Untangling Graphs and Curves on Surfaces via Local Moves Hsien-Chih - - PowerPoint PPT Presentation

Untangling Graphs and Curves

• n Surfaces via Local Moves

Hsien-Chih Chang

University of Illinois at Urbana-Champaign Dagstuhl seminar, Feb 12–17, 2017

1

How to simplify a doodle?

“Jazz Quintet”, Sir Shadow, 2005.

2

Homotopy moves

10 20 33

◮ How many homotopy moves does it take to simplify a

generic closed curve on a surface?

3

Homotopy moves

10 20 33

◮ How many homotopy moves does it take to simplify a

generic closed curve on a surface?

3

Previous work

◮ Finite

[Hass and Scott 1994, de Graaf and Schrijver 1997, Paterson 2002]

◮ Open question. Are polynomial many homotopy moves

sufficient?

◮ Untangling knot using polynomially many Reidemeister

moves [Lackenby 2015]

◮ Ω(n) moves are required

◮ at most two vertices removed at each move 4

Previous work

◮ Finite

[Hass and Scott 1994, de Graaf and Schrijver 1997, Paterson 2002]

◮ Open question. Are polynomial many homotopy moves

sufficient?

◮ Untangling knot using polynomially many Reidemeister

moves [Lackenby 2015]

◮ Ω(n) moves are required

◮ at most two vertices removed at each move 4

Previous work

◮ Finite

[Hass and Scott 1994, de Graaf and Schrijver 1997, Paterson 2002]

◮ Open question. Are polynomial many homotopy moves

sufficient?

◮ Untangling knot using polynomially many Reidemeister

moves [Lackenby 2015]

◮ Ω(n) moves are required

◮ at most two vertices removed at each move 4

Previous work

◮ Finite

[Hass and Scott 1994, de Graaf and Schrijver 1997, Paterson 2002]

◮ Open question. Are polynomial many homotopy moves

sufficient?

◮ Untangling knot using polynomially many Reidemeister

moves [Lackenby 2015]

◮ Ω(n) moves are required

◮ at most two vertices removed at each move 4

Special curves

◮ Contractible curve: O(n2) moves

[Steinitz 1916, Hass and Scott 1985]

◮ Actually, anything homotopic to simple curve

◮ Ω(n2) moves for non-contractible curves on torus

[C. and Erickson 2016]

5

Special curves

◮ Contractible curve: O(n2) moves

[Steinitz 1916, Hass and Scott 1985]

◮ Actually, anything homotopic to simple curve

◮ Ω(n2) moves for non-contractible curves on torus

[C. and Erickson 2016]

b c a n/8

{

n/8

{

n/8

{

n/8

{

b c a b c a b c a

5

Special surface: Plane and Sphere

◮ O(n2) moves are always enough

◮ regular homotopy (no 1

0 moves) [Francis 1969]

From “Generic and Regular Curves”, Jeff Erickson

6

Loop reductions

1 1 1 1 1 2 1 2 2 3 2 3 1 1 1 1 2 1 2

7

Loop reductions

1 1 1 1 1 2 1 2 2 3 2 3 1 1 1 1 2 1 2

y z ◮ at most O(n) moves to remove a loop ◮ O(n2) homotopy moves in total

8

Loop reductions

1 1 1 1 1 2 1 2 2 3 2 3 1 1 1 1 2 1 2

y z ◮ at most O(n) moves to remove a loop ◮ O(n2) homotopy moves in total

8

Loop reductions

1 1 1 1 1 2 1 2 2 3 2 3 1 1 1 1 2 1 2

y z ◮ at most O(n) moves to remove a loop ◮ O(n2) homotopy moves in total

8

Special surface: Plane and Sphere

◮ O(n2) moves are always enough

◮ regular homotopy (no 1

0 moves) [Francis 1969]

◮ O(n2) moves also follows from electrical transformations

[Steinitz 1916, Truemper 1989, Feo and Provan 1993]

9

Electrical transformations

degree-1 series-parallel ∆Y transformation

10

Resistor network

[Kennelly 1899]

From “Circuit Diagram”, xkcd 730 by Randall Munroe

11

Steinitz’s theorem

[Steinitz 1916, Steinitz and Rademacher 1934]

From “What the Bees Know and What They do not Know”, T´

• th, 1964

12

Steinitz’s theorem

[Steinitz 1916, Steinitz and Rademacher 1934]

From page “Steinitz’s theorem” in Wikipedia, David Eppstein

13

Many more examples

◮ Shortest paths and maximum flows [Akers, Jr. 1960] ◮ Estimating network reliability [Lehman 1963] ◮ Multicommodity flows [Feo 1985] ◮ Preserving cross metric on surfaces [Schrijver 1992] ◮ Construct link invariants [Goldman and Kauffman 1993] ◮ Counting spanning trees, perfect matchings, and cuts

