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On the Problem of Computing Zookeeper Routes

This paper discusses the Zookeeper Route Problem, a computational challenge in finding the shortest path within a simple polygon (zoo) that contains disjoint convex polygons (cages). It explores two main algorithms: the exact solution by Chin and Ntafos with O(n²) complexity and an approximate solution by Jonsson with O(n) complexity. The study delves into the reflection principle for determining optimal routes and presents various observations regarding the properties of routes as cage configurations change. The implications of these findings for route optimization in defined geometrical constraints are examined.

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On the Problem of Computing Zookeeper Routes

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  1. On the Problem of Computing Zookeeper Routes

  2. Hakan Jonsson & Sofia Sundberg 2004.07.01 ISSN 1402-1528 / ISRN LTU-FR--04/10--SE / NR 2004:10

  3. What’s Zookeeper’s Problem • Introduced by Chin & Ntafos • 1.A simple polygon(zoo) with a disjoint set of k convex polygons(cage) • 2.Every cage shares one edge with the zoo • 3.Find shostest route in the interior of the zoo without cross any cages.

  4. Ex

  5. FIXED • The route is forced to pass through a start point-s and s on the boundary of the zoo. • If zookeeper’s is non-fixed,it’s NP-hard

  6. Def • Z：the zoo,a simple polygon and remain the edge of cages • K：the number of cages • N： the size of zoo • P：all the simple polygon • Zopt： the shortest zookeeper route • Zapp： the approximation zookeeper route

  7. Def2 Zopt： the shortest zookeeper route Zc： the common part of Zopt and Zapp Zo,Za： the unique part of Zopt and Zapp

  8. The Algorithms • Chin & Ntafos：O(n^2) exact solution • Jonsson：O(n) approximate solution • P contain a set C of k edges denoted C1,C2,….Ck,and start point s on the boundary but not in any cage

  9. Exact Algorithms • They use the Reflection Principle,from a mirror to dash-b to find the shortest path (a to b) We unfolding it into an hourglass,then after adjustment ,can get the Zopt, it cost O(n)

  10. Reflection Principle

  11. Approximate solution-1 • Jonsson use a simpler approach,during a clockwise traversal of the boundary of the zoo,we gives each cage a unique first and last vertex • supporting chain is shortest path connect the first and last vertices of two consecutive cages.

  12. Approximate solution-2 • For each cage Ci has one supporting chain Si ,if two supporting intersect we give a signpost for the cage • The touch point of a cage Ci is the point on the boundary of the cage that lies closest to the sign post of the cage.

  13. Signpost

  14. Properties of zookeeper • If we chose different vertex of the cage, we will get the different length with other route. • Obstacle • Changing the Zoo

  15. Differences between cages

  16. Obstacle Obervation 1：If /(Za/Zc)/ is a constant, When /Zc/ increases,then (/Za/+ /Zc/)/ (/Zo/+ /Zc/) decreases. Geometric terms：to achieve a worst case for (/Za/+ /Zc/)/ (/Zo/+ /Zc/) , it should probably be a minimum of common parts between the routes and the zoo.

  17. Changing the Zoo • Obervation 2：Touch point ti of Zapp on a given cage Ci is unaffected by changes in cages other then Ci-1,Ci,Ci+1 • Obervation 3： Zopt remains that same as all tangents li are not changed

  18. Observed worst case-1

  19. Observed worst case-2 • DEF：A isoceles zoo is a zoo, with starting point and two cages, as an isosceles triangle with height h and top a • Lemma 1：In a isoceles zoo the Zapp is • Dapp(a,h) = {2hsina 0<a< /2 • 2h  /2<a< 

  20. Observed worst case-3 Lemma 2：In an isosceles zoo the length of Zapp is Dapp(a,h) =[2hsin(a/2)][(1+cos(a/2))] Lemma 3： In an isosceles zoo the quotient q(a,h)=Zapp/Zopt=Dapp/Dopt is maximized to for a = 2/3 and any h.

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