Transcript Chapter 5: Computational Complexity of Area Minimization in Multi-Layer

Chapter 5:
Computational Complexity of Area
Minimization in Multi-Layer
Channel Routing and an Efficient
Algorithm
Presented by
Md. Raqibul Hasan
Std No. 0409052016
Review: Basic Definition
 Horizontal Constraint Graph (HCG):
 HCG = (V,E)
 Where
corresponds to a net ni in the
vi  V
channel.
{vi , v j }  E
 An undirected edge
, if the intervals Ii and Ij
corresponding to nets ni and nj, intersect a column (overlap
horizontal expansion).
Review: Basic Definition
 Vertical Constraint Graph (VCG):
 VCG = (V,E)
 Where
corresponds to a net ni.
vi  V
 An directed edge
{vi , v j }  E , 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
 Dogleg
Dogleg
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Overview
 There are several polynomial time three-layer
channel routing algorithms that produce routing
solutions using dmax tracks under the VHV model.
 Under the HVH model the problem of routing a
channel with a minimum number tracks is NPComplete.
 The most notable characteristic of NP-complete
problems is that no fast solution to them is known.
Running time is exponential.
Overview
 It is not wise to waste time trying to solve a
problem which so far has eluded generations of
computer scientists. Instead, NP-complete
problems are often addressed by using
approximation algorithm in practice. We require
better heuristic for getting better performance
from the approximation algorithm.
Overview
 We require at least dmax /2 track in any
VHVH routing solution.
 An interesting open problem is that of
determining whether there is a no-dogleg
routing solution for a given channel
specification of multi terminal nets using
exactly dmax /2 tracks in VHVH model.
Algorithm for Multi-Layer Channel
Routing
 Complexity of area minimization in multi-layer
channel routing in the reserved layer Manhattan
routing model is NP-hard.
 Approximate solution: The drawback in using
unrestricted doglegging is the difficulty in
fabrication due to excessive via holes.
Algorithm for Multi-Layer Channel
Routing
 Chameleon uses short horizontal wires (jogs) in
vertical layers in order to reduce the number of via
holes and the total number of tracks.
jog
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 Complement of HCG is called horizontal non-
constraint graph (HNCG). A clique of the HNCG
corresponds to a set of non- overlapping intervals
that may safely be assigned to the same track.
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Complemented Graph
Density Routing Solution
 A routing solution is a density routing solution if
it requires
tracks
(lower bound), where “i” is
d max / i 
the number of horizontal layers.
Induced Graph
 G=(V, E) is a graph and V’ subset of V. Induced
graph G’=(V’, E’) where, E’ is a subset of E and
E’ is the set of edges between vertices V’.
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Vertex induced
V’ ={1,2,3}
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Edge induced
E’ ={(1,2),(1,4),(2,3)}
A simple framework for multi-layer
Channel Routing
 In the case of three layer VHV routing, all vertical
constraints between nets assigned to H can be resolved by
routing vertical wire segments using the two layers V1 and
V2.
 In case of VHVH routing, higher priority is given for
routing vertical wire segments through V1 when assigning
nets to H1.
 In case of VHVH routing we can not assign nets to H2
whose corresponding vertex set S introduce a cycle in the
induced subgraph VCs=(S, As) of the VCG. This is so
because such cycles can not be resolved using a single
adjacent vertical layer V2.
Reduced Vertical Constraint Graph
RVCG = (CC,A’)

CC = Set of clique covers of the HNCG.

A’ = Set of edges.

An directed edge would be introduced from clique
C  CC
to clique C  CC , if there are nets
and
ng 
Ci that there
nh  Cisj a directed edge from vg to vh in the
such
VCG.
j
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{1,4}
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{2,3}
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HNCG
RVCG
A simple framework for multi-layer
Channel Routing
 To obtain a VHVH routing solution we have to
assign set of nets to horizontal layer H2 in such a
way that net assigned to tracks of H2 do not
include cycle in VCG.
 Induced RVCG must be cycle free.
Some NP-Complete Problem
 MNVHVH (Multi-terminal no-dogleg VHVH channel
routing): We are given a channel specification of multiterminal nets. Is there a four-layer VHVH routing
solution for the given instance using
tracks?
 3SAT: Collection F={c1,
c2, c3,…,cq} of clauses on a
d
/
2


finite set U of variables such that |ci|=3 for 1<=i<=q. Is
there a truth assignment for U that satisfies all the clauses
in F?
F (A, B, C) = (A+B’+C).(A’+B+C’).
If A=0, B=1, C=1 then F is satisfied i.e. F=1.
max
Independent Set
 An independent set of a graph G=(V, E) is a subset
V’ of V such that each edge in E is incident on at
most one vertex in V’.
Independent
set2 {1,3}
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 IS2: An undirected graph G=(V, E). Here the
number of vertices in G is n. Is there an
independent set of size .
n / 2
 IS3: An undirected graph G=(V, E). Here the number of
vertices in G is n. Is there an independent set of size
.
 To prove a problem as NP-Complete,
we have to give a
n / 3
polynomial time verification algorithm and a polynomial
time reduction algorithm of a known NP-Complete to that
problem.
Thank You