Tight Bounds for Dynamic Convex Hull Queries Again

1
Tight Bounds for Dynamic Convex Hull
Queries(Again)

• Erik Demaine Mihai Patrascu

2
Dynamic Convex Hull

• Set S, Sn points in 2d
• insert point
• delete point

update time tu

• linear programming
• tangents

query time tq
3
History
p
p
p
So what are you going to improve?
4
O(lg n) Optimal?
bounded precision say, w bits

• NO! radix sort, hashing, closest pair in
  O(n)
• Sorting O(nvlglg n) n2O(vlglg n)
• Voronoi, segment intersection etc.
• Searching O(min lgwn, lg w) O(min lg n/lglg
  n, vw/lg w)

1d
2d
Patrascu FOCS06 Chan FOCS06 Chan, P. STOC07
predecessor search
point location
5
Motivation Information
O(lg n)

• binary search
• in each step, reduce entropy by 1 bit gt O(lg n)
• fusion trees a sketch of w bits allows
  search among vw values
• gt each step reduces entropy by ½lg w gt
  O(lgwn)
• different information concepts
• H(s1,s2)lg l lg r
• can sketch k segments, if all
  H(si,si1)H(s1,sk)/k

1d
2d
s1
r
l
s2
6
Dynamic Convex Hull
Static

• linear programminggt predecessor search e.g.
  O(lg w)lt Chazelle
• tangentsgt planar point location e.g. O(vw)

1
6
4
5
6
2
5
1
3
2
3
4
7
History
Updating
8
Review of Overmars, van Leeuwen

• split with vertical line
• compute 2 hulls recursively gt O(lg n) levels
• find bridges -- O(lg n)
• cutmerge hull trees -- O(lg n)
• gt tuO(lg2n)
• examine bridges
• recurse left or right
• gt tqO(lg n)

9
Proof sketch

• split into lg n subhulls gt depth O(lg n/lglg n)
• query
• remember can sketch k segments, if all
  H(si,si1)w/k gt superconstant time/level if
  some H is small
• information efficiency H only decreases through
  recursion
• info efficiency gt cannot be slow too many
  times H acts as potential, bounding running time

• locate among 2lg n bridges
• recurse

10
Summary Our Contribution

• dynamic geometry with bounded precision
• lots of geometry gt 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. ?(lgwn)
OPEN Improve updates. Can tu ltlt lg n ??
11
T
H
E
E
N
D