[Colbourn et al. 1995]

◮ Evaluation of spin models in statistical mechanics [Jaeger 1995] ◮ Solving generalized Laplacian linear systems

[Gremban 1996, Nakahara and Takahashi 1996]

◮ Kinematic analysis of robot manipulators

[Staffelli and Thomas 2002]

◮ Flow estimation from noisy measurements

[Zohar and Gieger 2007]

14

Previous work

◮ Finite [Epifanov 1966, Feo 1985] ◮ Bigon reduction [Steinitz 1916, Gr¨

unbaum 1967] — O(n2)

◮ Embed into grids as minor [Truemper 1989] — O(n3) ◮ Depth-sum potential [Feo and Provan 1993] — O(n2) ◮ Ω(n) moves are required

◮ at most one vertex removed at each move 15

Previous work

◮ Finite [Epifanov 1966, Feo 1985] ◮ Bigon reduction [Steinitz 1916, Gr¨

unbaum 1967] — O(n2)

◮ Embed into grids as minor [Truemper 1989] — O(n3) ◮ Depth-sum potential [Feo and Provan 1993] — O(n2) ◮ Ω(n) moves are required

◮ at most one vertex removed at each move 15

Previous work

◮ Finite [Epifanov 1966, Feo 1985] ◮ Bigon reduction [Steinitz 1916, Gr¨

unbaum 1967] — O(n2)

◮ Embed into grids as minor [Truemper 1989] — O(n3) ◮ Depth-sum potential [Feo and Provan 1993] — O(n2) ◮ Ω(n) moves are required

◮ at most one vertex removed at each move 15

Previous work

◮ Finite [Epifanov 1966, Feo 1985] ◮ Bigon reduction [Steinitz 1916, Gr¨

unbaum 1967] — O(n2)

◮ Embed into grids as minor [Truemper 1989] — O(n3) ◮ Depth-sum potential [Feo and Provan 1993] — O(n2) ◮ Ω(n) moves are required

◮ at most one vertex removed at each move 15

Previous work

◮ Finite [Epifanov 1966, Feo 1985] ◮ Bigon reduction [Steinitz 1916, Gr¨

unbaum 1967] — O(n2)

◮ Embed into grids as minor [Truemper 1989] — O(n3) ◮ Depth-sum potential [Feo and Provan 1993] — O(n2) ◮ Ω(n) moves are required

◮ at most one vertex removed at each move 15

Steinitz’s bigon reduction

[Steinitz 1916]

From “Convex polytopes”, Branko Gr¨ unbaum

16

Steinitz’s bigon reduction

[Steinitz 1916]

◮ Problem: No 02 moves x “Convex polytopes”, Branko Gr¨ unbaum

17

Minimal bigons

From “Convex polytopes”, Branko Gr¨ unbaum

18

Minimal bigons

From “Convex polytopes”, Branko Gr¨ unbaum

18

Minimal bigons

From “Convex polytopes”, Branko Gr¨ unbaum

18

Minimal bigons

◮ Any minimal bigon can be reduced

using only 10 and 33 move

[Steinitz 1916, Steinitz and Rademacher 1934]

19

What is the right answer?

Θ(n)? Θ(n2)?

20

The Feo and Provan Conjecture

Θ(n3/2)

electrical transformations

◮ Θ(n3/2) homotopy moves in the plane [C. and Erickson 2016]

21

The Feo and Provan Conjecture

Θ(n3/2)

electrical transformations

◮ Θ(n3/2) homotopy moves in the plane [C. and Erickson 2016]

21

# electrical transformations

• # homotopy moves

22

Medial graphs

[Tait 1876–7, Steinitz 1916]

“Blue Elephant”, Mick Burton, 1969

23

Medial graphs

[Tait 1876–7, Steinitz 1916]

From “Some Elementary Properties of Closed Plane Curves”, Tait, 1877

24

Medial graphs

[Tait 1876–7, Steinitz 1916]

From “Some Elementary Properties of Closed Plane Curves”, Tait, 1877

24

Electrical moves

10 21 33

25

# electrical moves # homotopy moves

◮ Follows from close reading of previous results

[Truemper 1989, Noble and Welsh 2000, C. and Erickson 2016]

◮ Replace 21 move with 20 move then apply

smoothing lemma

26

# electrical moves # homotopy moves

◮ Follows from close reading of previous results

[Truemper 1989, Noble and Welsh 2000, C. and Erickson 2016]

◮ Replace 21 move with 20 move then apply

smoothing lemma

26

Smoothing

minor in graphs = smoothing in medial graphs

27

Minor/smoothing lemma

◮ Any minor of an electrically reducible graph is also

electrically reducible [Truemper 1989]

