University of California Santa Barbara Department of Computer Science

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University of CaliforniaSanta BarbaraDepartment
of Computer Science

• Minimum Length Corridor and Related Problems
• Ph. D. Thesis Proposal

Committee Teofilo F. Gonzalez (Chair) Peter
Cappello Ömer Egecioglu
Arturo Gonzalez-Gutierrez
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Outline

• Minimum Length Corridor (MLC) Problem
• Related Problems
• NP-completeness
• Approximation Algorithm TRA-MLC-R
• Future Work

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Minimum Length Corridor (MLC) Problem

• Motivation
• Applications
• Formal Definition
• Background
• Hierarchy

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MLC Problem Motivation
MLC solution
TRA-MLC solution
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MLC Problem Applications

• Circuit Board Layout Design
• Wires for power supply
• Building Wiring Design
• Optical Fiber for Data Communication Networks
• Wires for Electrical Networks
• Trip Planning Queries in Spatial Databases

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MLC Problem Formal Definition

• INPUT Given a pair (F,R), where F is a
  rectangular boundary whose interior is
  partitioned into rectilinear polygons
  RR1,R2,,Rm.
• OUTPUT A corridor for (F,R) consisting of a set
  of connected line segments each of which
  completely overlap with the line segments that
  form F or the boundary of the rectilinear
  polygons, and include at least one point of F and
  at least one point from each of the m rooms.
• OBJECTIVE FUNCTION Minimize the total edge
  length of the corridor.

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MLC Problem Background

• Demaine, E.D., and ORourke, J. Open problems
  from CCCG 2000. In Proceedings of the 13th.
  Canadian Conference on Computational Geometry
  (2001), (by Naoki Katoh). http//theory.csail.mit.
  edu/edemaine/papers/CCCG2000Open/
• Eppstein, D. Some open problems in graph theory
  and computational geometry. http//www.ics.uci.edu
  /eppstein/200-f01.pdf, November 2001.
• Jin, L. Y., and Chong, O. W. The minimum touching
  tree problem. http//www.yewjin.com/research/Minim
  umTouchingTrees.pdf, National University of
  Singapore, School of Computing, 2003.
• No polynomial time algorithm known
• Not even a constant ratio approximation algorithm
• Seems likely to be NP-complete but no proof known

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MLC Problem Hierarchy
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Related Problems

• Tree Feedback Node Set (TFNS)
• Tree Vertex Cover (TVC)
• Group Steiner Tree (GST)
• Tree Errand Cover (TEC)

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Related Problems Tree Feedback Node Set
Generalization of the N-MLC

• INPUT A connected undirected edge-weighted graph
  G(V,E,w), where wE?? is an edge-weight
  function.
• OUTPUT A tree T(V,E), where E?E, and V?V is
  a feedback node set(every cycle in G includes at
  least one vertex in V) and the total edge-weight
  is minimized.

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Related Problems Tree Feedback Node Set Tree
Vertex Cover Problem

• INPUT A connected undirected edge-weighted graph
  G(V,E,w), where wE?? is an edge-weight
  function.
• OUTPUT A tree TG of G ? its vertices cover all
  the edges E, and the total length
  is minimized.

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Related Problems Tree Vertex Cover

• A polynomial approximation algorithm with
  performance 3.78 Arkin, E.M. et. al. (1993)
• Uses the approximation algorithm for the weighted
  vertex cover and then uses an approximation
  algorithm for the Steiner tree problem.
• Replace the approximation algorithm for the
  weighted vertex cover by a constant ratio
  approximation algorithm for the weighted FNS
  problem Chudak, F. A. et. al. (1998).

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Related Problems Group Steiner Tree
ProblemGeneralization of the MLC Problem

• INPUT A connected undirected edge-weighted graph
  G(V,E,w), where wE?? is an edge-weight
  function, a non-empty subset N, N?V, of
  terminals and a partition N1,N2,...,Nk of N.
• OUTPUT A tree TG(N) of G ? at least one terminal
  from each set Ni is in the tree TG(N) and the
  total length is minimized.

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Related Problems Group Steiner Tree
ProblemReducing the TRA-MLC to the GST
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Related Problems Group Steiner Tree
ProblemReducing the TRA-MLC to the GST
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Related Problems Group Steiner Tree
ProblemReducing the TRA-MLC to the GST
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Related Problems Group Steiner Tree Problem
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Related Problems Tree Errand Cover

• INPUT A connected undirected edge-weighted graph
  G(V,E,w), where wE?? is an edge-weight
  function, a non-empty set Ut1,t2,...,tm of
  errands. Associated with each vertex i?V is a set
  Si?U, where .
• OUTPUT A tree TG of G ? and
  the total length is minimized

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Related Problems Tree Errand CoverTRA-MLC?TEC
UR1,R2,R3 ,R4,R5,R6,R7,R8,R9
S1R1 S2R1,R2 S3R2,R3 S4R3 S5R1,R2,R5
S6R2,R3,R5 S7R3,R5,R6 S8R3,R6 S9R1,R4
S10R1,R4,R5
S11R5, R6, R8 S12R6,R8 S13R4,R7 S14R4,
R5, R7 S15R5,R8,R9 S16R8,R9 S17R7 S18R
7,R5 S19R5,R9 S20R9
S4R3 S5R1,R2,R5 S6R2,R3,R5 S7R3,R5,R6
S8R3,R6 S10R1,R4,R5 S12R6,R8 S14R4,
R5, R7 S16R8,R9
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Related Problems Tree Errand CoverPerformance
ratio

• Slavik, P. (1998)
• The TEC problem can be approximated to within a
  factor of 2? in polynomial time, when each errand
  is assigned to at most ? vertices.
• Still, applying this case even to the TRA-MLC-R
  problem does not give us a constant approximation
  algorithm.
• There are errands that may be assigned to any
  number of vertices.

