l emmata i

play

L EMMATA I r The wavefront propagation of strictly monotone polygonal - PowerPoint PPT Presentation

Apr 27, 2023 •203 likes •720 views

S TRAIGHT S KELETONS OF M ONOTONE P OLYGONS Therese Biedl Martin Held Stefan Huber Dominik Kaaser Peter Palfrader EuroCG 2014, Ein Gedi, Israel 1 S TRAIGHT S KELETONS r Aichholzer, Aurenhammer, Alberts, Grtner 1995 [2] . r Wavefront

S TRAIGHT S KELETONS OF M ONOTONE P OLYGONS Therese Biedl Martin Held Stefan Huber Dominik Kaaser Peter Palfrader EuroCG 2014, Ein Gedi, Israel 1
S TRAIGHT S KELETONS r Aichholzer, Aurenhammer, Alberts, Gärtner 1995 [2] . r Wavefront propagation : shrinking, mitered offset curves of polygon P . r Traces of wavefront vertices are the edges of the straight skeleton S ( P ) . 2
S TRAIGHT S KELETONS r Aichholzer, Aurenhammer, Alberts, Gärtner 1995 [2] . r Wavefront propagation : shrinking, mitered offset curves of polygon P . r Traces of wavefront vertices are the edges of the straight skeleton S ( P ) . 2
S TRAIGHT S KELETONS r Aichholzer, Aurenhammer, Alberts, Gärtner 1995 [2] . r Wavefront propagation : shrinking, mitered offset curves of polygon P . r Traces of wavefront vertices are the edges of the straight skeleton S ( P ) . 2
S TRAIGHT S KELETONS r Aichholzer, Aurenhammer, Alberts, Gärtner 1995 [2] . r Wavefront propagation : shrinking, mitered offset curves of polygon P . r Traces of wavefront vertices are the edges of the straight skeleton S ( P ) . 2
S TRAIGHT S KELETONS r Aichholzer, Aurenhammer, Alberts, Gärtner 1995 [2] . r Wavefront propagation : shrinking, mitered offset curves of polygon P . r Traces of wavefront vertices are the edges of the straight skeleton S ( P ) . 2
S TRAIGHT S KELETONS r Aichholzer, Aurenhammer, Alberts, Gärtner 1995 [2] . r Wavefront propagation : shrinking, mitered offset curves of polygon P . r Traces of wavefront vertices are the edges of the straight skeleton S ( P ) . 2
S TRAIGHT S KELETONS r Aichholzer, Aurenhammer, Alberts, Gärtner 1995 [2] . r Wavefront propagation : shrinking, mitered offset curves of polygon P . r Traces of wavefront vertices are the edges of the straight skeleton S ( P ) . 2
S TRAIGHT S KELETONS r Aichholzer, Aurenhammer, Alberts, Gärtner 1995 [2] . r Wavefront propagation : shrinking, mitered offset curves of polygon P . r Traces of wavefront vertices are the edges of the straight skeleton S ( P ) . 2
S TRAIGHT S KELETONS r Aichholzer, Aurenhammer, Alberts, Gärtner 1995 [2] . r Wavefront propagation : shrinking, mitered offset curves of polygon P . r Traces of wavefront vertices are the edges of the straight skeleton S ( P ) . 2
S TRAIGHT S KELETONS r Aichholzer, Aurenhammer, Alberts, Gärtner 1995 [2] . r Wavefront propagation : shrinking, mitered offset curves of polygon P . r Traces of wavefront vertices are the edges of the straight skeleton S ( P ) . 2
S TRAIGHT S KELETONS r Aichholzer, Aurenhammer, Alberts, Gärtner 1995 [2] . r Wavefront propagation : shrinking, mitered offset curves of polygon P . r Traces of wavefront vertices are the edges of the straight skeleton S ( P ) . 2
T OPOLOGY C HANGES – E DGE AND S PLIT E VENTS r Wavefront topology changes over time. r edge event : an edge of the wavefront vanishes. r split event : wavefront splits into two parts. r In S ( P ) , events (topology changes) are witnessed by nodes . 3
T OPOLOGY C HANGES – E DGE AND S PLIT E VENTS r Wavefront topology changes over time. r edge event : an edge of the wavefront vanishes. r split event : wavefront splits into two parts. r In S ( P ) , events (topology changes) are witnessed by nodes . edge events 3
T OPOLOGY C HANGES – E DGE AND S PLIT E VENTS r Wavefront topology changes over time. r edge event : an edge of the wavefront vanishes. r split event : wavefront splits into two parts. r In S ( P ) , events (topology changes) are witnessed by nodes . split event 3
T OPOLOGY C HANGES – E DGE AND S PLIT E VENTS r Wavefront topology changes over time. r edge event : an edge of the wavefront vanishes. r split event : wavefront splits into two parts. r In S ( P ) , events (topology changes) are witnessed by nodes . split event edge events 3
C ONSTRUCTING THE S TRAIGHT S KELETON Algorithm time space O ( n 3 log n ) Aichholzer, Aurenhammer ’98 [1] O ( n ) Eppstein, Erickson ’99 [5] ⋆ O ( n 17 / 11 + ǫ ) O ( n 2 log n ) Cacciola ’04 [3] O ( n 2 ) O ( n 2 log n ) Huber and Held ’10 [7] O ( n ) Vigneron and Yan ’13 [8] ⋆⋆ O ( n 4 / 3 + ǫ ) O ( n ) r Popular approach: Simulate the wavefront propagation. r Main Problem: Identify next event. r Edge events are cheap. Split events are expensive. Can we do better for specific input classes? Y ES , FOR ( STRICTLY ) MONOTONE POLYGONS . 4
