Traversing the Machining Graph

Danny Chen, Notre Dame

Rudolf Fleischer, Li Jian,

Wang Haitao,Zhu Hong, Fudan

Sep,2006

2D-Milling

Example[Arkin,Held,Smith’00]

Zigzag machining

Example[Tang,Joneja’03]:

Example[Tang,Joneja’03]:

The Model

The Model

The Model

We are stuck

Non-compulsory edge

(be traversed at most once)

Compulsory edge

(be traversed exactly once)

The Model

We are stuck: jump

The Model

Goal: minimize jumps

Greedy?

Greedy?

Greedy?

Greedy?

2 jumps

Greedy?

Greedy?

Greedy?

2 jumps

Greedy?

Greedy?

1 jump

Greedy?

1 jump

Greedy?

2 jumps

Greedy?

1 jump

Greedy?

1 jump

Greedy?

Greedy?

Greedy?

no jump

Greedy? May be exponential

What is Known

Simple polygon:NP-hard?Some heuristics [Held’91,

Tang,Chou,Chen’98]

Polygon with h holes:NP-hard

[Arkin,Held,Smith’00]5OPT+6h jumps [AHS’00]Opt+h+N jumps [Tang,Joneja’03]

What we Show

Simple polygon:NP-hard? No, linear time (DP)Some heuristics [Held’91,

Tang,Chou,Chen’98]

Polygon with h holes:NP-hard [Arkin,Held,Smith’00]5OPT+6h jumps [AHS’00]Opt+h+N jumps [Tang,Joneja’03]OPT+εh jumps in polynomial timeOpt jumps in linear+O(1)O(h) time (DP)

lemma

Lemma [Arkin,Held,Smith’00]: There exists a optimal solution s.t.

(1) every path starts and ends with compulsory edges.

(2) No two non-compulsory edges are traversed consecutively. (alternating lemma)

Simple Pocket: The Dual Tree

Simple Pocket:Dynamic Programming

start at the leaves

Simple Pocket:Dynamic Programming

Dynamic Programming

Does path end here? 5 cases

constant time per node

Polygon with h Holes

time O(n)+O(1)O(h)

Polygon with h Holes Identify O(h) pivotal nodes.

Polygon with h Holes

Using arbitrary strategy to cut all the cycles gives a (O(1)^O(h))*O(n) algorithms.

Identify O(h) pivotal node whose removal s.t. 1.break all cycles.

2.each remaining (dual) tree is adjacent to O(1) pivotal nodes.

Then, we can do it in (O(1)^O(h))+O(n) time.

Polygon with h Holes:Boundary graph

Polygon with h Holes:Minimum Restrict Path Cover

Boundary graphOriginal Pocket

Forbidden pairs:

(e_1,e_4) and (e_2,e_3)

e_1 e_2

e_4 e_3

A valid path: no forbidden pairs appear in one path.

MRPC: find min # valid paths cover all vertices.

Polygon with h Holes:Minimum Restrict Path Cover

Graph with Bounded Tree Width

(informal)

Polygon with h Holes:Minimum Restrict Path Cover

Tree

Graph with bound treewidth

O(1) communicatons

1 communicaton

Polygon with h Holes:Minimum Restrict Path

Cover(MRPC) It turns out MRPC can be solved in linear

time by dynamic programming if the boundary graph has bounded treewidth.

(assume its tree-decomposition is given)

Remark: If tree-decomposition is not given, find 3-approximation to treewidth in time O(n log n). [Reed’92]

Polygon with h Holes:

k-outerplanar graph:

Polygon with h Holes:

k-outerplanar graph:

Peel off the outer layer

Polygon with h Holes:

k-outerplanar graph:

Peel off the outer layer

Peel again

Polygon with h Holes:

k-outerplanar graph:

Theorem: if a graph is k-outerplanar, it has treewidth 3k-1 . [Bodlaender’88]

Peel off the outer layer

Peel again

Peel again

--nothing left…

A 3-outplanar graph

Polygon with h Holes

Lemma:(1) If dual graph has a bounded treewidth and bounded degree, its corresponding boundary graph has bounded treewidth.

(2) If dual graph is a k-outplanar graph, its corresponding boundary graph is a 2k-outerplanar graph.

Thus, if the dual graph is (1) a graph with bounded treewidth and

bounded degree, or (2)a k-outerplanar graph,MRPC can be solved in polynomial time.

Polygon with h Holes

Cut:

Polygon with h HolesApproximation for general planar graphs

Original Pocket After cut

Cut an edge (in the dual):

Polygon with h HolesApproximation for general planar graphs

Original dual After cut

Polygon with h HolesApproximation for general planar graphs

Decompose dual into a series of k-outerplanar graph

k

Baker’s technique

Polygon with h HolesApproximation for general planar graphs

Decompose dual into a series of k-outerplanar graph by cutting edges

Intuitively, cutting one edge reduce the number of face by one.

use 2h/k cuts to decompose the dual (planar) graph into series of (k+1)-outerplanar graphs

Polygon with h HolesApproximation for general planar graphs

solve these (k+1)-outerplanar graphs optimally, then put solutions together for a solution with at most OPT+4h/k jumps

choose k=4/ε

OPT+εh jumps in polynomial time

Polygon with h HolesApproximation for general planar graphs

Thank You!