5. Properties of Abstract Voronoi Diagrams

Transcription

1 5. Poeties of Abstact Voonoi Diagams 5.1 Euclidean Voonoi Diagams Voonoi Diagam: Given a set S of n oint sites in the lane, the Voonoi diagam V (S) of S is a lana subdivision such that Each site S is assigned a Voonoi egion denoted by VR(, S) All oints in VR(, S) shae the same neaest site in S Voonoi Edge: The common bounday between two adjacent Voonoi egions, VR(, S) and VR(, S), i.e., VR(, S) VR(, S),is called a Voonoi edge. Voonoi Vetex: The common vetex among moe than two Voonoi egions is called a Voonoi vetex.

2 The Euclidean Voonoi diagam can be comuted in O(n log n) time Line Segment Voonoi Diagam Cicle Voonoi Diagam Voonoi Diagam in the L 1 metic

3 5.2 Bisecting Systems Fo two sites, S, the bisecto J(, ) between and is defined as {x R 2 d(x, ) = d(x, ) J(, ) atitions the lane into two half-lanes D(, ) = {x R 2 d(x, ) < d(x, )} D(, ) = {x R 2 d(x, ) < d(x, )} VR(, S) = S\{} D(, ) V (, S) = R 2 \ S VR(, S) consists of Voonoi edges. 5.3 Abstact Voonoi Diagams A unifying aoach to comuting Voonoi diagams among diffeent geometic sites unde diffeent distance measues. A bisecting system J = {J(, ), S} fo a set S of sites (indices) A bisecting system J is admissible if J satisfies the following axioms (A1) Each bisecting cuve in J is homeomohic to a line (not closed) (A2) Fo each non-emty subset S of S and fo each S, VR(, S ) is ath-connected. (A3) Fo each non-emty subset S, R 2 = S VR(, S ) (A4) Any two cuves in J have only finitely many intesection oints, and these intesections ae tansvesal.

4 (A1) can be witten as Each cuve in J is unbounded. Afte steeogahic ojection to the shee, it can be comleted to a closed Jodan cuve though the noth ole. (A4) can be emoved though seveal comlicated oofs. Not Tavesal Tavesal Not Admissible Non-Man Land Disconnected

5 Thee ossibilities of an admissible system fo thee sites Abstact Voonoi Diagams A categoy of Voonoi diagams oints in any convex distance function Kalsuhe metic Line segments and convex olygons of constant size 5.3 Basic Poeties Lemma 1 Let (S, J ) be a bisecting cuve system. The the following assetions ae euivalent. 1. If,, and ae aiwise diffeent sites in S, then D(, ) D(, ) D(, ) (Tansitivity) 2. Fo each nonemty subset S S, R 2 = s VR(, S )

6 Poof: (2) (1) Let z be a oint in D(, ) D(, ). By (2), thee must be a site t S = {,, } such that z VR(t, S ). If t =, z VR(, S ) D(, ); othewise z VR(, S ) D(, ), contadicting z D(, ) z VR(, S ) D(, ), contadicting z D(, ) (1) (2) By induction on S. If S = 2, the assetion is immediate. The case whee S = 3 follows diectly fom (1) Let z be a oint in the lane. By induction hyothesis, to each S, thee exists a site c() such that z VR(c(), S \ {}) case 1: Thee exists v w such that c(v) = c(w). Then z VR(c(v), S \ {v}) VR(c(v), S \ {w} VR(c(v), S \ {v} D(c(v), v) = VR(c(v), S ) case 2 The maing c is injective. Let, v, w be scuh that {, c(), v, w} = 4. Since c(v) c(w), one of them is diffeent. We assume c(v) is diffeent fom. Since c(v) c() we obtain the contadiction: z VR(c(), S \ {}) D(c(), c(v)) z VR(c(v), S \ {v}) D(c(v), c())

7 Theoem A bisecting cuve system (S, J ) is admissible if and only if the following conditions ae fulfilled. 1. D(, ) D(, ) D(, ) holds fo any thee sites,,, in S 2. Any two cuve J(, ) and J(, ) coss at most twice and do not constitude a clockwise cycle in the lane oof By Lemma 1, concentate on the connectedness of Voonoi egions. Conside an infinitely lage bounded cuve Γ which contains all intesections among cuves in J Fo any,, S, V ({,, }) encicled by Γ is a lana gah with exacyly 4 faces each of whose vetices is of degee at least 3. By the Eule Fomula, the lana gah gas at most 4 vetices Since at least two edges of the oiginal diagam tend to infinity, two vetices must be situated in Γ. J(, ) and J(, ) coss at most twice since each intesection between them is a Voonoi vetex by definition. A simle case analysis shows no clockwise cycle aising fom J(, ) and J(, ) disconnected egion

8 The case analysis shows that fo any 3-element subset S of S, all Voonoi egions in V (S ) is connected. We ove by induction on m: If R = VR(, {, 1, 2,..., m }) is connected, then R D(, m+1 ) = VR(, {, 1, 2,..., m+1 }) is connected. Let J(, ) be oiented such that D(, ) is on its left side. Assume the contay that R D(, m+1 ) wee not connected. If R D(, m+1 ) is bounnded, C be R and J(, m+1 ) would fom a clockwise cycle. Fo i m, J(, i ) and J(, m+1 ) fom a clockwise cycle. Thee exists a contadiction Othewise, we intesect R with the inne domain of Γ, and C be its contou. The same easoning alies to C and J(, m+1 ) C m+1 i