In crem en tal an d Decrem en tal Main ten an ce of Plan ar Width - PDF document

In crem en tal an d Decrem en tal Main ten an ce of Plan ar Width David Eppstein Dept. In form ation & Com puter Scien ce Un iv. of Californ ia, Irvin e 1

Width & Diam eter Any con vex body has two tan gen ts for each possible slope Map slope → distan ce between tan gen ts Width = m in distan ce Diam eter = m ax distan ce If in put is n on con vex, use its con vex hull 2

Why Width? Stable grip for parallel jaw gripper (for parts orien tation or fixturin g) Statistics: L ∞ regression (m in im ize m ax distan ce from data to estim ator) 3

Old Results Static O ( n log n ) [Preparata & Sham os 1985] Dyn am ic Average case for ran dom update sequen ce O ( log n ) [Eppstein , 1996] Con stan t factor approxim ation O ( log 2 n ) [Jan ardan ; Rote, Schwarz, Sn oeyin k 1993] Offlin e, decision problem on ly O ( log 3 n ) [Agarwal an d Sharir 1991] Diam eter (but n ot width) O ( n ǫ ) [Agarwal, Eppstein , Matouˇ sek 1995] This paper: partially exten d diam eter result to width 4

New Results Width of a dyn am ic poin t set in tim e O ( kn ǫ ) per in sertion or deletion where k = n um ber of chan ges to con vex hull For in sertion s on ly (in crem en tal) or deletion s on ly (decrem en tal) total hull chan ge = O ( n ) so am ortized tim e/ update = O ( n ǫ ) 5

Main Idea Main tain set of con vex hull features (edges an d vertices) [Overm ars an d van Leeuwen 1981] Defin e distan ce between features such that closest pair = width Build data structures for n earest n eighbor queries Con vert n earest n eighbors ⇒ closest pair 6

Hull Feature Com patibility A vertex an d edge are com patible if som e tan gen t at the vertex has sam e slope as the edge Each edge has ≤ 2 com patible vertices but vertex m ay have m any com patible edges Previous average-case algorithm : m ain tain graph of all com patible pairs but can have Ω ( n ) chan ges in worst case 7

Hull Feature Distan ce Defin e distan ce between com patible features to be Euclidean distan ce from poin t to lin e But, defin e distan ce between in com patible features to be in fin ite Lemma: The width of a poin t set equals the m in im um distan ce between any two features So width becom es a closest pair problem Solve by reducin g closest pairs to n earest n eighbors 8

Nearest Com patible Feature Fin d n earest vertex to edge Bin ary search in hull for com patible vertex (If m ore than on e, both are equally close) Fin d n earest edge to vertex Bin ary search for the com patible edges But what if there are m any of them ? Use BB [ α ] tree to represen t com patible edges as un ion of O ( log n ) can on ical sets Search for n earest in each can on ical set 9

Nearest Com patible Feature (con tin ued) Given can on ical set of edges, all com patible with query vertex, which is the closest? Cut each hull edge by equal slope plan es in R 3 , then distan ce to edge = vertical distan ce to plan e Query becom es vertical ray shootin g! 10

Vertical Ray Shootin g Given set of plan es above query poin t Which plan e has sm allest vertical distan ce to query? Static: Fin d in tersection of halfspaces Build poin t location data structure O ( log n ) per query Dyn am ic: O ( n ǫ ) per update or query [Agarwal an d Matouˇ sek 1995] We n eed the dyn am ic version 11

Closest Pair from Nearest Neighbors Theorem: If we can perform dyn am ic n earest n eighbor queries in tim e T ( n ) per update or query, then we can m ain tain closest pairs in T ( n ) log 2 n per update. [Eppstein 1995] Even easier m ethod (n ot in proceedin gs): Each vertex rem em bers n earest com patible edge When an update chan ges an edge, use n eighbor query to fin d affected vertex For each n ew or affected vertex, look up n ew n earest edge 12

Con clusion s Reduce width → vertical ray shootin g O ( log n ) ray shootin g operation s per hull chan ge so O ( n ǫ ) tim e per hull chan ge For in crem en tal or decrem en tal width, O ( n ǫ ) tim e per poin t in sertion or deletion What about fully dyn am ic width? Worst case in stead of am ortized tim e? Gen eralization s to other problem s e.g. to m in m ax distan ce problem s? 13