Download Simple curves on surfaces

Algorithmic Problems for Curves
on Surfaces
Daniel Štefankovič
University of Rochester
outline
● Simple curves on surfaces
* representing surfaces, simple curves in surfaces
* algorithmic questions, history
● TOOL: (Quadratic) word equations
● Regular structures in drawings (?)
● Using word equations (Dehn twist, geometric intersection
numbers, ...)
● What I would like...
How to represent surfaces?
Combinatorial description of a surface
1. (pseudo) triangulation
b
a
bunch of triangles
+ description of how to glue them
c
Combinatorial description of a surface
2. pair-of-pants decomposition
bunch of pair-of-pants
+ description of how to glue them
(cannnot be used to represent: ball with 2 holes, torus)
Combinatorial description of a surface
3. polygonal schema
b
a =
a
b
2n-gon + pairing of the edges
Simple curves on surfaces
closed curve = homeomorphic image of circle S1
simple closed curve =  is injective (no self-intersections)
(free) homotopy equivalent
simple closed curves
How to represent simple curves
in surfaces (up to homotopy)?
(properly embedded arc)
Ideally the representation is “unique”
(each curve has a unique representation)
Combinatorial description of a (homotopy type of)
a simple curve in a surface
1. intersection sequence with
a triangulation
b
a
c
Combinatorial description of a (homotopy type of)
a simple curve in a surface
1. intersection sequence with
a triangulation
b
a
c
bc-1bc-1ba-1
almost unique if triangulation points on S
Combinatorial description of a (homotopy type of)
a simple curve in a surface
2. normal coordinates (w.r.t. a
triangulation)
(b)=3
(c)=2
(Kneser ’29)
unique if triangulation points on S
Combinatorial description of a (homotopy type of)
a simple curve in a surface
2. normal coordinates (w.r.t. a
triangulation)
(b)=300
(c)=200
a very concise representation!
(compressed)
Combinatorial description of a (homotopy type of)
a simple curve in a surface
3. weighted train track
5
10
13
5
10
3
Combinatorial description of a (homotopy type of)
a simple curve in a surface
4. Dehn-Thurston coordinates
● number of intersections
● “twisting number”
for each “circle”
(important for surfaces without boundary)
unique
outline
● Simple curves on surfaces
* representing surfaces, simple curves in surfaces
* algorithmic questions, history
● TOOL: (Quadratic) word equations
● Regular structures in drawings (?)
● Using word equations (Dehn twist, geometric intersection
numbers, ...)
● What I would like...
Algorithmic problems - History
Contractibility (Dehn 1912)
can shrink curve to point?
Transformability (Dehn 1912)
are two curves homotopy equivalent?
Schipper ’92; Dey ’94; Schipper, Dey ’95
Dey-Guha ’99 (linear-time algorithm)
Simple representative (Poincaré 1895)
can avoid self-intersections?
Reinhart ’62; Ziechang ’65; Chillingworth ’69
Birman, Series ’84
Algorithmic problems - History
Geometric intersection number
minimal number of intersections of two curves
Reinhart ’62; Cohen,Lustig ’87; Lustig ’87;
Hamidi-Tehrani ’97
polynomial only in explicit representations
Computing Dehn-twists
“wrap” curve along curve
Penner ’84; Hamidi-Tehrani, Chen ’96;
Hamidi-Tehrani ’01
polynomial in compressed representations, but
only for fixed set of curves
Algorithmic problems – will show
Geometric intersection number
minimal number of intersections of two curves
Reinhart ’62; Cohen,Lustig ’87; Lustig ’87;
Hamidi-Tehrani ’97, Schaefer-Sedgewick-Š ’08
polynomial in explicit compressed representations
Computing Dehn-twists
“wrap” curve along curve
Penner ’84; Hamidi-Tehrani, Chen ’96;
Hamidi-Tehrani ’01, Schaefer-Sedgewick-Š ’08
polynomial in compressed representations, for
fixed set of curves any pair of curves
outline
● Simple curves on surfaces
* representing surfaces, simple curves in surfaces
* algorithmic questions, history
● TOOL: (Quadratic) word equations
● Regular structures in drawings (?)
● Using word equations (Dehn twist, geometric intersection
numbers, ...)
● What I would like...
Word equations
xabx =yxy
x,y – variables
a,b - constants
Word equations
x,y – variables
a,b - constants
xabx =yxy
a solution:
x=ab
y=ab
Word equations with given lengths
xayxb = axbxy
additional constraints:
|x|=4, |y|=1
x,y – variables
a,b - constants
Word equations with given lengths
xayxb = axbxy
additional constraints:
|x|=4, |y|=1
a solution:
x=aaaa
y=b
x,y – variables
a,b - constants
Word equations
word equations
word equations with given lengths
Word equations
In NP ???
word equations - NP-hard
decidability – Makanin 1977
PSPACE – Plandowski 1999
word equations with given lengths
Plandowski, Rytter ’98 – polynomial time algorithm
Diekert, Robson ’98 – linear time for quadratic eqns
(quadratic = each variable occurs  2 times)
Word equations
OPEN:
In NP ???
word equations - NP-hard
MISSING:
decidability
– Makanin
1977 on
exponential
upper bound
PSPACE
– Plandowski
1999
the length
of a minimal
solution
word equations with given lengths
Plandowski, Rytter ’98 – polynomial time algorithm
Diekert, Robson ’98 – linear time for quadratic eqns
(quadratic = each variable occurs  2 times)
outline
● Simple curves on surfaces
* representing surfaces, simple curves in surfaces
* algorithmic questions, history
● TOOL: (Quadratic) word equations
● Regular structures in drawings (?)
● Using word equations (Dehn twist, geometric intersection
numbers, ...)
● What I would like...
Shortcut number (g,k)
k curves on surface of genus g
intersecting another curve 

