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 channel.

An undirected edge , if the intervals Ii and Ij corresponding to nets ni and nj, intersect a column (overlap horizontal expansion).

Evv ji },{

Vvi

Review: Basic Definition

Vertical Constraint Graph (VCG): VCG = (V,E)

Where corresponds to a net ni. 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.

Vvi

Evv ji },{

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6

Review: Basic Definition

Dogleg

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Dogleg

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 NP-Complete.

The most notable characteristic of NP-complete problems is that no fast solution to them is known. Running time is exponential.

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.

Overview

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.

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jog

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

A routing solution is a density routing solution if it requires tracks (lower bound), where “i” is the number of horizontal layers.

id /max

Density Routing Solution

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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3

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3 4Vertex inducedV’ ={1,2,3}

Edge inducedE’ ={(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 to clique , if there are nets and such that there is a directed edge from vg to vh in the VCG.

CCCi CCC j

ig Cn jh Cn

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HNCG

{1,4}

{2,3}

RVCG

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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 multi-terminal 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 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.

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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 set {1,3} IS2: An undirected graph G=(V, E). Here the

number of vertices in G is n. Is there an independent set of size .

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2/n

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 polynomial time verification algorithm and a polynomial time reduction algorithm of a known NP-Complete to that problem.

3/n

Thank You