Stable Roommates for Weighted Straight Skeletons Therese Biedl 1 Stefan Huber 2 Peter Palfrader 3 1 David R. Cheriton School of Computer Science University of Waterloo, Canada 2 Institute of Science and Technology Austria 3 FB 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 P W P ( t ) ◮ 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 edge vertex S ( P ) face f ( e ) e ◮ 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 W P . ◮ 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, 2 k (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, 2 k (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 ) matching partner of M ( ℓ ) 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 ) matching partner of M ( ℓ ) ℓ b ( ℓ ) matching tail of 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 { ℓ i , ℓ j } : 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 ) matching partner of M ( ℓ ) ℓ Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 14 of 17
Stable partitions Stable partition: ◮ Permutation π of ℓ 1 , . . . , ℓ N . ◮ In each cycle of size ≥ 3: each ℓ prefers π ( ℓ ) over π − 1 ( ℓ ). ◮ There is no party-blocking pair { ℓ i , ℓ j } : they prefer each other over π − 1 ( ℓ i ) and π − 1 ( ℓ j ). Therese Biedl, Stefan Huber, Peter Palfrader: Stable Roommates for Weighted Straight Skeletons Preliminaries 15 of 17
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