◮ A proper smoothing requires strictly less electrical moves

[C. and Erickson 2016]

28

Minor/smoothing lemma

◮ Any minor of an electrically reducible graph is also

electrically reducible [Truemper 1989]

◮ A proper smoothing requires strictly less electrical moves

[C. and Erickson 2016]

1→0 2→1 = 1→0 3→3 2→1 = = 1→2 = =

28

Truemper’s grid reduction

[Truemper 1989]

29

Feo and Provan’s depth-sum potential

[Feo and Provan 1993]

1 1 1 1 1 2 1 2 2 3 2 3 ◮ Theorem. Any plane graph always has a positive move

with respect to depth-sum potential [Feo and Provan 1993]

◮ Question. A better proof using curve language?

30

Feo and Provan’s depth-sum potential

[Feo and Provan 1993]

1 1 1 1 1 2 1 2 2 3 2 3 ◮ Theorem. Any plane graph always has a positive move

with respect to depth-sum potential [Feo and Provan 1993]

◮ Question. A better proof using curve language?

30

Open question

◮ Prove (or disprove) the Feo and Provan conjecture! ◮ Problems of using the early techniques:

◮ Steinitz’s bigon reduction — no small bigons ◮ Feo and Provan’s potential — no positive moves ◮ Truemper’s grid embedding — inefficient 31

Open question

◮ Prove (or disprove) the Feo and Provan conjecture! ◮ Problems of using the early techniques:

◮ Steinitz’s bigon reduction — no small bigons ◮ Feo and Provan’s potential — no positive moves ◮ Truemper’s grid embedding — inefficient 31

Open question

◮ Prove (or disprove) the Feo and Provan conjecture! ◮ Problems of using the early techniques:

◮ Steinitz’s bigon reduction — no small bigons ◮ Feo and Provan’s potential — no positive moves ◮ Truemper’s grid embedding — inefficient 31

Open question

◮ Prove (or disprove) the Feo and Provan conjecture! ◮ Problems of using the early techniques:

◮ Steinitz’s bigon reduction — no small bigons ◮ Feo and Provan’s potential — no positive moves ◮ Truemper’s grid embedding — inefficient 31

Open question

◮ Prove (or disprove) the Feo and Provan conjecture! ◮ Problems of using the early techniques:

◮ Steinitz’s bigon reduction — no small bigons ◮ Feo and Provan’s potential — no positive moves ◮ Truemper’s grid embedding — inefficient 31

Might not be easy

◮ Ω(n2) moves for non-contractible curves on torus

b c a n/8

{

n/8

{

n/8

{

n/8

{

b c a b c a b c a

32

Might not be easy

◮ In fact, Ω(n2) moves in the annulus, even for

contractible curves [C. and Erickson 2017]

xi yi xʹ

zi ai 33

Against Feo and Provan

◮ Ω(n2) electrical moves on the annulus

[C. and Erickson 2017]

◮ Follows from the inequality ◮ Need generalized smoothing lemma in annulus ◮ Bullseye as the only target 34

Against Feo and Provan

◮ Ω(n2) electrical moves on the annulus

[C. and Erickson 2017]

◮ Follows from the inequality? ◮ Need generalized smoothing lemma in annulus ◮ Bullseye as the only target 34

Against Feo and Provan

◮ Ω(n2) electrical moves on the annulus

[C. and Erickson 2017]

◮ Follows from the inequality? ◮ Need generalized smoothing lemma in annulus ◮ Bullseye as the only target 34

Against Feo and Provan

◮ Ω(n2) electrical moves on the annulus

[C. and Erickson 2017]

◮ Follows from the inequality? ◮ Need generalized smoothing lemma in annulus ◮ Bullseye as the only target 34

Against Feo and Provan?

◮ Ω(n2) electrical moves on the annulus

[C. and Erickson 2017]

◮ Follows from the inequality? ◮ Need generalized smoothing lemma in annulus ◮ Bullseye as the only target 34

Against Feo and Provan?

◮ Ω(n2) electrical moves on the annulus

[C. and Erickson 2017]

◮ Follows from the inequality? ◮ Need generalized smoothing lemma in annulus ◮ Bullseye as the only target

◮ Non-facial electrical transformations are required for

Feo-Provan conjecture

35

Questions?

36

Thank you!

37

• Q. How do you prove the O(n3/2) upper bound on

homotopy moves in the plane?

