General Routing Overview and Channel Routing

Shantanu Dutt

ECE Dept.

UIC

References and Copyright (cont.)

• Slides used: (Modified by Shantanu Dutt when necessary)– [©Sarrafzadeh] © Majid Sarrafzadeh, 2001;

Department of Computer Science, UCLA– [©Sherwani] © Naveed A. Sherwani, 1992

(companion slides to [She99])– [©Keutzer] © Kurt Keutzer, Dept. of EECS,

UC-Berekeleyhttp://www-cad.eecs.berkeley.edu/~niraj/ee244/index.htm

– [©Gupta] © Rajesh Gupta UC-Irvinehttp://www.ics.uci.edu/~rgupta/ics280.html

– [©Kang] © Steve Kang, UIUC http://www.ece.uiuc.edu/ece482/– [©Bazargan] © Kia Bazargan

Routing

• Problem– Given a placement, and a fixed number of metal layers, find a

valid pattern of horizontal and vertical wires that connect the terminals of the nets

– Levels of abstraction:• Global routing• Detailed routing

• Objectives– Cost components:

• Area (channel width) – min congestion in prev levels helped• Wire delays – timing minimization in previous levels• Number of layers (fewer layers less expensive)• Additional cost components: number of bends, vias

©Bazargan

Metal layer 1

Via

Routing Anatomy

Topview

3Dview

Metal layer 2

Metal layer 3

Sym

bolic

Layou

t

Note: Colors usedin this slide are notstandard

©Bazargan

Global vs. Detailed Routing• Global routing

– Input: detailed placement, with exact terminal locations

– Determine “channel” (routing region) for each net

– Objective: minimize area (congestion), and timing (approximate)

• Detailed routing– Input: channels and approximate routing from

the global routing phase– Determine the exact route and layers for each

net– Objective: valid routing, minimize area

(congestion), meet timing constraints– Additional objectives: min via, power

Figs. [©Sherwani]

Taxonomy of VLSI Routers

[©Keutzer]

Graph Search

Steiner

Iterative

Hierarchical Greedy Left-Edge

River

Switchbox

Channel

Maze

Line Probe

Line Expansion

RestrictedGeneral Purpose

Clock

Specialized

Power/Gnd

Routers

DetailedGlobal

Maze

Global Routing• Stages

– Routing region definition– Routing region ordering– Steiner-tree / area routing

• Grid– Tiles super-imposed on placement– Regular or irregular– Smaller problem to solve,

higher level of abstraction– Terminals at center of grid tiles

• Edge capacity– Number of nets that can pass a certain

grid edge (aka congestion)– On edge Eij,

Capacity(Eij) Congestion(Eij)• Steiner routing is generally performed on the

routing graph using edge lengths as cost and considering edge capacities

M1M2

M3

[©Sarrafzadeh]

Grid Graph• Course or fine-grain• Vertices: routing regions, edges: route exists?• Weights on edges

– How costly is to use that edge – Could vary during the routing (e.g., for congestion)– Horizontal / vertical might have different weights

[©Sherwani]

t1 t2 t3t4

t1 t2 t3t4

1 1 1

1 1 1

2 2 1 1

t1 t3t4

t2

Global Routing – Graph Search

• Good for two-terminal nets• Build grid graph (Coarse? Fine?)• Use graph search algorithms, e.g., Dijkstra• Iterative: route nets one by one• How to handle:

– Congestion?– Critical nets?

• Order of the nets to route?– Net criticality– Half-perimeter of the bounding box– Number of terminals

©Bazargan

(4-side routing, requires Steiner routing, NP-hard)

(2-side routing, solvable optimally in linear time w/o vertical constraints)

channels (blue) covering adjacent overlapping module boundary pairs

switch-boxes (maroon) in rest if the areas

©Dutt for channel & sw-box definition in the left figure

• A cycle in the VCG an unroutable placement unless a net can be routed on more than 1 track• Otherwise, depth of VCG is lower bound on channel density

Case 2a:Closest non-ovnet to e crosses L

Case 2b:Closest non-ovnet to e does notcross L

L

e: Most recentlyrouted net

L’

e: Most recentlyrouted net

L

Optimality of the Left Edge Algorithm

Case 1: Max density line L cuts e Case 2: Max density line L does not cut e • In Case 1, the density of L reduces by 1 after current track t (e is on t) is routed• In Case 2, let e’ be the net not overlapping e & whose s(e’) is closest to e(e).

• Case 2a: If e’ crosses L, then since e’ will be on t, density of L reduces by 1 after t is routed• Case 2b: If not, then the set S(L) of all other nets crossing L are overlapping w/ e (otherwise one of them will be e’ and crossing L, and we will not be in Case 2b). Then there exists another cut line L’ that cuts S(L) and e, and thus have density > density of L, and we reach a contradiction (that L is the max density line)

• Thus after current track t is routed, the density of L reduces by 1. This applies to all max density lines. Thus # of tracks needed = density of initial max density line which is a lower bound on # tracks. Hence the Left-Edge algorithm is optimal in the # of tracks

e’

e’

s(e) e(e)

s(e’)

s(e’)

S(L)

©Dutt

Update the VCG by deleting all Ij ‘’s (and their arcs) routed in track t-1 > 0;

(no arcs in the VCG incoming to Ij)

1a

2

1bb

a

Acyclic VCGCyclic VCG

w/ the added flexibility that the new net e’s s(e’) can be = watermark if current net e and e’ belong to the same net