C ONSTRUCTING THE S TRAIGHT S KELETON Algorithm time space O ( n 3 log n ) Aichholzer, Aurenhammer ’98 [1] O ( n ) Eppstein, Erickson ’99 [5] ⋆ O ( n 17 / 11 + ǫ ) O ( n 2 log n ) Cacciola ’04 [3] O ( n 2 ) O ( n 2 log n ) Huber and Held ’10 [7] O ( n ) Vigneron and Yan ’13 [8] ⋆⋆ O ( n 4 / 3 + ǫ ) O ( n ) r Popular approach: Simulate the wavefront propagation. r Main Problem: Identify next event. r Edge events are cheap. Split events are expensive. Can we do better for specific input classes? Y ES , FOR ( STRICTLY ) MONOTONE POLYGONS . 4
C ONSTRUCTING THE S TRAIGHT S KELETON Algorithm time space O ( n 3 log n ) Aichholzer, Aurenhammer ’98 [1] O ( n ) Eppstein, Erickson ’99 [5] ⋆ O ( n 17 / 11 + ǫ ) O ( n 2 log n ) Cacciola ’04 [3] O ( n 2 ) O ( n 2 log n ) Huber and Held ’10 [7] O ( n ) Vigneron and Yan ’13 [8] ⋆⋆ O ( n 4 / 3 + ǫ ) O ( n ) r Popular approach: Simulate the wavefront propagation. r Main Problem: Identify next event. r Edge events are cheap. Split events are expensive. Can we do better for specific input classes? Y ES , FOR ( STRICTLY ) MONOTONE POLYGONS . 4
M ONOTONE P OLYGONS r Strictly monotone chain C (monotone w.r.t. to a line m ): Polygonal chain that intersects normals of m in at most one point. r Strictly monotone polygon P (monotone w.r.t. to a line m ): Simple polygon that can be split into two strictly monotone chains. C n P C s m 5
M ONOTONE P OLYGONS - P RIOR W ORK Das et al. claim O ( n log n ) time algorithm [4] : r Requires general position. r Correctness? 6
L EMMATA I r The wavefront propagation of strictly monotone polygonal chain C changes only when edges collapse. r In particular, the wavefront never splits into parts. r Consequence: We can construct S ( C ) in O ( n log n ) time. 7
L EMMATA I r The wavefront propagation of strictly monotone polygonal chain C changes only when edges collapse. r In particular, the wavefront never splits into parts. r Consequence: We can construct S ( C ) in O ( n log n ) time. 7
L EMMATA I r The wavefront propagation of strictly monotone polygonal chain C changes only when edges collapse. r In particular, the wavefront never splits into parts. r Consequence: We can construct S ( C ) in O ( n log n ) time. 7
L EMMATA I r The wavefront propagation of strictly monotone polygonal chain C changes only when edges collapse. r In particular, the wavefront never splits into parts. r Consequence: We can construct S ( C ) in O ( n log n ) time. 7
L EMMATA I r The wavefront propagation of strictly monotone polygonal chain C changes only when edges collapse. r In particular, the wavefront never splits into parts. r Consequence: We can construct S ( C ) in O ( n log n ) time. 7
L EMMATA I r The wavefront propagation of strictly monotone polygonal chain C changes only when edges collapse. r In particular, the wavefront never splits into parts. r Consequence: We can construct S ( C ) in O ( n log n ) time. 7
L EMMATA I r The wavefront propagation of strictly monotone polygonal chain C changes only when edges collapse. r In particular, the wavefront never splits into parts. r Consequence: We can construct S ( C ) in O ( n log n ) time. 7
L EMMATA I r The wavefront propagation of strictly monotone polygonal chain C changes only when edges collapse. r In particular, the wavefront never splits into parts. r Consequence: We can construct S ( C ) in O ( n log n ) time. 7
L EMMATA I r The wavefront propagation of strictly monotone polygonal chain C changes only when edges collapse. r In particular, the wavefront never splits into parts. r Consequence: We can construct S ( C ) in O ( n log n ) time. 7
L EMMATA II r In S ( P ) , a unique chain M of arcs connects west to east. r S ( P ) north of M is not influenced by the south chain. r S ( P ) north of M is identical to S ( C n ) between C n and M . r Given C n , C s , S ( C n ) and S ( C s ) , we can find M in time O ( n log n ) . r S ( P ) comprises M and parts of S ( C n ) and S ( C s ) 8
L EMMATA II r In S ( P ) , a unique chain M of arcs connects west to east. r S ( P ) north of M is not influenced by the south chain. r S ( P ) north of M is identical to S ( C n ) between C n and M . r Given C n , C s , S ( C n ) and S ( C s ) , we can find M in time O ( n log n ) . r S ( P ) comprises M and parts of S ( C n ) and S ( C s ) 8
L EMMATA II r In S ( P ) , a unique chain M of arcs connects west to east. r S ( P ) north of M is not influenced by the south chain. r S ( P ) north of M is identical to S ( C n ) between C n and M . r Given C n , C s , S ( C n ) and S ( C s ) , we can find M in time O ( n log n ) . r S ( P ) comprises M and parts of S ( C n ) and S ( C s ) 8
L EMMATA II r In S ( P ) , a unique chain M of arcs connects west to east. r S ( P ) north of M is not influenced by the south chain. r S ( P ) north of M is identical to S ( C n ) between C n and M . r Given C n , C s , S ( C n ) and S ( C s ) , we can find M in time O ( n log n ) . r S ( P ) comprises M and parts of S ( C n ) and S ( C s ) 8

Download Presentation

Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

More recommend