Touring a Sequence of Polygons

Transcription

1 Tourng a Sequence of Polygon Mohe Dror Alon Efrat Anna Lubw Joe Mtchell Unverty of Arzona Unverty of Arzona Unverty of Waterloo Stony Brook Unverty

2 Tourng Polygon Problem Gven: a equence of convex polygon, a tart pont and a target pont t Fnd: a hortet path that tart at, vt the polygon n equence, and end at t P 3 P 1 P 2 t

3 Tourng Polygon Problem Gven: a equence of convex polygon, a tart pont and a target pont t Fnd: a hortet path that tart at, vt the polygon n equence, and end at t P 3 P 1 P 2 t

4 Tourng Polygon Problem Gven: a equence of convex polygon, a tart pont and a target pont t Fnd: a hortet path that tart at, vt the polygon n equence, and end at t P 3 - the path may be contraned by fence P 1 P 2 F 3 P 4 t'

5 Tourng Polygon Problem Gven: a equence of convex polygon, a tart pont and a target pont t Fnd: a hortet path that tart at, vt the polygon n equence, and end at t P 3 - the path may be contraned by fence P 1 P 2 F 3 P 4 t'

6 Tourng Polygon Problem Gven: a equence of convex polygon, a tart pont and a target pont t Fnd: a hortet path that tart at, vt the polygon n equence, and end at t P 3 - the path may be contraned by fence - only polygon facade mut be convex P 1 P 2 F 3 P 4 t'

7 Tourng Polygon Problem Gven: a equence of convex polygon, a tart pont and a target pont t Fnd: a hortet path that tart at, vt the polygon n equence, and end at t P 3 - the path may be contraned by fence - only polygon facade mut be convex P 1 P 2 P 4 F 3 - polygon may nterect t' P 5

8 Our Algorthm n = ze of polygon and fence k = number of polygon uncontraned Tourng Polygon Problem (TPP) wth djont convex polygon general TPP P 3 P 3 P 1 P 2 F 3 P 1 P 2 t P 5 P 4 t' 2 O(kn log n) O(k n log n) for fxed, hortet path quere take O(k log n + output-ze)

9 Applcaton: part cuttng

10 Applcaton: part cuttng TPP djont convex polygon no fence ("uncontraned")

11 Applcaton: part cuttng O(kn log n) k = number of polygon n = total ze

12 Applcaton: afar problem

13 Applcaton: afar problem TPP djont convex polygon polygon order wth fence

14 Applcaton: afar problem problem from Ntafo `92 3 O(n ) `92 2 O(n ) `94 O(n 3 ) Tan and Hrata `01 2 O(n log n)

15 Applcaton: zookeeper problem TPP djont convex polygon polygon order wth fence

16 Applcaton: zookeeper problem problem from Chn and Ntafo `92 O(n log n) Bepamyatnkh `02

17 Applcaton: watchman route problem

18 Applcaton: watchman route problem eental pocket TPP polygon = eental pocket polygon order wth fence

19 Applcaton: watchman route problem problem from Chn and Ntafo `91 4 O(n ) `91 O(n 4 ) Tan, Hrata, Inagak `99 3 O(n log n)

20 Idea of Algorthm: (1) Local Optmalty a path locally optmal f movng any one bend of the path doe not mprove t a

21 Idea of Algorthm: (1) Local Optmalty a path locally optmal f movng any one bend of the path doe not mprove t a b

22 Idea of Algorthm: (1) Local Optmalty a path locally optmal f movng any one bend of the path doe not mprove t a b

23 Idea of Algorthm: (1) Local Optmalty a path locally optmal f movng any one bend of the path doe not mprove t a

24 Idea of Algorthm: (1) Local Optmalty a path locally optmal f movng any one bend of the path doe not mprove t Theorem. Locally optmal = globally optmal for TPP. Proof: Theorem. A locally optmal path unque.

25 Idea of Algorthm: (2) Shortet Path Map hortet path map: dvde plane nto regon by combnatorc of hortet path u v

26 Idea of Algorthm: (2) Shortet Path Map hortet path map: dvde plane nto regon by combnatorc of hortet path, q q u v

27 Idea of Algorthm: (2) Shortet Path Map hortet path map: dvde plane nto regon by combnatorc of hortet path u v, v, q q

28 Idea of Algorthm: (2) Shortet Path Map hortet path map: dvde plane nto regon by combnatorc of hortet path u v q, (u,v), q

29 Idea of Algorthm: (2) Shortet Path Map hortet path map: dvde plane nto regon by combnatorc of hortet path u v

30 Idea of Algorthm: (2) Shortet Path Map hortet path map: dvde plane nto regon by combnatorc of hortet path q u v

31 Idea of Algorthm: (2) Shortet Path Map hortet path map: dvde plane nto regon by combnatorc of hortet path u v q

32 Shortet Path Map Theorem: The wort-cae complexty of the hortet path map k ((n k) 2 )

33 Shortet Path Map Theorem: The wort-cae complexty of the hortet path map O((n k) 2 k ). We can compute t n output entve tme, then do effcent quere. (For the zoo-keeper problem, the hortet path map ha polynomal ze.)

