Stable Roommates for Weighted Straight Skeletons Therese Biedl 1 - - PowerPoint PPT Presentation

Stable Roommates for Weighted Straight Skeletons

Therese Biedl1 Stefan Huber2 Peter Palfrader3

1David R. Cheriton School of Computer Science

University of Waterloo, Canada

2Institute of Science and Technology Austria 3FB Computerwissenschaften

Universit¨ at Salzburg, Austria

EuroCG 2014 — Dead Sea, Israel March 3–5, 2014

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons 1 of 17

Straight skeletons — a brief introduction

◮ Introduced by [Aichholzer et al., 1995].

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 2 of 17

Straight skeletons — a brief introduction

WP(t) P

◮ Introduced by [Aichholzer et al., 1995].

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 2 of 17

Straight skeletons — a brief introduction

edge event

◮ Introduced by [Aichholzer et al., 1995].

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 2 of 17

Straight skeletons — a brief introduction

split event

◮ Introduced by [Aichholzer et al., 1995].

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 2 of 17

Straight skeletons — a brief introduction

S(P)

◮ Introduced by [Aichholzer et al., 1995].

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 2 of 17

Straight skeletons — a brief introduction

vertex edge face f (e) e S(P)

◮ Introduced by [Aichholzer et al., 1995].

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 2 of 17

Straight skeletons — with weights

◮ Introduced in [Eppstein and Erickson, 1999]. ◮ To every edge e of P a weight σ(e) is assigned, its speed.

1 1 1 2 −1

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 3 of 17

Straight skeletons — with weights

◮ Introduced in [Eppstein and Erickson, 1999]. ◮ To every edge e of P a weight σ(e) is assigned, its speed.

1 1 1 2 −1

Weighted straight skeletons are “quite established”:

◮ Algorithms were published. ◮ Implementations are available. ◮ Used in theory & applications.

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 3 of 17

Straight skeletons — with weights

◮ Introduced in [Eppstein and Erickson, 1999]. ◮ To every edge e of P a weight σ(e) is assigned, its speed.

1 1 1 2 −1

Weighted straight skeletons are “quite established”:

◮ Algorithms were published. ◮ Implementations are available. ◮ Used in theory & applications.

Still no rigorous definition is known so far!

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 3 of 17

Prior work

Only since recently we know: weighted straight skeletons can behave very differently.

Simple polygon Polygon with holes Property σ ≡ 1 σ pos. σ arb. σ ≡ 1 σ pos. σ arb. S(P) is connected

• ×

S(P) has no crossing

• ×
• ×

f (e) is monotone w.r.t. e

• ×

×

• ×

× bd f (e) is a simple polygon

• ×
• ×

× T (P) is z-monotone

• ×
• ×

S(P) has n(S(P)) − 1 + h edges

• ×
• ×

S(P) is a tree

• ×

Table : [Biedl et al., 2013]

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 4 of 17

Prior work — ambiguity of edge events

Ambiguity for parallel edges of different weights become adjacent.

Figure : Resolution methods proposed in [Biedl et al., 2013].

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 5 of 17

Prior work — ambiguity of edge events

Ambiguity for parallel edges of different weights become adjacent.

Figure : Resolution methods proposed in [Biedl et al., 2013].

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 5 of 17

Prior work — ambiguity of edge events

Ambiguity for parallel edges of different weights become adjacent.

Figure : Resolution methods proposed in [Biedl et al., 2013].

Still open: How to handle split events?

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 5 of 17

Split events

The standard scheme works for unweighted straight skeletons.

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 6 of 17

Split events

The standard scheme works for unweighted straight skeletons.

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 6 of 17

Split events

But for arbitrary weights the standard scheme may fail.

u v e

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 7 of 17

Split events

How to handle this?

p

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 8 of 17

Guiding principle

At all times between events, the wavefront shall be planar.

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 9 of 17

Pairing edges

p B(p, ǫ)

First:

◮ Remove collapsed edges.

Task: Find a pairing of remaining edges to restore planarity of WP.

◮ Is this always possible? Uniquely?

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 10 of 17

Directed pseudo-line arrangements

p

◮ We have k involved chains.

◮ Hence, 2k (non-collapsed) edges. ◮ Assign direction to each edge. Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 11 of 17

Directed pseudo-line arrangements

p

◮ We have k involved chains.