(the
curves do not intersect)
Shortcut number (g,k)
k curves on surface of genus g
intersecting another curve 
4
1

3
1
1
6
8
4
Shortcut number (g,k)
k curves on surface of genus g
intersecting another curve 
4
1

3

1
1
6
8
4
Shortcut number (g,k)
k curves on surface of genus g
intersecting another curve 
smallest n such that n intersections
 reduced drawing
(g,1) = 2
Shortcut number
3
2
4
1
3
2
4
1
(1,2) > 6
Shortcut number
3
5
5
2
2
4
4
6
6
1
1
3
Shortcut number
Conjecture:
(g,k)  Ck
Experimentally:
(,2) = 7
(,3) = 31 (?)
Known [Schaefer, Š ‘2000]:
(0,k)  2k
(1,2) > 6
Directed shortcut number d(g,k)
k curves on surface of genus g
intersecting another curve 
4
1

3
1
1
6
8
4
BAD

Directed shortcut number d(g,k)
upper bound must depend on g,k
Experimentally: d(0,2) = 20
finite?
Directed shortcut number d(g,k)
finite?
interesting?
quadratic word equation  drawing problem
bound on d(,)  upper bound on word eq.
x=yz
z=wB
x=Aw
y=AB
A
x
y
z
A
B
w
B
Spirals

spiral of depth 1
(spanning arcs, 3 intersections)

interesting for word equations


Unfortunately: Example with no spirals
[Schaefer, Sedgwick, Š ’07]
Spirals and folds
spiral of depth 1

(spanning arcs, 3 intersections)

fold of width 3
Pach-Tóth’01: In the plane (with puncures) either
a large spiral or a large fold must exist.
Unfortunately: Example with no spirals, no folds
[Schaefer, Sedgwick, Š ’07]
Embedding on torus
outline
● Simple curves on surfaces
* representing surfaces, simple curves in surfaces
* algorithmic questions, history
● TOOL: (Quadratic) word equations
● Regular structures in drawings (?)
● Using word equations (Dehn twist, geometric intersection
numbers, ...)
● What I would like...
Geometric intersection number

minimum number of intersections
achievable by continuous
deformations.

