The Rectilinear Steiner Arborescence Problem is NPComplete

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The Rectilinear Steiner Arborescence Problem is
NP-Complete

• Weiping Shi (Dept. of Electrical Engg. TAMU)
• Chen Su (Inet Technologies, TX)
• SIAM Journal of Computing 2006
• Presenter Vishal Kapoor

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Getting Started Background and Definitions
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Steiner Tree

• A minimal length tree connecting a set of points
  in a plane
• May have Steiner Points

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Rectilinear Steiner Tree

• A Steiner tree in which every path is made up of
  straight line segments

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Rectilinear Steiner Arborescence (RSA)

• Directed Steiner tree rooted at origin
• Points are in the first quadrant of the plane
• Every segment in the tree is directed left to
  right or bottom to top

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Difference?

• RSA is a shortest distance tree with respect to
  the origin

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Rectilinear Steiner Minimum Arborescence (RSMA)

• A minimum length RSA
• Difference between RSA and RSMA

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Applications

• Useful in VLSI routing
• It has been shown that routing trees based on
  such structures may have much lesser delays than
  those based on traditional Steiner trees

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The Proof
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My thoughts about the proof

• Very artistic (geometric) proof
• Cited in literature as pretty
• Uses graph drawing techniques
• One of the clearest papers that I have read
  provides details very clearly!

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The Decision Problem

• A Set of points P p1, p2 pn in the plane
  and a positive number k
• Is there an RSA of total edge length k or less?
• Proof Reduction from 3-SAT

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3-SAT

• A set of variables V v1, v2 vn and clauses
  C c1, c2 cn
• Each clause has at most 3 literals.
• Is there an assignment of variables so that all
  clauses are satisfied?
• Satisfying the 3-CNF form is NP-Complete
• Eg
• (v1 OR v2 OR v6) AND (v1 OR v8)

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3-SAT problem as a planar graph (Planar 3-SAT)
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(Planar graph used in the paper)
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A Quick Recap

• Steiner Tree / Points
• Rectilinear Steiner Tree
• RSA, RSMA differences
• The Decision Problem
• 3-SAT, Planar 3-SAT

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Reduction Technique

• Based on Component Design
• Steps
• 3-SAT graph is reduced to another planar graph H
• H is embedded into a grid (as graph R)
• Each vertex of R is replaced by a tile

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First Step Transforming G to H

• H is planar with maximum degree 3
• Each vertex of G contribute deg(v) vertices in H
• Eg

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Properties of HNew Vertices OR, NOT

• Nice form
• If not in Nice form
• Insert NOT vertex wij
• Suppose the clause looks like below
• Insert CLAUSE vertex cj for each clause
• Insert OR vertex cj to connect big clauses

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Example
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Second Step Embedding H as graph R in a grid

• Can be done in polynomial time O(V2) - Valiant

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Properties of RNice drawing techniques

• Vertices only unit distance apart are connected
• For each CLAUSE vertex
• Positive variable enters from left
• Negative variable enters from below
• Eg -gt

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contd

• 3. Each OR vertex occupies 2 horizontal
  vertices in R

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Final Step Covering R with tilesQuadrupeds and
forbidden regions

• These are a set of RSAs that connect the white
  points each RSA is rooted at a black point
• Has 2 min forests, both of length
• Black points are interface points, white points
  are inside points

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Another Recap

• 3 steps for reduction
• 1. Transforming G into H
• Always looking out for "nice" clauses
• Adding OR, NOT, CLAUSE vertices to make "nice"
  form
• 2. Embedding H into a grid as R
• Adding OR vertices having 2-horizontal-vertices
• Quadrupeds, Forbidden regions, minimum forests
  made of RSAs
• Now we have to see what is tiling

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Tiling
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A basic tile

• Represents a single variable
• Made up of overlapping quadrupeds
• Has height and width 96 units
• Only OR tile has height 96 and width 2x96
• Has

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Parity of a tile

• 1 if the rightmost vertex connected by horizontal
  edge
• else 0
• Tiles enforce (propagate) parities on neighboring
  tiles

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Parity Enforcement (Propagation)
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Flipping Parity NOT Tiles

• A horizontal NOT tile with and

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CLAUSE tiles

• Have the same interface to the left and to the
  bottom (Why?)
• Min forest (black)
• has length 108
• achieved iff tile to left
• has parity 1
• and tile below has 0

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OR tile

• Total width 192
• Always tries to achieve parity 1

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Another Recap

• A basic tile that represents ordinary variables
• Definition of Parity and Parity enforcement
  (propagation)
• NOT, CLAUSE and OR tiles and their properties and
  dimensions

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Lets go to the Final Result

• We are done we have all the structures that we
  need
• All structures till were made / manipulated
  locally and can be constructed in polynomial time
• Thus the final reduction should be polynomial time

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Connecting tiles togetherBasic Tiles and Trials

• Intuitively, trials close the basic tiles
• Are not in any forbidden region, so dont affect
  how white points are connected

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RSA problem is NP-Complete

• L sum of min edge lengths to connect all black
  points min forest length of each type of tile
• A planar 3-SAT having m clauses has a satisfying
  assignment iff the set of points has an RSA of
  length L108m

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Questions / Comments?
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Homework A Descriptive Question

• RSA is NP-Complete by reduction to 3-SAT
• 3-SAT is Strongly NP-Complete
• What does Strongly NPC mean in a general sense?
  Give a precise mathematical definition and 1
  example explaining the idea
• What types of problems are not Strongly
  NP-complete? What are they called?
• Is there a complexity class/classes for such
  languages?
• What does this mean in terms of circuit
  complexity?
• A PTIME algorithm for solving the homework
• 1 Choose any 1 color and answer questions of
  that color only
• 2 Google for a column (and a small paper) on
  NP-hardness by Garey Johnson for checking this
  out
• Evaluation I will grade based on the quality of
  your answer not quantity ?

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Have a nice weekend!