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Cayley Complexity of One Degree of Freedom Linkages in 2D Meera - - PowerPoint PPT Presentation

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary Cayley Complexity of One Degree of Freedom Linkages in 2D Meera Sitharam Menghan Wang Heping Gao University of Florida Department of Computer Information

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Cayley Complexity of One Degree of Freedom Linkages in 2D

Meera Sitharam Menghan Wang Heping Gao

University of Florida Department of Computer Information Science & Engineering

2011

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

1-dof Linkages

1 2 3 4 5

One degree of freedom (1-dof) linkage (mechanism) in 2D Linkage (G, δ): G = (V , E), δ : E → R

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Cayley Configuration Space

How to describe the space of configurations (2D realizations) for a 1-dof linkage (G, δ)?

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Cayley Configuration Space

How to describe the space of configurations (2D realizations) for a 1-dof linkage (G, δ)? Cayley Configuration Space of (G, δ) on non-edge f = (u, v): the set of possible distances between u and v Φf (G, δ):= {δ∗(f ) : linkage (G ∪ f , δ, δ∗) has realization}

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Cayley Configuration Space

How to describe the space of configurations (2D realizations) for a 1-dof linkage (G, δ)? Cayley Configuration Space of (G, δ) on non-edge f = (u, v): the set of possible distances between u and v Φf (G, δ):= {δ∗(f ) : linkage (G ∪ f , δ, δ∗) has realization} Φf (G, δ) is a set of intervals on the real line

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Cayley Configuration Space

How to describe the space of configurations (2D realizations) for a 1-dof linkage (G, δ)? Cayley Configuration Space of (G, δ) on non-edge f = (u, v): the set of possible distances between u and v Φf (G, δ):= {δ∗(f ) : linkage (G ∪ f , δ, δ∗) has realization} Φf (G, δ) is a set of intervals on the real line Each point δ∗(f ) in Φf (G, δ) is a Cayley configuration

1 2 3 4 5 f 1 2 3 4 5 f 1 2 3 4 5 f (a) (b) (c)

2

δ∗(f ) = 0 δ∗(f ) = √ 2

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Complexity of Cayley Configuration Spaces

How to measure the complexity of Cayley configuration space?

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Complexity of Cayley Configuration Spaces

How to measure the complexity of Cayley configuration space? (a) Cayley complexity: algebraic complexity of interval endpoint values Definition Quadratically Solvable (QS) values: solutions to triangularized quadratic system with coefficient in Q (in extension field over Q by nested square-roots)

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Complexity of Cayley Configuration Spaces

How to measure the complexity of Cayley configuration space? (a) Cayley complexity: algebraic complexity of interval endpoint values Definition Quadratically Solvable (QS) values: solutions to triangularized quadratic system with coefficient in Q (in extension field over Q by nested square-roots) (b) Cayley size: number of intervals

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Complexity of Cayley Configuration Spaces

How to measure the complexity of Cayley configuration space? (a) Cayley complexity: algebraic complexity of interval endpoint values Definition Quadratically Solvable (QS) values: solutions to triangularized quadratic system with coefficient in Q (in extension field over Q by nested square-roots) (b) Cayley size: number of intervals (c) Cayley computational complexity: time complexity of

  • btaining all intervals (as function of Cayley size)

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

A Natural Class of Graphs

Cayley configurations δ∗(f ) can be efficiently converted to Cartesian configurations provided: Completeness: G ∪ f minimally rigid (implies (G ∪ f , δ, δ∗(f )) has finitely many realizations for each δ∗(f )) Low realization complexity: linear realization complexity if local

  • rientations are specified

1 2 4 3 5 6 7 8 f 1 2 4 3 5 6 7 8 f

  • rientation 1
  • rientation 2

1 2 4 3 5 6 7 8 f

Note: any f = (i, i + 2) guarantees both properties

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Quadratically Solvable Graphs