Geometric intersection number

minimum number of intersections
achievable by continuous
deformations.

i(,)=2
EXAMPLE: Geometric intersection numbers
are well understood on the torus
(3,5)
3
5
det
= -13
2 -1
(2,-1)
Recap:
1) how to represent them?
1. intersection sequence with a triangulation
bc-1bc-1ba-1
2. normal coordinates (w.r.t. a triangulation)
(a)=1 (b)=3 (c)=2
2) what/how to compute?
geometric intersection number
STEP1: Moving between the representations
1. intersection sequence with a triangulation
bc-1bc-1ba-1
2. normal coordinates (w.r.t. a triangulation)
(a)=1 (b)=3 (c)=2
Can we move between these two
representations efficiently?
(a)=1+2100
(b)=1+3.2100
(c)=2101
Theorem (SSS’08):
normal coordinatescompressed intersection sequence
in time O( log (e))
compressed intersection sequencenormal coordinates
in time O(|T|.SLP-length(S))
compressed = straight line program (SLP)
X0 := a
X1 := b
X2 := X1X1
X3 := X0X2
X4 := X2X1
X5 := X4X3
X5 = bbbabb
compressed = straight line program (SLP)
X0 := a
X1 := b
X2 := X1X1
X3 := X0X2
X4 := X2X1
X5 := X4X3
X5 = bbbabb
OUTPUT OF:
Plandowski, Rytter ’98 – polynomial time algorithm
Diekert, Robson ’98 – linear time for quadratic eqns
CAN DO (in poly-time):
● count the number of occurrences of a symbol
● check equaltity of strings given by two SLP’s
(Miyazaki, Shinohara, Takeda’02 – O(n4))
● get SLP for f(w) where f is a substitution *
and w is given by SLP
Simulating curve using quadratic word equations
X
z
z
u
v
y
w
|u|=|v|=(u) |x|=(|z|+|u|-|w|)/2
...
number of
u=xy
Diekert-Robson components
...
v=u
Moving between the representations
1. intersection sequence with a triangulation
bc-1bc-1ba-1
2. normal coordinates (w.r.t. a triangulation)
(a)=1 (b)=3 (c)=2
Theorem:
normal coordinatescompressed intersection sequence
in time O( log (e))
“Proof”:
X
z
u
y
v
u=xy
...
av=ua
|u|=|v|=|T| (u)
Dehn twist of  along 


Dehn twist of  along 

D()
Dehn twist of  along 


D()
Geometric intersection numbers
i(,Dn())/i(,) ! i(,)
n¢ i(,)i(,) -i(,)
 i(,Dn()) 
n¢ i(,)i(,)+i(,)
Computing Dehn-Twists (outline)
1. normal coordinates ! word equations
with given lengths
2. solution = compressed intersection
sequence with triangulation
3. sequences ! (non-reduced) word for
Dehn-twist (substitution in SLPs)
4. Reduce the word ! normal coordinates
(only for surfaces with S  0)
outline
● Simple curves on surfaces
* representing surfaces, simple curves in surfaces
* algorithmic questions, history
● TOOL: (Quadratic) word equations
● Regular structures in drawings (?)
● Using word equations (Dehn twist, geometric intersection
numbers, ...)
● What I would like...
PROBLEM #1: Minimal weight representative
2. normal coordinates (w.r.t. a
triangulation)
(b)=3
(c)=2
unique if triangulation points on S
PROBLEM #1: Minimal weight representative
INPUT: triangulation + gluing
normal coordinates of 
edge weights
OUTPUT: ’
minimizing  ’(e)
eT
PROBLEM #2: Moving between representations
4. Dehn-Thurston coordinates
(Dehn ’38, W.Thurston ’76)
unique representation for closed surfaces!
PROBLEM
normal coordinatesDehn-Thurston coordinates
in polynomial time? linear time?
PROBLEM #3: Word equations
NP-hard
decidability – Makanin 1977
PSPACE – Plandowski 1999
PROBLEM:
are word equations in NP?
are quadratic word equations in NP?
PROBLEM #4: Computing Dehn-Twists faster?
1. normal coordinates ! word equations
with given lengths
2. solution = compressed intersection
sequence with triangulation
3. sequences ! (non-reduced) word for
Dehn-twist (substitution in SLPs)
4. Reduce the word ! normal coordinates
O(n3) randomized, O(n9) deterministic
PROBLEM #5: Realizing geometric intersection #?
our algorithm is very indirect
can compress drawing realizing geometric intersection #?
can find the drawing?