On the Problem of Computing Zookeeper
Routes
Hakan Jonsson & Sofia Sundberg
2004.07.01
ISSN 1402-1528 / ISRN LTU-FR--04/10--SE / NR 2004:10
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.
Ex
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
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
Def2
Zopt : the shortest zookeeper route
Zc : the common part of Zopt and Zapp
Zo,Za : the unique part of Zopt and Zapp
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
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)
Reflection Principle
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.
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.
Signpost
Properties of zookeeper
• If we chose different vertex of the cage, we will get the different length with other route.
• Obstacle
• Changing the Zoo
Differences between cages
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.
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
Observed worst case-1
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<
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.