Useful cycle technique

38

Loop reduction

1 1 1 1 1 2 1 2 2 3 2 3 1 1 1 1 2 1 2

y z ◮ At most O(A) moves, where A is number of interior faces ◮ Depth-sum potential Φ decreases by at least A

39

Loop reduction

1 1 1 1 1 2 1 2 2 3 2 3 1 1 1 1 2 1 2

y z ◮ At most O(A) moves, where A is number of interior faces ◮ Depth-sum potential Φ decreases by at least A

39

Loop reduction

1 1 1 1 1 2 1 2 2 3 2 3 1 1 1 1 2 1 2

y z ◮ At most O(A) moves, where A is number of interior faces ◮ Depth-sum potential Φ decreases by at least A

39

Useful cycle technique

From “Choking Loops on Surfaces”, Feng and Tong, 2013

40

Tangle

41

Tangle

41

Tangle

m vertices, s strands, max-depth d

42

Tangle reductions

◮ First, remove all the self-loops in O(md) moves

43

Tangle reductions

◮ Second, straighten all strand in O(ms) moves

44

Tangle reductions

◮ Second, straighten all strand in O(ms) moves

45

Algorithm

◮ A tangle is useful if s ≤ O(m1/2) and d ≤ O(m1/2) ◮ Algorithm: Tighten any useful tangle until the curve is simple

46

Algorithm

◮ A tangle is useful if s ≤ O(m1/2) and d ≤ O(m1/2) ◮ Algorithm: Tighten any useful tangle until the curve is simple

46

Amortized analysis

◮ Tightening one useful tangle:

O(md + ms) ≤ O(m3/2) moves

◮ At least Ω(m) vertices removed ◮ Each charged with O(m1/2) ≤ O(n1/2) moves ◮ In total O(n3/2) homotopy moves ◮ How do we know that there is always a useful tangle?

47

Amortized analysis

◮ Tightening one useful tangle:

O(md + ms) ≤ O(m3/2) moves

◮ At least Ω(m) vertices removed ◮ Each charged with O(m1/2) ≤ O(n1/2) moves ◮ In total O(n3/2) homotopy moves ◮ How do we know that there is always a useful tangle?

47

Amortized analysis

◮ Tightening one useful tangle:

O(md + ms) ≤ O(m3/2) moves

◮ At least Ω(m) vertices removed ◮ Each charged with O(m1/2) ≤ O(n1/2) moves ◮ In total O(n3/2) homotopy moves ◮ How do we know that there is always a useful tangle?

47

Amortized analysis

◮ Tightening one useful tangle:

O(md + ms) ≤ O(m3/2) moves

◮ At least Ω(m) vertices removed ◮ Each charged with O(m1/2) ≤ O(n1/2) moves ◮ In total O(n3/2) homotopy moves ◮ How do we know that there is always a useful tangle?

47

Finding useful tangle

z

One of the level curve is useful

48

• Q. How do you prove the Ω(n3/2) lower bound on

homotopy moves in the plane?

Defect of flat torus knots

49

Defect

[Arnold 1994, Aicardi 1994]

δ(γ) := −2

• x≬y

sgn(x) · sgn(y)

[Polyak 1998]

◮ x ≬ y means x and y are interleaved — x, y, x, y ◮ sgn(·) follows Gauss convention

1 2 1 2

50

Defect

[Arnold 1994, Aicardi 1994]

δ(γ) := −2

• x≬y

sgn(x) · sgn(y)

[Polyak 1998]

a b c d e f g h i j k

51

Defect

[Arnold 1994, Aicardi 1994]

defect changes by at most 2 under any homotopy moves

52

Flat torus knots T(p, q)

(p − 1)q intersection points

53

Flat torus knots T(p, q)

T(7, 8) T(8, 7) δ(T(p, p + 1)) = 2 p+1

3

• δ(T(q + 1, q)) = −2

q

3

• [Even-Zohar et al. 2016]

[Hayashi et al. 2012]

54

Defect of T(p, ap + 1)

δ(T(p, ap + 1)) = δ(T(p, (a − 1)p + 1)) + 2 p − 1 2

• + 2(p − 1)

= 2a p + 1 3

• 55

Defect of T(p, ap + 1)

δ(T(p, ap + 1)) = δ(T(p, (a − 1)p + 1)) + 2 p − 1 2

• + 2(p − 1)

= 2a p + 1 3

• 55

Defect of T(p, ap + 1)

δ(T(p, ap + 1)) = δ(T(p, (a − 1)p + 1)) + 2 p − 1 2

• + 2(p − 1)

= 2a p + 1 3

• 55

Defect of T(p, ap + 1)

δ(T(p, ap + 1)) = δ(T(p, (a − 1)p + 1)) + 2 p − 1 2

• + 2(p − 1)

= 2a p + 1 3

• 55

Curve with Ω(n3/2) defect

T(√n + 1, √n) has n vertices and Ω(n3/2) defect

56