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NP-Completeness

• TRA-MLC-R
• MLC-R
• MLCf-R
• MA- MLCf-R
• TRA-MLC
• MLC
• MLCf
• MA-MLCf

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NP-Completeness

• Complexity of the Minimum-Length Corridor
  Problem, Submitted to Publication, March 2005.

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NP-Completeness
Theorem 1. TRA-MLC-R a MLC-R
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NP-Completeness
Theorem 2. TRA-MLC-R a MLCf-R. TRA-MLC-R a
MA-MLCf-R with k access points, k2.
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NP-completeness

• Theorem 3. The MLC, MLC-R, MLCf, MLCf-R, MA- MLCf
  and MA- MLCf-R problems are NP-complete.

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NP-Completeness

• Theorem 4. The TRA-MLC problem is NP-complete.
• Theorem 5. The TRA-MLC-R problem is NP-complete

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NP-Completeness TRA-MLC
I ? 3SAT is satisfiable ? f(I)?TRA-MLC has a
corridor with length at most B
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NP-Completeness TRA-MLC
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NP-Completeness TRA-MLC-R
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NP-Completeness TRA-MLC-R
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NP-Completeness TRA-MLC-R
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Approximation Algorithm TRA-MLC-R

• Approximation Ratio
• Approach by Restriction
• Restriction to four points
• Restriction to five points
• Restriction to one and two points
• Restriction to three points

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Approximation Algorithm Approximation Ratio

• Use Slaviks Algorithm for the TEC problem for a
  given ?

S?(I) Solution to the TRA-MLC-R? generated by
using Slaviks Algorithm O?(I) Optimal solution
to the TRA-MLC-R? S(I) Any solution to the
TRA-MLC-R O(I) Optimal solution to the TRA-MLC-R
For every corridor S(I) we can construct S?(I) ?
S?(I) c S(I) Applying this to O(I), S?(I) c
O(I) We know that O?(I) S?(I) Therefore,
O?(I) S?(I) c O(I) Slaviks solution shows
that S?(I)2? O?(I) And, S?(I) 2? O?(I) 2? c
O(I)
?maxV(Ri)
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Approximation Algorithm Restriction to one
pointOne out of four corners
TRA-MLC-R ? TRA-MLC-R4
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Approximation Algorithm Restriction to five
pointsFour corners and special point
TRA-MLC-R ? TRA-MLC-R5
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Approximation Algorithm Restriction to five
points Basic definitions

• SP(p,q) Length of the shortest path from vertex
  p to vertex q using only boundary F and rectangle
  line segments.
• MinLen(p,Ri) Minimum edge length such that every
  vertex q of rectangle Ri satisfies
  SP(p,q)MinLen(p,Ri)
• diminMinLen(vi,Rj)ri?j Min-connectivity
  distance for rectangle Ri through vertex vi to
  another rectangle Rx

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Approximation Algorithm Restriction to five
points Min connectivity distance for R5
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Approximation Algorithm Restriction to five
pointscpe and ncpe rectangles
V(Ri)5?Ri is a cpes
V(Ri)gt5?Ri is a ncpes
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Approximation Algorithm Restriction to five
points ncpes and pre-order traversal
S5(I)3S(I)
FEtraversing?Econexions of sp
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Approximation Algorithm Restriction to five
points Regions type 2,1,0.
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Approximation Algorithm Restriction to five
points
0
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Approximation Algorithm Restriction to five
points From here on we assume the first ncpe
rectangle is already connected
0 001
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Approximation Algorithm Restriction to five
points
0 0010010
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Approximation Algorithm Restriction to five
points
0 001001000100
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Approximation Algorithm Restriction to five
points
0 001001000100001000
0 0010
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Approximation Algorithm Restriction to five
points
0 0010 0011
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Approximation Algorithm Restriction to five
points
0 0010 001100111
000100011
0001(01)
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Approximation Algorithm Restriction to five
points
0001(01) 01
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Approximation Algorithm Restriction to five
points
0001(01) 01010
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Approximation Algorithm Restriction to five
points
0001(01) 01(01)
0001(01) 01010 011
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Approximation Algorithm Restriction to five
points Rules

• 0?OK
• 1 ?OK
• 1x?x
• 02x?x
• 0001(01) ?OK
• 01(01) ?OK
• 0001(01)2x ?x
• 01(01)2x ?x

S5(I)3S(I) S5253O(I)30O(I)
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Approximation Algorithm Restriction to five
pointsBy rule 5 0001(01) ?OK
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Approximation Algorithm Restriction to one and
two points
Special point and bottom-left corner
Special point
Special point and top-left corner
Special point and top-right corner
Special point and bottom-right corner
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Future work

• Is there a provably constant 30-approximation
  algorithm for the TRA-MLC-R problem by
  restricting maxV(Ri) to five vertices?
• Is the decision version of TRA-MLC-R5 problem
  NP-complete?
• Is there a provably constant approximation
  algorithm for the TRA-MLC-R problem by
  restricting maxV(Ri) to three vertices
  (TR,BL,SP)?
• ?3,c5?23530
• Is the decision version of TRA-MLC-R3 problem
  NP-complete?
• Is it possible to have a O(constant)-approximation
  algorithm for a given number of critical points
  bounded by a constant l.

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Thank you!

• Questions
• Comments
• Suggestions