Computational Complexity of Area Minimization in Multi-Layer Channel Routing

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Computational Complexity of Area Minimization in
Multi-Layer Channel Routing

• Nafize Rabbani Paiker
• Std. ID 0409052057

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Introduction
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Outline of the Talks

• Review Basic Definition
• Motivation to VHVH Routing
• Complexity Results for Multi-Layer Channel
  Routing
• Some Algorithms for Multi-Layer Channel Routing
• Definitions and Notations
• A Simple Framework for Multi-Layer Channel
  Routing
• NP-Completeness of Multi-Layer No-Dogleg Routing

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Review Basic Definition

• Horizontal Constraint Graph (HCG)
• HCG, HC (V,E)
• Where corresponds to a the interval Ii of the
  net ni in the channel.
• An undirected edge , if the intervals Ii
  and Ij corresponding to nets ni and nj, intersect
  a column.

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Review Basic Definition

• Vertical Constraint Graph (VCG)
• VCG, VC (V,A)
• Where corresponds to a the interval Ii of the
  net ni in the channel.
• An directed edge , in the VCG indicates
  that the net ni has to connect a top terminal and
  the net nj is connected to a bottom terminal at
  the same column position.

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Review Basic Definition

• dmax Channel Density
• vmax The length of the longest path of VCG.
• Dogleg Routing
• Horizontal wire segment of a net is split into
  two (or more) parts and assigned to different
  tracks.
• No-Dogleg Routing
• A net is allowed to have only one horizontal wire
  segment.
• Restricted Dogleg Routing
• Dogleg only allowed only in those columns in
  which it contains a terminal.
• Unrestricted Dogleg Rouging
• Dogleg allowed in any columns.

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Review Basic Definition
Restricted Dogleg Routing
Unrestricted Dogleg Routing
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Review VHV Routing

• Assign all horizontal wire segment with LEA.
• Assign all upward vertical wire segment to a
  vertical layer (say V1).
• Assign all downward vertical wire segment to a
  vertical layer (say V2).

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Review HVH Routing

• Assign all vertical wire segment to a vertical
  layer (V).
• Segment the horizontal wire segments into two
  groups.
• Assign each group of horizontal to different
  vertical layers.

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Motivation to VHVH Routing

• VHV routing model is suitable where VCG has a
  cycle or dmax lt vmax.
• Because we can always get routing solution using
  exact dmax tracks.

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Motivation to VHVH Routing

• HVH routing model is suitable where VCG is
  cycle-free or dmax vmax.
• Because there is a likelihood of getting a
  routing solution with number of tracks lower than
  dmax due to two horizontal layers.

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Motivation to VHVH Routing

• The benefits of both VHV and HVH model can be
  achieved using VHVH routing model.
• Least number of tracks required

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Complexity Results for Multi-Layer Channel Routing

• Problem
• Is there any no-dogleg routing solution for a
  given channel specification of multi-terminal
  nets using exactly tracks in VHVH routing?
• Complexity NP-complete
• Problem
• Is there any no-dogleg routing solution for a
  given channel specification of multi-terminal
  nets using exactly tracks in ViHi routing?
• Complexity NP-complete

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Complexity Results for Multi-Layer Channel Routing

• Problem
• Is there any no-dogleg routing solution for a
  given channel specification of multi-terminal
  nets using exactly tracks in ViHi1 routing?
• Complexity NP-complete

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Some Algorithms for Multi-Layer Channel Routing

• Algorithm proposed by Enbody and Du
• This model considered ViHi and ViHi1 routing
  model and use unrestricted dogleg routing.
• Problem Excessive via holes.
• Enbody R. J. and H. C. Du, Near-Optimal
  n-Layer channel Routing, Proc. of 23rd ACM/IEEE
  Design Automation Conf., pp. 708-714, 1986.

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Some Algorithms for Multi-Layer Channel Routing

• Algorithm used in Chameleon
• Use no-dogleg routing.
• Use short horizontal wires (jogs) on vertical
  layers in order to reduce the number of via holes
  and total number of track.
• Problems
• Excessive via holes.
• The use of wires may lead to undesirable routing
  of wires outside the left and right end of the
  channels.
• The use of jogs results in overlapping wire
  segments in adjacent layers.
• Barun D., J.L. Baruns, S. Devadas, H.K. Ma,
  K. Mayaram, F. Romeo and A. Sangiovanni-Vincentell
  i, Chameleon A New Multi-Layer Channel Router,
  Proc. of 23rd ACM/IEEE Design Automation Conf.,
  pp. 495-502, 1986.

