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.