Definition G ∪ f Quadratically Solvable (QS) from f : ∃ a ruler and compass realization of (G ∪ f , δ, δ∗(f )) starting from f Hence: Cayley configuration δ∗(f )

efficient conversion

− − − − − − − − − − − → Cartesian configuration

1 2 4 3 5 6 7 8 f

Note: for any f = (i, i + 2), G ∪ f is QS starting from f

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

A Class of Quadratically Solvable Graphs

Definition

G is △-decomposable if it is a single edge, or can be divided into 3 △-decomposable subgraphs s.t. every two of them share a single vertex. 1-dof △-decomposable graph: drop an edge f from a △-decomposable graph Note: △-decomposable implies minimally rigid Graph construction from f : each step appends a new vertex shared by 2 △-decomposable subgraphs This is also a (unique) QS realization sequence of corresponding linkage starting from f Hence △-decomposable = ⇒ QS

1 2 4 3 5 6 7 8 f

△-decomposable subgraph

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

A Class of Quadratically Solvable Graphs

Theorem (Owen & Power, 2005) QS = ⇒ △-decomposable for planar graphs Strong conjecture: △-decomposable implies QS for general graphs In this talk, we only consider △-decomposable graphs Will refer to them as QS graphs

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

QS Cayley complexity

Definition G has QS Cayley complexity with respect to non-edge f : all interval endpoints – of Φf (G, δ) – are QS Extreme graphs: O(n) of them, one per step of QS realization sequence,

  • btained by adding an extreme edge

1 2 4 3 5 6 7 f 1 2 4 5 6 7 8 f

Theorem A 1-dof QS graph G has QS Cayley complexity on f ⇐ ⇒ all of its extreme graphs starting from f are QS. This is probably folklore. For completeness, formally proven in (Gao & Sitharam, 2008).

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Outline

1

Characterizing QS Cayley complexity It is a Property of G Independent of Choice of Non-edge f Algorithmic Characterization (4-cycle Theorem) Finite Forbidden-Minor Characterization

2

Cayley Size & Cayley Computational Complexity Guaranteeing Computational Complexity O(n) & Cayley size O(1)

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary Independent of Choice of f Algorithmic Characterization (4-cycle Theorem) Finite Forbidden-Minor Characterization

Outline

1

Characterizing QS Cayley complexity It is a Property of G Independent of Choice of Non-edge f Algorithmic Characterization (4-cycle Theorem) Finite Forbidden-Minor Characterization

2

Cayley Size & Cayley Computational Complexity Guaranteeing Computational Complexity O(n) & Cayley size O(1)

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary Independent of Choice of f Algorithmic Characterization (4-cycle Theorem) Finite Forbidden-Minor Characterization

Choice of f

Possible f : (i, i + 2) for any i By possible f we mean any non-edge f s.t. G ∪ f is QS.

1 2 4 3 5 6 7 8 f

Does Cayley complexity depend on choice of f ?

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary Independent of Choice of f Algorithmic Characterization (4-cycle Theorem) Finite Forbidden-Minor Characterization

Choice of f

Possible f : (i, i + 2) for any i By possible f we mean any non-edge f s.t. G ∪ f is QS.

1 2 4 3 5 6 7 8 f

Does Cayley complexity depend on choice of f ?

  • NO.

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary Independent of Choice of f Algorithmic Characterization (4-cycle Theorem) Finite Forbidden-Minor Characterization

Independent of Choice of f

Theorem (Sitharam, Wang, Gao) 1-dof QS graph G either has QS Cayley complexity on all possible f or on none

  • f them.

Proof is non-trivial.

Thus: our measure of QS Cayley complexity is robust. Characterizing G

  • f QS Cayley complexity with a specific

f is sufficient.

1 2 4 3 5 6 7 8 f

G has QS Cayley complexity

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary Independent of Choice of f Algorithmic Characterization (4-cycle Theorem) Finite Forbidden-Minor Characterization

Outline

1

Characterizing QS Cayley complexity It is a Property of G Independent of Choice of Non-edge f Algorithmic Characterization (4-cycle Theorem) Finite Forbidden-Minor Characterization

2

Cayley Size & Cayley Computational Complexity Guaranteeing Computational Complexity O(n) & Cayley size O(1)

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary Independent of Choice of f Algorithmic Characterization (4-cycle Theorem) Finite Forbidden-Minor Characterization

Algorithmic Characterization (4-cycle Theorem)

Theorem (Sitharam, Wang) 1-dof QS graph G has QS Cayley complexity ⇐ ⇒ ∃ non-edge f (∀f ) each construction step from f is based on a pair