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Some Algorithms for Multi-Layer Channel Routing

• MulCh and M3CR
• Use unreserved layer.
• Problems
• Electrical interference between the layers.
• Greenberg R.I. and A. Sangiovanni-Vincetelli
  , MulCh A Multi-Layer Channel Router using One-,
  Two- and Three- Layer Partitions, Proc. of IEEE
  Int. conf. on Computer Aided Design, pp. 88-91,
  1988.
• Fang S.-C., W.-S. Feng and S.-L. Lee, A New
  Efficient Approach to Multi-Layer Channel Routing
  Problem, Proc. of 29th ACM/IEEE Design Automation
  Conf., pp. 579-584, 1992.

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Definitions and Notations

• Density Routing Solution
• A routing solution is a density routing solution
  if it requires tracks, where i
  number of horizontal layers.
• Induced Subgraph
• An induced undirected (directed) subgraph,
  Gv(V,EV) of a undirected (directed) graph
  G(V,E) is defined over set of vertices V where
  and a set of vertices EV,
  where EV is the set of undirected (directed)
  edges between the vertices in V.
• Induced Set of Edges
• The set of edges, EV in an induced subgraph GV
  is referred to as induced set of edges.

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Definitions and Notations

• Reduced Vertical constraint Graph, RVCG
• RVCG, RVC (CC,A)
• CC Set of clique covers of the HNCG.
• A Set of edges.
• Rule of initiating directed edge in RVCG
• An directed edge would be introduced from clique
  to clique , if there are nets
  and such that there is a directed
  edge from vg to vh in the VCG, VC (V,A).

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Definitions and Notations

• Reduced Vertical constraint Graph, RVCG

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A Simple Framework for Multi-Layer Channel Routing

• VHVH routing model.
• Lower bound of the number of tracks required for
  routing any channel of density dmax

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A Simple Framework for Multi-Layer Channel Routing

• All vertical constraints between nets assigned to
  H1 can be resolved by routing vertical wire
  segments using the two layers V1 and V2 ?VHV
  routing.
• For VHVH routing, higher priority is given for
  routing vertical wire segments through V1.
• Leaves the vertical layer V2 free for routing a
  vertical wire segment for a net assigned to H2.

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A Simple Framework for Multi-Layer Channel Routing

• Which nets can be assigned to H2?
• Whose corresponding vertex set S does not contain
  a cycle in the induced subgraph VCs (S,As) of
  the VCG
• ?Cycles cannot be resolved using a single
  adjacent vertical layer.
• Induced RVCG, RVCCCH2(CCH2,ACCH2) must be cycle
  free.

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Problem MNVHVH

• Problem Multi-terminal No-dogleg VHVH channel
  routing (MNVHVH)
• Instance Channel specification of multi-terminal
  nets.
• Question Is there a four-layer VHVH routing
  solution for the given instance using
  tracks?
• Complexity NP-complete

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Problem 3SAT

• Problem 3SAT
• Instance Collection F c1, c2, , cq of
  clauses on a finite set U of variables such that
  ci 3 for 1 i q.
• Question Is there a truth assignment for U that
  satisfies all the clauses in F?

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Problem IS3

• Problem IS3
• Instance An undirected graph G (V,E). Here the
  number of the vertices in G is n.
• Question Is there any independent set of size
  ?
• Complexity NP-complete

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Theorem 5.1 IS3 is NP-Complete

• Given a guess , where
• The verification of whether V is an independent
  set of G (V,E) requires polynomial time.
• Transformation 3SAT to the maximum clique
  problem.
• Given
• 3SAT instance
• An undirected graph G (V,E) of n3q vertices
  is constructed such that
• G has a clique of size q if and only if the
  formula F of 3SAT instance is satisfiable.
• ?The complement G (V,E).of the graph G has an
  independent set of size if and
  only if F is satisfiable.
• G can never have an independent set larger than
  q the independent set has exactly q elements
  whenever F is satisfiable.

? IS3 is NP-complete.
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Problem IS2

• Problem IS2
• Instance An undirected graph G (V,E). Here the
  number of the vertices in G is n.
• Question Is there any independent set of size
  ?
• Complexity NP-complete

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Theorem 5.2 IS2 is NP-Complete

• Given a guess , where
• The verification of whether V is an independent
  set of G (V,E) requires polynomial time.
• Given
• 3SAT instance
• An undirected graph G (V,E) of n4q vertices.
• q vertices are isolated ? vertex set A.
• Remaining 3q vertices are as they are in theorem
  5.1 ? vertex set B.
• F of 3SAT instance has a satisfiable truth
  assignment if and only if the graph the graph of
  n4q vertices has an independent set of size
• Suppose, G has an impendent set S of size 2q.
• Induced subgraph GB con not have an independent
  set of size larger q.
• q vertices of S must belong to the A of the
  isolated vertices and the remaining q vertices of
  S must belong to the set B.
• GB has an independent set of size q ? F has a
  satisfiable truth assignment.

? IS2 is NP-complete.
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Conclusion
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Question
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Thank You