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Touring a Sequence of Polygons

Touring a Sequence of Polygons. Moshe Dror, Alon Efrat, Anna Lubiw, Joseph S. B. Michel STOC’03. Outline. Introduction Unconstrained TPP for disjointed convex polygons General TPP algorithm TPP on nonconvex polygons Open problems. Problem.

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Touring a Sequence of Polygons

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  1. Touring a Sequence of Polygons Moshe Dror, Alon Efrat, Anna Lubiw, Joseph S. B. Michel STOC’03

  2. Outline • Introduction • Unconstrained TPP for disjointed convex polygons • General TPP algorithm • TPP on nonconvex polygons • Open problems

  3. Problem • Given a sequence of polygons in the plane, a start point s, and a target point t, we seek a shortest path that starts at s, visits in order each of the polygons, and ends at t.

  4. Related problem • If the order to visit {P1, P2, …, Pk}is not specified, we get the NP-hard TSP with Neighborhoods problem.

  5. The fenced problem

  6. Applications: Parts Cutting Problem O(knlog(n/k))

  7. Applications: Safari Problem

  8. Applications: Zookeeper Problem

  9. Previous result • Zookeeper: O(nlogn),Bespamyatnikh, ’02 • Safari: O(n3), Tan et al, ’01 • A shortest tour must visit Pi’s in the same order as they meet the boundary of P Safari: O(knlogn)

  10. Application: The Watchman Route Problem O(n4) Tan’99 O(n3logn)

  11. The unconstrained TPP for disjoint convex polygons • Unconstrained : no fence • “Disjoint”

  12. Idea of Algorithm: Local Optimality • a path is locally optimal if moving any one bend of the path does not improve it

  13. Idea of Algorithm: Local Optimality • Theorem. Locally optimal  globally optimal

  14. Idea of Algorithm: last step shortest path map • Given s, divide plane into regions by combinatorics of shortest path, Si, by Ti

  15. v’=r(q) πi(q)=πi-1(q) v’=q v’=v7 Queryi(q) for πi(q) • πi(q): thelast segment for tour π(q): sP1 … Pi q • Locate q in Si • Recursive call Queryi-1(v’) by different cases

  16. Complexity of Queryi(q) • Given S1, …, Sk, the path πi(q) can be determined in time O(klog(n/k)) • Maximum when each |Pi|=n/k

  17. For i = 1 to k //Construct the subdivision of plane Si, for polygon Pi For every vertex of Pi, Queryi-1(v) to get πi-1(v) To determine the subdivision by vertex v Construct S1,…, Sk

  18. Theorom • For the unconstrained TPP for disjoint convex polygons with input size n, a data structure of size O(n) can be built in time O(kn(log(n/k))) that enables shortest i-path queries to any query point q to be answered in O(ilog(n/k))

  19. fences Intersecting polygons The General TPP algorithm

  20. Ti: first contact set of shortest paths s, P1 , . . . , Pi-1with Pi Ri :shortest path rays leaving Pi Theorom Ti is a tree Ri is a starburst General TPP

  21. Last step shortest path map • Several maps • : by Ti, Ri

  22. Last step shortest path map • Several maps • : by Ti, Ri, and Fi

  23. Last step shortest path map • Several maps • : by rays that arrive at Pi+1 after traveling through fence Fi

  24. Query(q, SiX) • Case 1. q is in a cell whose root is a vertex v ∈ Fior a vertex v ∈ Ti. • Case 2. q is in the “pass-through” region. • Case 3. q is in a cell whose root is an edge e of Ti.

  25. Construction

  26. Theorem • The TPP for arbitrary convex polygons Pi and fences Fi can be solved in time O(nk2 log n), using O(nk) space.

  27. TPP on Nonconvex polygons • Theorem. TPP is NP hard for non-convex polygons (even without fences) • From 3-SAT, based on a careful adaptation of the Canny-Reif proof.

  28. Open problems • TPP for disjoint noncovex simple polygons

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