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
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