  • f vertices taken from two adjacent QS

subgraphs, from a 4-cycle of QS subgraphs Gives O(n) time algorithm to recognize QS Cayley complexity graphs

v1 T2 v2 p3 p2 T1 p1 T3 T4 f p4 vk u w G_k-1 T5 T6

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary Independent of Choice of f Algorithmic Characterization (4-cycle Theorem) Finite Forbidden-Minor Characterization

Outline

1

Characterizing QS Cayley complexity It is a Property of G Independent of Choice of Non-edge f Algorithmic Characterization (4-cycle Theorem) Finite Forbidden-Minor Characterization

2

Cayley Size & Cayley Computational Complexity Guaranteeing Computational Complexity O(n) & Cayley size O(1)

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary Independent of Choice of f Algorithmic Characterization (4-cycle Theorem) Finite Forbidden-Minor Characterization

Finite Forbidden-Minor Characterization

Can there exist finite forbidden-minor characterization for general 1-dof QS graphs?

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary Independent of Choice of f Algorithmic Characterization (4-cycle Theorem) Finite Forbidden-Minor Characterization

Finite Forbidden-Minor Characterization

Can there exist finite forbidden-minor characterization for general 1-dof QS graphs?

  • NO.

Will show counterexamples later.

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary Independent of Choice of f Algorithmic Characterization (4-cycle Theorem) Finite Forbidden-Minor Characterization

1-Path & △-Free

Need to look at natural subclasses: 1-Path & △-Free Definition 1-Path: ∃ only one “last vertex” v, that is, v is shared by exactly 2 QS subgraphs, each of them share only one vertex with the rest of the graph.

1 1 2 2 3 3 4 4 5 5 6 A B A B C D 6

Definition △-Free: no subgraph of G is a triangle

1 2 4 7 5 3 6 1 2 4 7 5 3 6 8

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary Independent of Choice of f Algorithmic Characterization (4-cycle Theorem) Finite Forbidden-Minor Characterization

Equivalence to Planarity

Theorem (Sitharam, Wang) A 1-path, △-free, 1-dof QS graph G has QS Cayley complexity ⇐ ⇒ G is planar

1 2 4 7 5 3 1 2 5 4 3 6 (a) (b)

  • Ex. (b) has QS Cayley

complexity, (a) doesn’t

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary Independent of Choice of f Algorithmic Characterization (4-cycle Theorem) Finite Forbidden-Minor Characterization

Equivalence to Planarity

Theorem (Sitharam, Wang) A 1-path, △-free, 1-dof QS graph G has QS Cayley complexity ⇐ ⇒ G is planar

1 2 4 7 5 3 1 2 5 4 3 6 (a) (b)

  • Ex. (b) has QS Cayley

complexity, (a) doesn’t

1-path & △-free are necessary. Otherwise no finite forbidden-minor characterization exists

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary Independent of Choice of f Algorithmic Characterization (4-cycle Theorem) Finite Forbidden-Minor Characterization

1-Path & △-Free are Necessary

Counter example 1: not △-free

v3(u2) v2 G2 G1 v6 v1(u1) v5(w3)

u3 u4 w1 w2 v4

Has QS Cayley complexity since the sole extreme graph is QS. Can extend the graph to make G1 have an arbitrary clique as minor

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary Independent of Choice of f Algorithmic Characterization (4-cycle Theorem) Finite Forbidden-Minor Characterization

v3(u2) v2 G2 G1 v6 v1(u1) v5(w3)

u3 u4 w1 w2 v4

v3(u2) v2 G2 G1 v6 v1(u1) u3 v4 v3(u2) v2 G2 G1 v6 v1(u1) u3 w2 v4 w1 v3(u2) v2 G2 G1 v6 v1(u1) v5(w3) u3 u4 w1 w2 v4

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary Independent of Choice of f Algorithmic Characterization (4-cycle Theorem) Finite Forbidden-Minor Characterization

1-Path & △-Free are Necessary

Counter example 2: not 1-path

v1 v2 f u1 u2 u4 u3 w12 w13 w14 w23 w24 w34 u5 w25 w15 w35 w45

Has QS Cayley complexity (can be checked using the 4-cycle theorem). Can be made to have an arbitrary clique as minor.

