Probability-based%20approach%20for%20solving%20the%20Rectilinear%20Steiner%20tree%20problem

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Probability-based approach for solving the
Rectilinear Steiner tree problem

• Jayakrishnan Iyer

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Steiner Tree

• Steiner Tree Given a set of points in a plane
  N1,2,,n and graph G (N,E) wherein
• Here we associate a weight we for all e in E.
• A Steiner Tree for G is a subgraph St (N,E)
  for all
• such that
• 1)St is a tree and
• 2)For all s,t in T, there exists only one path
  from s to t in St

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Steiner tree problem

• Minimize Length of the steiner tree.
• It is an NP-complete problem
• Rectilinear Steiner Trees Distance between 2
  points is the sum of distances in the horizontal
  and vertical dimensions
• Very Important in VLSI routing.
• Some good theoretical bounds
• O(n22.62n) by Ganley/Cohoon
• nO(vn) by Smith quite impractical due to large
  exponent.
• O(n22.38n) by Fobermeir/Kaufmann and an for
  random instances where alt2

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Probabilistic Model

• Grid graph Define a grid for any set of points
  Pp1,p2,,pn where each pi is (xi,yi) and xi?xj
  and yi ?yj for i ? j.
• Optimal RST for segments in tree T such that wire
  length for all segments over T is minimum.
• Define Horizontal and Vertical Grid distances
• for a given grid.

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Probabilistic Model(contd)

• An Eg. Grid graph for a set of 3 points
• Optimal Steiner tree corresponding to this

2
1
3
p2
3
l2
l1
p1
2
1
p3
p2
(Steiner Point)
Sl
p1
p3
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Probabilistic Model (contd)

• A generic grid graph (ref. chen et. al.)

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Probabilistic Model (contd)

• Here m and n are the vertical and horizontal grid
  sizes respectively.
• M number of all possible shortest paths from pi
  to pj
• Number of paths througth R(I,J-1) i.e. R2
  F(m,n)
• Number of paths througth C(I,J) i.e. C2 G(m,n)
• F(1,q) G(q,1) 1 and F(q,1) G(1,q) q

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Probabilistic Model (contd)

• Probability of Shortest path through R(I,J-1)
  F(m,n)/M. Similarly for path through C(I,J)
  G(m,n)/M.
• F(m,n) F(m-1,n) G(m-1,n) and
• G(m,n) F(m,n-1) G(m,n-1)
• Also, F(m,n-1) F(m-1,n-1) G(m-1,n-1)
• And G(m-1,n) G(m-1,n-1) F(m-1,n-1)

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Probabilistic model (contd)

• Theorem If for qgt0 F(q,0)G(0,q) 1 then
• Where m,ngt1.

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Probabilistic analysis (contd)

• F(m,n) and G(m,n) can be recursively defined as,

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Probabilistic model (contd)

• Probability Matrix
• For any row segment R(Ik,J-l-1) where 0ltkltn and
  0l m-1, we have for M shortest paths, number of
  paths passing through the given segment,
• The probability of a shortest path passing
  through these segments is given by,
• Similarly for a column segment the probability
  is,

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Algorithms

• Pure Probabilistic - steps
• 1)Compute r(i),c(i),H(k) and V(k) for given
• Pp1,p2,,pn where i is in 1,2,,n and
• K1,2,,n-1
• 2)Compute F(m,n) and G(m,n).
• 3)Obtain PR and PC for all the N2 pairs and
  compute the matrix and normalize them.
• PR(i,j) PR(i,j)/H(j) for 1ltiltn and 1ltjltn-1
• PC(i,j) PC(i,j)/V(i) for 1ltiltn-1 and 1ltjltn
• 4)Ignore segments leading to a cycle.
• 5)Delete redundant and degree-one segments not
  connected to any point in P.
• 6)Obtain the set of Steiner points S and compute
  wire-length.

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Algorithm (contd)

• Time Complexity of this algorithm is O(N4) due to
  the time taken for nomalizing the N2 pairs for
  each of the row and column probability matrices.
• Other algorithms have only a slight modification
  the previous algorithm and reduce the time
  complexity to O(N3) by using MST to get the
  N-1edge set.

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Certain dilemmas

• Although wire length computation results have
  been provided in the paper, it does not hint as
  to how accurately the problem is solved (upper
  bound on success).
• Time taken for a particular point set instance of
  the problem.

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• Questions ??