Algorithmic Problems for Curves on Surfaces

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


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)


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



a

b

c

bc-1bc-1ba-1

almost unique if triangulation points on S


2. normal coordinates (w.r.t. a triangulation)

a)=1

b)=3

c)=2

(Kneser ’29) unique if triangulation points on S



a)=100

b)=300

c)=200

a very concise representation!(compressed)


3. weighted train track

5

10

3

1310

5


4. Dehn-Thurston coordinates

● number of intersections ● “twisting number”for each “circle”

unique

(important for surfaces without boundary)








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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








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


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




(quadratic = each variable occurs 2 times)

In NP ???

exponential upper bound on the length of a minimal solution

MISSING:

OPEN:








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Shortcut number (g,k) k curves on surface of genus gintersecting another curve

(the curves do not intersect)


141

41

3 86


141

41

3 86


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








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?



bc-1bc-1ba-1

a)=1 b)=3 c)=2

geometric intersection number

STEP1: Moving between the representations



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


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)








outline

PROBLEM #1: Minimal weight representative


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


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?

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Algorithmic Problems for Curves on Surfaces