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary Independent of Choice of f Algorithmic Characterization (4-cycle Theorem) Finite Forbidden-Minor Characterization

v1 v2 f u1 u2 u4 u3 w12 w13 w14 w23 w24 w34 u5 w25 w15 w35 w45

v1 v2 f u1 u2 u4 u3 w12 w13 w14 w23 w24 w34 u5 w25 w15 w35 w45 v1 v2 f u1 u2 u4 u3 w12 w13 w14 w23 w24 w34 u5 w25 w15 w35 w45

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Outline

1

Characterizing QS Cayley complexity It is a Property of G Independent of Choice of Non-edge f Algorithmic Characterization (4-cycle Theorem) Finite Forbidden-Minor Characterization

2

Cayley Size & Cayley Computational Complexity Guaranteeing Computational Complexity O(n) & Cayley size O(1)

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Cayley size & Cayley computational complexity

Recall the three aspects of complexity of Cayley Configuration Spaces (a) Cayley complexity (b) Cayley size: number of intervals (c) Cayley computational complexity: complexity of obtaining all intervals Have characterization of (a) Let’s consider (b) and (c)

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Cayley size & Cayley computational complexity

Suppose G has QS Cayley complexity

1 2 4 3 5 6 7 8 f

Are we guaranteed to have small Cayley size & low Cayley computational complexity?

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Cayley size & Cayley computational complexity

Suppose G has QS Cayley complexity

1 2 4 3 5 6 7 8 f

Are we guaranteed to have small Cayley size & low Cayley computational complexity? Only if we specify necessary orientations of the realizations

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Necessary Orientations

A natural, minimal set of local orientations for both forward & backward QS realization sequences forward orientations from f

1 2 4 3 5 6 7 8 f 1 2 4 3 5 6 7 8 f

backward orientations for all extreme linkages

1 2 4 3 5 6 7 f 1 2 4 3 5 6 7 f

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Blow-up of Cayley Size & Computational Complexity without Orientations

Is either type of orientations sufficient without the other?

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Blow-up of Cayley Size & Computational Complexity without Orientations

Is either type of orientations sufficient without the other?

  • NO.

Can adapt existing examples of Borcea & Streinu to show exponential blow-up

1 2 4 3 5 6 7 8 f

Already so for our standard example.

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Efficient Cayley Configuration Space

Theorem (Sitharam, Wang) For 1-dof QS graph G with QS Cayley complexity, given both forward and backward orientations, the Cayley size is O(1) and the Cayley computational complexity is O(|V |) Proof non-trivial & based on the 4-cycle theorem Yields straightforward algorithm using quadrilateral interval mapping via 4-cycles.

1 2 4 3

7.0 7.5 8.0 8.5 9.0 6.5 7.0 7.5 8.0 8.5 9.0

(3,4) (1,2) max min I1 I2

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Summary

Cayley configuration space & measure of complexity Choice of base non-edge does not affect QS Cayley complexity. Algorithmic characterization (4-cycle Theorem) For 1-path, △-free, 1-dof QS graphs: QS Cayley complexity ⇐ ⇒ planarity Low Cayley size & computational complexity in the presence

  • f necessary orientations

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Proof of Planarity Theorem

Theorem A 1-path, △-free, 1-dof QS graph G has QS Cayley complexity ⇐ ⇒ G is planar

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Proof of Planarity Theorem

Lemma (1) Given a 1-path, △-free, 1-dof QS graph G with non-edge f = (v1, v2). If 3 or more vertices are directly constructed on f OR exactly 2 vertices are directly constructed on f & deg(v1) ≥ 3,deg(v2) ≥ 3 We have

1

G has a K3,3 minor

2

G does not have QS Cayley complexity on f

1 2 4 7 5 3 (a) 1 2 (b) 5 4 3 6 6 7 8

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Proof of Planarity Theorem

Lemma (2) Given a 1-path 1-dof QS graph G with non-edge f = (v1, v2) s.t. u1, u2 are the only 2 vertices directly constructed on f , if v1 is a “last vertex” & v2 is not (resp. both v1 and v2 are “last vertices”), then

1

G ′ = G \ {v1} (resp. G ′ = G \ {v1, v2}) is 1-path 1-dof QS graph on f ′ = (u1, u2).