34 Shortet Path Map lat tep hortet path map: dvde plane nto regon by combnatorc of lat tep of hortet path x u v y

35 Shortet Path Map lat tep hortet path map: dvde plane nto regon by combnatorc of lat tep of hortet path x through u v y

36 Shortet Path Map lat tep hortet path map: dvde plane nto regon by combnatorc of lat tep of hortet path x u v bounce (x,y) y

37 Shortet Path Map anwerng quere ung the lat tep hortet path map x through u v q y

38 Shortet Path Map anwerng quere ung the lat tep hortet path map u v q

39 Shortet Path Map anwerng quere ung the lat tep hortet path map u v q

40 Shortet Path Map anwerng quere ung the lat tep hortet path map Example 2 x q q' u v bounce (x,y) y

41 Shortet Path Map anwerng quere ung the lat tep hortet path map Example 2 q' u v

42 Shortet Path Map anwerng quere ung the lat tep hortet path map Example 2 q q' u v

43 Shortet Path Map anwerng quere ung the lat tep hortet path map Example 2 q q' u v

44 Shortet Path Map Lemma: Ung lat tep hortet path map, we can anwer hortet path quere n O(k log n).

45 Idea of Algorthm: (1) Local Optmalty (2) Lat Step Shortet Path Map

46 Uncontraned TPP for djont convex polygon R 1 u T 1 v w T frt contact et of hortet path, P,..., P 1 1 R hortet path ray leavng P wth P

47 Uncontraned TPP for djont convex polygon R 2 x u v w y T 2 z T frt contact et of hortet path, P,..., P 1 1 R hortet path ray leavng P wth P

48 Uncontraned TPP for djont convex polygon R 2 x Structural Reult u v w y Lemma: T a chan on the boundary of P. T 2 z Lemma: R a tarburt.e. there a unque ray to every pont of the plane. Corollary: Locally hortet path are unque. T frt contact et of hortet path, P,..., P 1 wth P R hortet path ray leavng P 1

49 Uncontraned TPP for djont convex polygon u v w R 2 T frt contact et wth P R ray leavng P T 2 x y z Algorthm T 0 = for = 1.. k compute T and R for each vertex v of P fnd d -1 (v) f t arrve at v from outde P then v a vertex of T ue d -1 (v) to compute ray of R at v d (v) = hortet path, P 1,.., P, v

50 Uncontraned TPP for djont convex polygon Analy - hortet path query: O(k log n) - algorthm total: O(n k log n) Algorthm T 0 = for = 1.. k compute T and R for each vertex v of P fnd d -1 (v) f t arrve at v from outde P then v a vertex of T ue d -1 (v) to compute ray of R at v d (v) = hortet path, P 1,.., P, v

51 General TPP: local optmalty fence nterectng polygon

52 General TPP: local optmalty fence nterectng polygon

53 General TPP: local optmalty fence nterectng polygon

54 General TPP: local optmalty fence nterectng polygon

55 General TPP: fence R 1 u v w A 1 v

56 General TPP: nterectng polygon R 1 T 1 T frt contact et of hortet path, P,..., P 1 1 R hortet path ray leavng P wth P

57 General TPP: nterectng polygon T 2 x y R 2 T frt contact et of hortet path, P,..., P 1 1 R hortet path ray leavng P wth P

58 General TPP Structural Reult T 2 Lemma: T a tree. x y Lemma: R a tarburt.e. there a unque ray to every pont of the plane. R 2 Cor. Locally hortet path are unque. T frt contact et of hortet path, P,..., P 1 wth P R hortet path ray leavng P 1 A ray arrvng at P +1 after travellng through fence F

59 General TPP Algorthm T 2 x y T =, R = all ray from 0 0 A = ray nde F 0 0 for = 1.. k compute T, R, and A R 2 O(nk 2log n) T frt contact et wth P R ray leavng P A ray arrvng at P +1 d (v) = hortet path, P 1,.., P, v after travellng through fence F

60 Extenon remnder of TPP: Gven: a equence of pobly nterectng, convex [facade] polygon, a tart pont and a target pont t Fnd: a hortet path that tart at, vt the polygon n equence, repectng the fence, and end at t non-convex polygon Theorem. TPP NP-hard for non-convex polygon (even wthout fence). Proof. From 3-SAT, baed on a careful adaptaton of the Canny-Ref proof.

61 Extenon TPP a a 3-D hortet path problem. Thu there a fully polynomal tme approxmaton cheme (even for non-convex polygon).

62 Extenon TPP a a 3-D hortet path problem. t Thu there a fully polynomal tme approxmaton cheme (even for non-convex polygon).

63 Extenon TPP a a 3-D hortet path problem. t Thu there a fully polynomal tme approxmaton cheme (even for non-convex polygon).

64 Extenon Open. What the complexty of TPP for djont non-convex polygon.

65 The End