updates. Can tu << lg n ??
TH
E
EN
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