Touring a Sequence of Polygons Moshe Dror Alon Efrat Anna Lubiw Joe Mitchell

Transcription

1 Touring a Sequence of Polygons Moshe Dror Alon Efrat Anna Lubiw Joe Mitchell University of Arizona University of Waterloo Stony Brook University

2 Problem: Given a sequence of polygons in the plane, a start point, and a target point,, we seek a shortest path that starts at, visits in order each of the polygons, and ends at. s P3 P 4 P 1 P 2 t

3 Related Problem: TSPN: If the order to visit is not specified, we get the NP-hard TSP with Neighborhoods problem. TSPN:!"$#&%(' -approx in general )*' -approx, PTAS in special cases

4 The Fenced Problem: s F 1 P 1

5 The Fenced Problem: s F 1 P 1 F 2 P 2

6 The Fenced Problem: s F 1 P 1 P 3 F 2 P 2 F 3

7 The Fenced Problem: s F 1 F 4 P 4 P 1 P 3 F 2 P 2 F 3

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

9 Applications: Parts Cutting Problem: s 8 5 6

10 Applications: Safari Problem: s

11 Applications: Zookeeper Problem: s

12 Applications: Watchman Route Problem: Essential cut s

13 ' Summary of Results: Disjoint convex polygons: ),-%."$#/0%12'3' time, )0%(' space (For fixed, ) 4"5#60%17'3' shortest path queries to.) Arbitrary convex polygons: )0% "$#&%(' time, )0%7' space Full combinatorial map: worst-case size 8930%;: 7'3< Output-sensitive algorithm; ),>="$#&%(' -time shortest path queries. TPP for nonconvex polygons: NP-hard FPTAS, as special case of 3D shortst paths

14 ' + Applications: Safari: )0% "5#&%(' vs. )0% Watchman: )0% "$#?%(' vs. )0%A@B' floating watchman: vs. )0%DCE' We avoid use of complicated path adjustments arguments, DP Parts cutting: -%."$#/0%12'F'

15 Unconstrained TPP: Disjoint Convex Polygons: Given:,, sequence of disjoint convex polygons ' Goal: Find a shortest -path from HG JI to. Local Optimality Conditions:

16 Unconstrained TPP: Disjoint Convex Polygons: Lemma: For any K L and any M K ON P Q, R unique shortest M -path, S/TUWVX', from HG I to. Thus, local optimality is equivalent to global optimality. Lemma: In the TPP for disjoint convex polygons ', each first contact set Y T is a (connected) chain on Z[ T. Lemma: For any V K L and any M, there is a unique point V6\[K that S/TU]VX'^G S/T`_ WV \ 'ba V \ V. Y T such

17 General Approach: Build a Shortest Path Map: SPM $' : a decomposition of the plane into cells according to the combinatorial type of a shortest -path to t r r t s s Bad news: worst-case size can be huge: Theorem: The worst-case complexity of SPM $' is cd30%e: 7'3< '

18 s

19 s

20 2 i s 2 i+1

21 ' Good news: worst-case size cannot be bigger than huge : Theorem: The worst-case complexity of SPMf $' is )30%;: 7'3< Size g T satisfies g T(h <g Ti_ = )Ejk T jl'. Output-sensitive algorithm to build SPM: Theorem: One can compute SPM $' in time ),nmoj SPM $'Pjl', after which a shortest -path from to a query point can be computed in time ) >="5#&%('.

22 t p p Last Step Shortest Path Map: Y T G first contact set of T : points where a shortest 0MJ: *' -path first enters T after visiting B Ti_ For V K Y T : poq T ]Vr'^G set of rays of locally shortest M -paths going straight through V : a single ray pfs T ]Vr'^G set of rays of locally shortest M -paths properly reflecting at V a single ray (V interior to an edge of YXT ), or a cone (V a vertex of YXT ) T ]VX'^G p q T WVX'Aa pfs T ]Vr' T G aouwvyxbz T ]VX' (an infinite family of rays) is the starburst with source Y T

23 { { + + The Last Step Shortest Path Map: T G the last step shortest path map, subdivision according to the combinatorial type of the rays of t T passing through points V K L T decomposes the plane into cells of two types: (1) cones with an apex at a vertex } of Y T, whose bounding rays are reflection rays p \ 0}~' and p \ }~' } is the source of cell (2) unbounded 3-sided regions associated with edge of Y T, classified as reflection cells or pass-through cells is the source of cell The pass-through region is the union of all pass-through cells

24 Last Step Shortest Path Map: v 1 Pass through Region e 1 v 6 v 5 v 1v2 P 1 v 2 v 3 v 4 v 4 s e 2 e 3 v 3

25 P 1 v 8 e 8 e 7 s v 8 v 7 v 9 v 10 P 2 v 11 v 7 v 9 Pass through region

26 + + Using the Last Step Shortest Path Map: Find a shortest M -path to query point : Locate in { T [)!"$# jk T jl' ] cell rooted at vertex } of YXT :Aƒ last segment of S T ' is } recursively compute S Ti_ 0}2' (locate } in { Ti_, etc) cell rooted at edge of Y T is pass-through: S T ' G S T`_ ', so recursively compute shortest 0M: *y' -path to is a reflection cell: recursively compute shortest 0Mˆ: *' - path to 5\, the reflection of wrt Lemma: Given { { T, S T ' can be determined in time ), "$#Q0%12'F'

27 Algorithm: Construct each of the subdivisions { { { iteratively: For each vertex }Š of TW, we compute S T 0}ŠB'. If this path arrives at }Š from the inside of TW, then }Š is not a vertex of Y TW. Otherwise it is, and the last segment of S T 0}ŠŒ' determines the rays p T s 0}ŠŒ' and p T q 0} Š ' that define the subdivision { TW. Theorem: For a given sequence ' of disjoint convex polygons having a total of % vertices, a data structure of size )%o' can be constructed in time ),-%."$#/0%12'3' that enables shortest M -path queries to any query point to be answered in time 0M "$#/0%17'3'.

28 TPP for Fenced, Arbitrary Convex Polygons: Use Last Step Shortest Path Maps, but combinatorics and algorithm are substantially more complex. Thus, Alon gets to try to describe them...;-) Needed for Safari, Watchman Route, Zookeeper.

29 TPP on Nonconvex Polygons: Proposition: The TPP in the Ž metric is polynomially solvable (in )0% ' time and space) for arbitrary rectilinear polygons T and arbitrary fences T. The result lifts to any fixed dimension if the regions T and the constraining regions JT are orthohedral.

30 TPP on Nonconvex Polygons: Theorem: The touring polygons problem is NP-hard, for any Ž u metric (V * ), in the case of nonconvex polygons (T, even in the unconstrained ( T G L ) case with obstacles bounded by edges having angles 0, 45, or 90 degrees with respect to the -axis. Proof: from 3-SAT based on a careful adaptation of Canny-Reif proof

31 Splitter Gadgets: P 1 P 2 P 1 P 4 P 3 P 3 P

32 Shuffle Gadget: P 3 P P 1 P

33 Ganging Three Shuffle Gadgets:

34 Literal Filter: n 2 shortest path classes i 1 shufflers b = most significant bit i (n i+1) shufflers

35 Clause Filter: 3 way splitter 3 way splitter

36 Open Problem: What is the complexity of the TPP for disjoint non-convex simple polygons?