Algorithmic Problems for Curves

1
Algorithmic Problems for Curves on Surfaces
Daniel tefankovic University of Rochester
2
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...
3
How to represent surfaces?
4
Combinatorial description of a surface
1. (pseudo) triangulation
b
a
c
bunch of triangles description of how to glue
them
5
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)
6
Combinatorial description of a surface
3. polygonal schema
b

a
a
b
2n-gon pairing of the edges
7
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
8
How to represent simple curves in surfaces (up to
homotopy)?
(properly embedded arc)
Ideally the representation is unique (each
curve has a unique representation)
9
Combinatorial description of a (homotopy type of)
a simple curve in a surface

• intersection sequence with
• a triangulation

b
c
a
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Combinatorial description of a (homotopy type of)
a simple curve in a surface

• intersection sequence with
• a triangulation

b
c
a
bc-1bc-1ba-1
almost unique if triangulation points on ?S
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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
?(a)1
(Kneser 29)
unique if triangulation points on ?S
12
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)100
a very concise representation!
(compressed)
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Combinatorial description of a (homotopy type of)
a simple curve in a surface
3. weighted train track
5
10
13
5
10
3
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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
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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...
16
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
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Algorithmic problems - History
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,
but only for fixed set of curves
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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- 08 Computing
Dehn-twists wrap curve along curve Penner
84 Hamidi-Tehrani, Chen 96 Hamidi-Tehrani
01, Schaefer-Sedgewick- 08
polynomial in explicit compressed representations
polynomial in compressed representations, for
fixed set of curves any pair of curves
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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...
20
Word equations
x,y variables a,b - constants
xabx yxy
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Word equations
x,y variables a,b - constants
xabx yxy
a solution xab yab
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Word equations with given lengths
x,y variables a,b - constants
xayxb axbxy
additional constraints x4, y1
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Word equations with given lengths
x,y variables a,b - constants
xayxb axbxy
additional constraints x4, y1
a solution xaaaa yb
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Word equations
word equations
word equations with given lengths
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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)
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Word equations
OPEN
In NP ???
word equations - NP-hard
MISSING
decidability Makanin 1977 PSPACE Plandowski
1999
exponential upper bound on 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)
27
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...
28
Shortcut number ?(g,k)
k curves on surface of genus g intersecting
another curve ?
?
(the curves do not intersect)
29
Shortcut number ?(g,k)
k curves on surface of genus g intersecting
another curve ?
4
1
?
4
8
1
3
6
1
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Shortcut number ?(g,k)
k curves on surface of genus g intersecting
another curve ?
4
1
?
4
8
1
3
6
1
?
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Shortcut number ?(g,k)
k curves on surface of genus g intersecting
another curve ?
smallest n such that ?n intersections
? reduced drawing
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Shortcut number ?(g,1) 2
3
2
4
1
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Shortcut number ?(1,2) gt 6
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Shortcut number ?(1,2) gt 6
Conjecture
?(g,k) ? Ck
Experimentally ?(?,2) 7
?(?,3) 31 (?)
Known Schaefer, 2000
?(0,k) ? 2k
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Directed shortcut number ?d(g,k)
k curves on surface of genus g intersecting
another curve ?
4
1
?
4
8
1
3
6
1
BAD
?
36
Directed shortcut number ?d(g,k)
upper bound must depend on g,k
?d(0,2) 20
Experimentally
finite?
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Directed shortcut number ?d(g,k)
finite?
interesting?
quadratic word equation ? drawing problem bound
on ?d(?,?) ? upper bound on word eq.
xyz zwB xAw yAB
A
B
y
A
x
w
z
B
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Spirals
spiral of depth 1
?
(spanning arcs, 3 intersections)
?
interesting for word equations
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Unfortunately Example with no spirals
Schaefer, Sedgwick, 07
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Spirals and folds
spiral of depth 1
?
(spanning arcs, 3 intersections)
?
fold of width 3
Pach-Tóth01 In the plane (with puncures) either
a large spiral or a large fold must exist.
41
Unfortunately Example with no spirals, no folds
Schaefer, Sedgwick, 07
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Embedding on torus
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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...
44
Geometric intersection number
minimum number of intersections
achievable by continuous
deformations.
?
?
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Geometric intersection number
minimum number of intersections
achievable by continuous
deformations.
?
?
i(?,?)2
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EXAMPLE Geometric intersection numbers are well
understood on the torus
(2,-1)
(3,5)
3 5
det
-13
2 -1
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Recap

• how to represent them?
• 2) what/how to compute?

• intersection sequence with a triangulation

bc-1bc-1ba-1
2. normal coordinates (w.r.t. a triangulation)
?(a)1
?(b)3
?(c)2
geometric intersection number
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STEP1 Moving between the representations

• 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?
?(c)2101
?(a)12100
?(b)13.2100
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Theorem (SSS08)
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
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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 SLPs (Miyazaki,
Shinohara, Takeda02 O(n4)) ? get SLP for
f(w) where f is a substitution ??? and w
is given by SLP
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Simulating curve using quadratic word equations
u
X
z
v
z
y
w
uv?(u) ...
x(zu-w)/2
number of components
uxy ... vu
Diekert-Robson
52
Moving between the representations

• 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
uxy ... avua
u
X
v
z
uv??T? ?(u)
y
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Dehn twist of ? along ?
?
?
54
Dehn twist of ? along ?
?
D?(?)
55
Dehn twist of ? along ?
?
?
D?(?)
56
Geometric intersection numbers
i(a,Dng(b))/i(a,g) ! i(g,b)
n i(a,g)i(g,b) -i(a,b) ? i(a,Dng(b)) ? n
i(a,g)i(g,b)i(a,b)
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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)
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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...
59
PROBLEM 1 Minimal weight representative
2. normal coordinates (w.r.t. a
triangulation)
?(b)3
?(c)2
?(a)1
unique if triangulation points on ?S
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PROBLEM 1 Minimal weight representative
INPUT triangulation gluing normal
coordinates of ? edge
weights OUTPUT ??? minimizing ?
?(e)
e?T
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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?
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PROBLEM 3 Word equations
NP-hard
decidability Makanin 1977 PSPACE Plandowski
1999
PROBLEM are word equations in NP? are
quadratic word equations in NP?
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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
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PROBLEM 5 Realizing geometric intersection ?
our algorithm is very indirect can compress
drawing realizing geometric intersection ?
can find the drawing?