algorithmic problems for curves on surfaces daniel Štefankovič university of rochester
TRANSCRIPT
Algorithmic Problems for Curveson Surfaces
Daniel ŠtefankovičUniversity of Rochester
● 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...
outline
How to represent surfaces?
Combinatorial description of a surface
1. (pseudo) triangulation
bunch of triangles + description of how to glue them
a
b
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
2n-gon + pairing of the edges
=a a
b
b
Simple curves on surfacesclosed curve homeomorphic image of circle S1
simple closed curve = is injective (no self-intersections)
(free) homotopy equivalentsimple closed curves
How to represent simple curvesin surfaces (up to homotopy)?
Ideally the representation is “unique” (each curve has a unique representation)
(properly embedded arc)
Combinatorial description of a (homotopy type of) a simple curve in a surface
1. intersection sequence with a triangulation
a
b
c
Combinatorial description of a (homotopy type of) a simple curve in a surface
1. intersection sequence with a triangulation
a
b
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)
a)=1
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)
a)=100
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
3
1310
5
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”
unique
(important for surfaces without boundary)
● 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...
outline
Algorithmic problems - HistoryContractibility (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
Geometric intersection number minimal number of intersections of two curves
Reinhart ’62; Cohen,Lustig ’87; Lustig ’87; Hamidi-Tehrani ’97
Computing Dehn-twists “wrap” curve along curve
Penner ’84; Hamidi-Tehrani, Chen ’96; Hamidi-Tehrani ’01
polynomial only in explicit representations
polynomial in compressed representations, butonly for fixed set of curves
Algorithmic problems - History
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
Computing Dehn-twists “wrap” curve along curve
Penner ’84; Hamidi-Tehrani, Chen ’96; Hamidi-Tehrani ’01, Schaefer-Sedgewick-Š ’08
polynomial in explicit compressed representations
polynomial in compressed representations, for fixed set of curves any pair of curves
● 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...
outline
Word equations
xabx =yxy x,y – variablesa,b - constants
xabx =yxy x,y – variablesa,b - constants
a solution:
x=ab y=ab
Word equations
Word equations with given lengths
x,y – variablesa,b - constantsxayxb = axbxy
additional constraints: |x|=4, |y|=1
Word equations with given lengths
x,y – variablesa,b - constantsxayxb = axbxy
additional constraints: |x|=4, |y|=1
a solution:
x=aaaa y=b
Word equations
word equations
word equations with given lengths
Word equations
word equations - NP-hard
word equations with given lengths
Plandowski, Rytter ’98 – polynomial time algorithmDiekert, Robson ’98 – linear time for quadratic eqns
decidability – Makanin 1977PSPACE – Plandowski 1999
(quadratic = each variable occurs 2 times)
In NP ???
Word equations
word equations - NP-hard
word equations with given lengths
Plandowski, Rytter ’98 – polynomial time algorithmDiekert, Robson ’98 – linear time for quadratic eqns
decidability – Makanin 1977PSPACE – Plandowski 1999
(quadratic = each variable occurs 2 times)
In NP ???
exponential upper bound on the length of a minimal solution
MISSING:
OPEN:
● 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...
outline
Shortcut number (g,k) k curves on surface of genus gintersecting another curve
(the curves do not intersect)
Shortcut number (g,k) k curves on surface of genus gintersecting another curve
141
41
3 86
Shortcut number (g,k) k curves on surface of genus gintersecting another curve
141
41
3 86
Shortcut number (g,k) k curves on surface of genus gintersecting another curve
smallest n such that n intersectionsreduced drawing
Shortcut number (g,1) 2
4 1
3 2
4 1
3 2
Shortcut number (1,2) 6
4 6 6 1 1 3
3 5 5 2 2 4
Shortcut number (1,2) 6
Conjecture: g,k) Ck
Experimentally:,2) 7,3) 31 (?)
Known [Schaefer, Š ‘2000]: (0,k) 2k
Directed shortcut number d(g,k) k curves on surface of genus gintersecting another curve
141
41
3 86
BAD
Directed shortcut number d(g,k)
d(0,2) = 20
upper bound must depend on g,k
finite?
Experimentally:
Directed shortcut number d(g,k)
finite?
quadratic word equation drawing problembound on d(,) upper bound on word eq.
x=yzz=wBx=Awy=AB
x y
zw
A
B
AB
interesting?
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
● 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...
outline
Geometric intersection numberminimum number of intersections achievable by continuous deformations.
Geometric intersection numberminimum number of intersections achievable by continuous deformations.
i(,)=2
EXAMPLE: Geometric intersection numbersare well understood on the torus
(3,5) (2,-1)
3 5 2 -1
det = -13
Recap:
1) how to represent them?
2) what/how to compute?
1. intersection sequence with a triangulation
2. normal coordinates (w.r.t. a triangulation)
bc-1bc-1ba-1
a)=1 b)=3 c)=2
geometric intersection number
STEP1: Moving between the representations
1. intersection sequence with a triangulation
2. normal coordinates (w.r.t. a triangulation)
bc-1bc-1ba-1
a)=1 b)=3 c)=2
Can we move between these two representations efficiently?
a)=1+2100 b)=1+3.2100 c)=2101
compressed = straight line program (SLP)
X0 a X1 b X2 X1X1
X3 X0X2
X4 X2X1
X5 X4X3
Theorem (SSS’08): normal coordinatescompressed intersection sequence in time O( log (e))
compressed intersection sequencenormal coordinates in time O(|T|.SLP-length(S))
X5 = bbbabb
compressed = straight line program (SLP)
X0 a X1 b X2 X1X1
X3 X0X2
X4 X2X1
X5 X4X3
X5 = bbbabb
Plandowski, Rytter ’98 – polynomial time algorithmDiekert, Robson ’98 – linear time for quadratic eqns
OUTPUT OF:
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
yz
u v
u=xy...v=u
|u|=|v|=(u)...
Diekert-Robsonnumber ofcomponents
w
z
|x|=(|z|+|u|-|w|)/2
Moving between the representations1. intersection sequence with a triangulation
2. normal coordinates (w.r.t. a triangulation)bc-1bc-1ba-1
a)=1 b)=3 c)=2
Theorem: normal coordinatescompressed intersection sequence in time O( log (e))
“Proof”:
X
yz
u 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
n¢ i(,)i(,) -i(,) i(,Dn
()) n¢ i(,)i(,)+i(,)
i(,Dn())/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)
● 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...
outline
PROBLEM #1: Minimal weight representative
2. normal coordinates (w.r.t. a triangulation)
a)=1
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)
eT
PROBLEM #2: Moving between representations
4. Dehn-Thurston coordinates(Dehn ’38, W.Thurston ’76)
unique representation for closed surfaces!
PROBLEM normal coordinatesDehn-Thurston coordinates
in polynomial time? linear time?
PROBLEM #3: Word equations
PROBLEM: are word equations in NP? are quadratic word equations in NP?
NP-hard
decidability – Makanin 1977PSPACE – Plandowski 1999
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?