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

34

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

D