Touring a Sequence of Polygons

Transcription

1 Touring a Sequence of Polygon Mohe Dror (1) Alon Efrat (1) Anna Lubiw (2) Joe Mitchell (3) (1) Univerity of Arizona (2) Univerity of Waterloo (3) Stony Brook Univerity

2 Problem: Given a equence of k polygon in the plane, a tart point, and a target point, t, we eek a hortet path that tart at, viit in order each of the polygon, and end at t. P3 P 4 P 1 P 2 t

3 Related Problem: TSPN: If the order to viit {P 1, P 2,..., P k } i not pecified, we get the NP-hard TSP with Neighborhood problem. TSPN: O(log n)-approx in general O(1)-approx, PTAS in pecial cae

4 The Fenced Problem: Here that part of the path connecting P i to P i+1 mut lie inide a a imple polygon F i, called the fence. F 1 P 1

5 The Fenced Problem: F 1 P 1

6 The Fenced Problem: F 1 P 1 F 2 P 2

7 The Fenced Problem: F 1 P 1 P 3 F 2 P 2 F 3

8 The Fenced Problem: F 1 F 4 P 4 P 1 P 3 F 2 P 2 F 3

9 The Fenced Problem: F 1 F 4 P 4 P 1 P 3 F 5 F 2 P 2 t F 3

10 Application: Part Cutting Problem:

11 Application: Safari Problem:

12 Application: Zookeeper Problem:

13 Application: Watchman Route Problem: Eential cut Fact: The optimal path viit the eential cut in the order they appear along P.

14 Summary of Reult: Dijoint convex polygon: O(kn log(n/k)) time, O(n) pace (For fixed, {P 1, P 2,..., P k }, O(k log(n/k)) hortet path querie to t.) Arbitrary convex polygon: O(nk 2 log n) time, O(nk) pace Full combinatorial map: wort-cae ize Θ((n k)2 k ) Output-enitive algorithm; O(k + log n)-time hortet path querie. TPP for nonconvex polygon: NP-hard FPTAS, a pecial cae of 3D hortt path

15 Application: Safari: O(n 2 log n) v. O(n 3 ) Watchman: O(n 3 log n) v. O(n 4 ) floating watchman: O(n 4 log n) v. O(n 5 ) We avoid ue of complicated path adjutment argument, DP Part cutting: O(kn log(n/k))

16 Uncontrained TPP: Dijoint Convex Polygon: Given:, t, equence of dijoint convex polygon {P 1,..., P k } Goal: Find a hortet k-path from = P 0 to t. Local Optimality Condition: Aume that optimal path bounce from P i. Can bounce from an edge (incoming angle = bouncing angle) or bounce from a vertex.

17 Uncontrained TPP: Dijoint Convex Polygon: Lemma: For any t R 2 and any i {0,..., k}, unique hortet i-path, π i (p), from = P 0 to t. Thu, local optimality i equivalent to global optimality. Lemma: In the TPP for dijoint convex polygon {P 1,..., P k }, each firt contact et T i i a (connected) chain on P i. Lemma: For any p R 2 and any i, there i a unique point p T i uch that π i (p) = π i 1 (p ) p p.

18 General Approach: Build a Shortet Path Map: SPM k (): a decompoition of the plane into cell according to the combinatorial type of a hortet k-path to t t r r t Bad new: wort-cae ize can be huge: Theorem: The wort-cae complexity of SPM k () i Ω((n k)2 k )

19

20

21 2 i 2 i+1

22 Good new: wort-cae ize cannot be bigger than huge : Theorem: The wort-cae complexity of SPM k () i O((n k)2 k ) Size m i atifie m i 2m i 1 + O( P i ). Output-enitive algorithm to build SPM: Theorem: One can compute SPM k () in time O(k SPM k () ), after which a hortet k-path from to a query point t can be computed in time O(k + log n).

23 Lat Step Shortet Path Map: T i = firt contact et of P i : point where a hortet (i 1)-path firt enter P i after viiting P 1,..., P i 1 For p T i : r i (p) = et of ray of locally hortet i-path going traight through p: a ingle ray r b i(p) = et of ray of locally hortet i-path properly reflecting at p a ingle ray (p interior to an edge of T i ), or a cone (p a vertex of T i ) r i (p) = r i (p) rb i (p) R i = p Ti r i (p) (an infinite family of ray) i the tarburt with ource T i

24 The Lat Step Shortet Path Map: S i = the lat tep hortet path map, ubdiviion according to the combinatorial type of the ray of R i paing through point p R 2 S i decompoe the plane into cell σ of two type: (1) cone with an apex at a vertex v of T i, whoe bounding ray are reflection ray r 1(v) and r 2(v) v i the ource of cell σ (2) unbounded 3-ided region aociated with edge e of T i, claified a reflection cell or pa-through cell e i the ource of cell σ The pa-through region i the union of all pa-through cell

25 Lat Step Shortet Path Map: v 1 Pa through Region e 1 v 6 v 5 v 1v2 P 1 v 2 v 3 v 4 v 4 e 2 e 3 v 3

26 P 1 v 8 e 8 e 7 v 8 v 7 v 9 v 10 P 2 v 11 v 7 v 9 Pa through region

27 Uing the Lat Step Shortet Path Map: Find a hortet i-path to query point q: Locate q in S i [O(log P i )] cell σ rooted at vertex v of T i lat egment of π i (q) i vq recurively compute π i 1 (v) (locate v in S i 1, etc) cell σ rooted at edge e of T i σ i pa-through: π i (q) = π i 1 (q), o recurively compute hortet (i 1)-path to q σ i a reflection cell: recurively compute hortet (i 1)- path to q, the reflection of q wrt e Lemma: Given S 1,..., S i, π i (q) can be determined in time O(k log(n/k))

28 Algorithm: Contruct each of the ubdiviion S 1, S 2,..., S k iteratively: For each vertex v j of P i+1, we compute π i (v j ). If thi path arrive at v j from the inide of P i+1, then v j i not a vertex of T i+1. Otherwie it i, and the lat egment of π i (v j ) determine the ray r b i(v j ) and r i (v j ) that define the ubdiviion S i+1. Theorem: For a given equence {P 1,..., P k } of k dijoint convex polygon having a total of n vertice, a data tructure of ize O(n) can be contructed in time O(kn log(n/k)) that enable hortet i-path querie to any query point q to be anwered in time O(i log(n/k)).

29 TPP on Nonconvex Polygon: Propoition: The TPP in the L 1 metric i polynomially olvable (in O(n 2 ) time and pace) for arbitrary rectilinear polygon P i and arbitrary fence F i. The reult lift to any fixed dimenion d if the region P i and the contraining region F i are orthohedral. TPP on Nonconvex Polygon: Theorem: The touring polygon problem (TPP) i NP-hard, for any L p metric (p 1), in the cae of nonconvex polygon P i, even in the uncontrained (F i = R 2 ) cae with obtacle bounded by edge having angle 0, 45, or 90 degree with repect to the x-axi. Open Problem: What i the complexity of the TPP for dijoint non-convex imple polygon?