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

Untangling Graphs and Curves on 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 1 � 0 2 � 0 3 � 3 ◮ How many homotopy moves does it take to simplify a generic closed curve on a surface? 3

Homotopy moves 1 � 0 2 � 0 3 � 3 ◮ 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 ( n 2 ) moves [Steinitz 1916, Hass and Scott 1985] ◮ Actually, anything homotopic to simple curve ◮ Ω ( n 2 ) moves for non-contractible curves on torus [C. and Erickson 2016] 5

Special curves ◮ Contractible curve: O ( n 2 ) moves [Steinitz 1916, Hass and Scott 1985] ◮ Actually, anything homotopic to simple curve ◮ Ω ( n 2 ) moves for non-contractible curves on torus [C. and Erickson 2016] a a a a b b b b c c c c { { { { n /8 n /8 n /8 n /8 5

Special surface: Plane and Sphere ◮ O ( n 2 ) moves are always enough ◮ regular homotopy (no 1 � � 0 moves) [Francis 1969] From “Generic and Regular Curves”, Jeff Erickson 6

Loop reductions 0 0 1 1 1 1 2 1 2 1 2 2 1 1 3 3 2 2 1 1 1 7

Loop reductions 0 0 1 1 1 1 2 2 1 1 2 2 1 1 3 3 2 2 1 1 1 y z ◮ at most O ( n ) moves to remove a loop ◮ O ( n 2 ) homotopy moves in total 8

Loop reductions 0 0 1 1 1 1 2 2 1 1 2 2 1 1 3 3 2 2 1 1 1 y z ◮ at most O ( n ) moves to remove a loop ◮ O ( n 2 ) homotopy moves in total 8

Loop reductions 0 0 1 1 1 1 2 2 1 1 2 2 1 1 3 3 2 2 1 1 1 y z ◮ at most O ( n ) moves to remove a loop ◮ O ( n 2 ) homotopy moves in total 8

Special surface: Plane and Sphere ◮ O ( n 2 ) moves are always enough ◮ regular homotopy (no 1 � � 0 moves) [Francis 1969] ◮ O ( n 2 ) 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´ oth, 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 ( n 2 ) ◮ Embed into grids as minor [Truemper 1989] — O ( n 3 ) ◮ Depth-sum potential [Feo and Provan 1993] — O ( n 2 ) ◮ 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 ( n 2 ) ◮ Embed into grids as minor [Truemper 1989] — O ( n 3 ) ◮ Depth-sum potential [Feo and Provan 1993] — O ( n 2 ) ◮ 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 ( n 2 ) ◮ Embed into grids as minor [Truemper 1989] — O ( n 3 ) ◮ Depth-sum potential [Feo and Provan 1993] — O ( n 2 ) ◮ 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 ( n 2 ) ◮ Embed into grids as minor [Truemper 1989] — O ( n 3 ) ◮ Depth-sum potential [Feo and Provan 1993] — O ( n 2 ) ◮ 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 ( n 2 ) ◮ Embed into grids as minor [Truemper 1989] — O ( n 3 ) ◮ Depth-sum potential [Feo and Provan 1993] — O ( n 2 ) ◮ 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 0 � 2 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 1 � 0 and 3 � 3 move [Steinitz 1916, Steinitz and Rademacher 1934] 19

What is the right answer? Θ ( n ) ? Θ ( n 2 ) ? 20

The Feo and Provan Conjecture Θ ( n 3 / 2 ) electrical transformations ◮ Θ ( n 3 / 2 ) homotopy moves in the plane [C. and Erickson 2016] 21

The Feo and Provan Conjecture Θ ( n 3 / 2 ) electrical transformations ◮ Θ ( n 3 / 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 1 � 0 3 � 3 2 � 1 25

# electrical moves � # homotopy moves ◮ Follows from close reading of previous results [Truemper 1989, Noble and Welsh 2000, C. and Erickson 2016] ◮ Replace 2 � 1 move with 2 � 0 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 2 � 1 move with 2 � 0 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 1 → 2 = = 3 → 3 2 → 1 = 28

Truemper’s grid reduction [Truemper 1989] 29

Feo and Provan’s depth-sum potential [Feo and Provan 1993] 0 1 1 2 1 2 1 3 3 2 2 1 1 ◮ 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] 0 1 1 2 1 2 1 3 3 2 2 1 1 ◮ 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