◮ Hence, 2k (non-collapsed) edges. ◮ Assign direction to each edge.

◮ Consider supporting lines of edges, after the event.

◮ → pseudo-line arrangement L of directed pseudo-lines. Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 11 of 17

Planar matchings

B(p, ǫ)

◮ Every pair intersects, in a single unique point, within B(p, ǫ).

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 12 of 17

Planar matchings

B(p, ǫ) ℓ matching partner of M(ℓ)

◮ Every pair intersects, in a single unique point, within B(p, ǫ). ◮ Matching: grouping into pairs.

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 12 of 17

Planar matchings

B(p, ǫ) ℓ matching tail of ℓ matching partner of M(ℓ) b(ℓ)

◮ Every pair intersects, in a single unique point, within B(p, ǫ). ◮ Matching: grouping into pairs. ◮ Planar matching: matching tails do not cross.

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 12 of 17

Planar matchings

B(p, ǫ)

◮ Every pair intersects, in a single unique point, within B(p, ǫ). ◮ Matching: grouping into pairs. ◮ Planar matching: matching tails do not cross.

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 12 of 17

Planar matchings

Theorem

Every directed pseudo-line arrangement has a planar matching.

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 13 of 17

Stable roommates

◮ Every pseudo-line has a preference list (ranking) of all others. ◮ Blocking pair {ℓi, ℓj}: They prefer each other over their matching partners. ◮ Matching is stable if there are no blocking pairs.

Lemma

L has a planar matching if and only if there is a stable matching.

B(p, ǫ) ℓ matching partner of M(ℓ)

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 14 of 17

Stable partitions

Stable partition:

◮ Permutation π of ℓ1, . . . , ℓN. ◮ In each cycle of size ≥ 3: each ℓ prefers π(ℓ) over π−1(ℓ). ◮ There is no party-blocking pair {ℓi, ℓj}: they prefer each other over π−1(ℓi)

and π−1(ℓj).

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 15 of 17

Stable partitions

Stable partition:

◮ Permutation π of ℓ1, . . . , ℓN. ◮ In each cycle of size ≥ 3: each ℓ prefers π(ℓ) over π−1(ℓ). ◮ There is no party-blocking pair {ℓi, ℓj}: they prefer each other over π−1(ℓi)

and π−1(ℓj).

Theorem ([Tan and Hsueh, 1995])

• 1. There is a stable partition, and it can be found in polynomial time.
• 2. There is a stable matching if and only if there is a stable partition with no

cycles of odd size.

Theorem

There are no parties of odd size for directed pseudo-line arrangements.

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 15 of 17

Odd parties do not exist

Lemma

The tails of ℓ and ℓ′ do not intersect, unless π(ℓ) = ℓ′ or π(ℓ′) = ℓ.

Lemma

There cannot be two parties of size at least three.

R1 R2 ℓ1 ℓ0 ℓ2 ℓ′

2

ℓ′

1

ℓ′ b(ℓ0) b(ℓ′

0)

b(ℓ′

2)

b(ℓ′

1)

b(ℓ1) b(ℓ2)

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 16 of 17

Acknowledgments

We would like to thank David Eppstein for mentioning this problem to us, and for the idea of interpreting the edge-pairing problem as a stable roommate problem.

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 17 of 17

Bibliography I

Aichholzer, O., Alberts, D., Aurenhammer, F., and G¨ artner, B. (1995). A novel type of skeleton for polygons.

• J. Universal Comp. Sci., 1(12):752–761.

Biedl, T., Held, M., Huber, S., Kaaser, D., and Palfrader, P. (2013). Weighted straight skeletons in the plane. In Proc. 25th Canad. Conf. on Comp. Geom. (CCCG ’13), pages 13–18, Waterloo, Canada. Eppstein, D. and Erickson, J. (1999). Raising roofs, crashing cycles, and playing pool: Applications of a data structure for finding pairwise interactions. Discrete Comp. Geom., 22(4):569–592. Tan, J. J. and Hsueh, Y.-C. (1995). A generalization of the stable matching problem. Discrete Applied Mathematics, 59(1):87–102.

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 18 of 17

Non-Uniqueness

B(p, ǫ) B(p, ǫ)

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons 19 of 17

No planar wavefront

p v u v ′′ v ′ p p (a) (b) (c) e e t + ǫ

Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons 20 of 17