UNIONS OF ONIONS Maarten L offler Wolfgang Mulzer Universiteit - PowerPoint PPT Presentation

RECONSTRUCTION ALGORITHM First, locate P in R and T . Remove any empty leaves of T . Then, recursively unite the onions. P 8

RECONSTRUCTION ALGORITHM First, locate P in R and T . Remove any empty leaves of T . Then, recursively unite the onions. P 8

RECONSTRUCTION ALGORITHM First, locate P in R and T . Remove any empty leaves of T . Then, recursively unite the onions. Done! P 8

BUT! 9

BUT! ...how do you unite two onions? 9

BUT! ...how do you unite two onions? 9

BUT! ...how do you unite two onions? OBSERVATION: We can find the convex hull of two onions in O (log n ) time. 9

BUT! ...how do you unite two onions? OBSERVATION: We can find the convex hull of two onions in O (log n ) time. 9

BUT! ...how do you unite two onions? OBSERVATION: We can find the convex hull of two onions in O (log n ) time. Let’s remove it and recurse. 9

BUT! ...how do you unite two onions? OBSERVATION: We can find the convex hull of two onions in O (log n ) time. Let’s remove it and recurse. 9

BUT! ...how do you unite two onions? OBSERVATION: We can find the convex hull of two onions in O (log n ) time. Let’s remove it and recurse. Unfortunately, what is left are no longer proper onions... 9

ONION RESTAURATION & UNIFICATION 10

ONION RESTAURATION & UNIFICATION Compute convex hulls of the individual onions. 10

ONION RESTAURATION & UNIFICATION Compute convex hulls of the individual onions. 10

ONION RESTAURATION & UNIFICATION Compute convex hulls of the individual onions. 10

ONION RESTAURATION & UNIFICATION Compute convex hulls of the individual onions. Recurse once again. 10

ONION RESTAURATION & UNIFICATION Compute convex hulls of the individual onions. Recurse once again. 10

ONION RESTAURATION & UNIFICATION Compute convex hulls of the individual onions. Recurse once again. 10

ONION RESTAURATION & UNIFICATION Compute convex hulls of the individual onions. Recurse once again. 10

ONION RESTAURATION & UNIFICATION Compute convex hulls of the individual onions. Recurse once again. 10

ONION RESTAURATION & UNIFICATION Compute convex hulls of the individual onions. Recurse once again. Etc... 11

ONION RESTAURATION & UNIFICATION Compute convex hulls of the individual onions. Recurse once again. Etc... 11

ONION RESTAURATION & UNIFICATION Compute convex hulls of the individual onions. Recurse once again. Etc... 11

ONION RESTAURATION & UNIFICATION Compute convex hulls of the individual onions. Recurse once again. Etc... 11

ONION RESTAURATION & UNIFICATION Compute convex hulls of the individual onions. Recurse once again. Etc... 11

ONION RESTAURATION & UNIFICATION Compute convex hulls of the individual onions. Recurse once again. Etc... 11

ONION RESTAURATION & UNIFICATION Compute convex hulls of the individual onions. Recurse once again. Etc... 11

ONION RESTAURATION & UNIFICATION Compute convex hulls of the individual onions. Recurse once again. Etc... 11

ONION RESTAURATION & UNIFICATION Compute convex hulls of the individual onions. Recurse once again. Etc... 11

ONION RESTAURATION & UNIFICATION Compute convex hulls of the individual onions. Recurse once again. Etc... 11

ONION RESTAURATION & UNIFICATION Compute convex hulls of the individual onions. Recurse once again. Etc... 11