2

G ′ = G \ {v1} (resp. G ′ = G \ {v1, v2}) has QS Cayley complexity on f ′ ⇐ ⇒ G has QS Cayley complexity on f

1 2 4 7 5 3 5 4 3 6 8 9 1 2 6 7 (a) (b)

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Proof of Planarity Theorem. By the two lemmas, the only interesting case is where G has exactly 2 vertices u1, u2 directly constructed on f = (v1, v2), and either (a) deg(v1) = 2, deg(v2) > 2, or (b) deg(v1) = 2, deg(v2) = 2. Define G ′ as in Lemma (2). 1 G is planar = ⇒ G has QS Cayley complexity on f : Prove by contradiction. Assume G is the minimum QS graph s.t. G does not have QS Cayley complexity on f and is planar. Clearly G ′ contradicts the assumption of minimality of G.

(b) (a)

u1 v2 u1 u2 v2 v1 u2 v1

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Proof of Planarity Theorem (cont.) 2 G has QS Cayley complexity on f = ⇒ G has no K3,3: Prove by contradiction. Assume G is the minimum QS graph s.t. G has QS Cayley complexity on f and G has a K3,3 minor. In case (a), either (v1, u1) or (v1, u2) must be

  • contracted. Either case we obtain the graph on
  • right. ( v3 is the first vertex constructed after u1

and u2)

v2 u1 u2 v3

K3,3 contains no triangles. Every way to eliminate the two triangles will produce a subgraph of G ′. In case (b), similar argument applies.

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Proof of Planarity Theorem (cont.) 3 1-path, △-free, 1-dof QS graph G has K5 = ⇒ G has K3,3: To keep G △-free, some vertices of K5 must be contracted from more than one vertices from G. To keep G 1-path we will get a K3,3. Therefore we have: G has QS Cayley complexity on f = ⇒ G has no K3,3 or K5. Thus completes the proof.

s1 s2 s3 s4 s5 sa sb s2 s3 s4 s5 sa sb sa sb sa sb sc sa v sb s2’ s3’ s4’ s1’ v1 s5’ v2 v3 v4 v5 (a) (b) (c) (d) (e) (f) (g) a b c d

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Proof of O(1) Cayley Size Theorem

Theorem For 1-dof QS graph G with QS Cayley complexity, given both forward and backward orientations, the Cayley size is O(1) and the Cayley computational complexity is O(|V |)

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Proof of O(1) Cayley Size Theorem

Levels

Definition Levels: construction partial order of a 1-dof QS graph from non-edge f L0: endpoints of f . L1: can directly construct on f . Li(i ≥ 2): can directly construct given L0 ∼ Li−1, cannot construct without Li−1.

2 3 7 0’ 4 5 6 8 1

t1 t2 t3 t4 t5 t6 t7 t8

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Proof of O(1) Cayley Size Theorem

Lemma For a 1-path G with QS Cayley complexity,

1

Each level has one or two construction steps.

2

If Lk has two construction steps, they are based the same pair

  • f vertices.

3

From Lk+1 on, each construction step must be based on QS subgraphs in Lk or higher levels.

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Proof of O(1) Cayley Size Theorem. 1 For the 1-path case, the chain of quadrilateral is obvious. The algorithm maps the attainable interval of one diagonal of a quadrilateral to the attainable interval of the other diagonal. By Lemma (2) this mapping process can be repeated, so we can finally get the interval of f . Since both forward and backward orientations are fixed, each mapping step is projection on a monotonic function. Therefore the Cayley size is O(1) .

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Proof of O(1) Cayley Size Theorem (cont.) 2 For graphs with two “last vertices”, we can find a base 4-cycle at the common “root” of both paths. Each path maps to a single interval of a diagonal of the root 4-cycle. Considering the constraints from both paths together, the result is the intersection of two intervals. 3 For graphs with more than 2 paths, we can prove by induction

  • n number of paths.

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D

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Characterizing QS Cayley complexity Cayley Size & Computational Complexity Summary

Thank you!

Meera Sitharam, Menghan Wang, Heping Gao Cayley Complexity of 1-dof Linkages in 2D