1. General Routing Overview and Channel Routing
Shantanu Dutt
ECE Dept.
UIC

2. 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, (companion slides to [She99])
[©Keutzer] © Kurt Keutzer, Dept. of EECS, UC-Berekeley
[©Gupta] © Rajesh Gupta UC-Irvine
[©Kang] © Steve Kang, UIUC
[©Bazargan] © Kia Bazargan

3. 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

4. Routing Anatomy 3D view Top view Symbolic Layout Metal layer 3 Via
Top layers have more spacing between wires
Top layers higher aspect ratio (like walls)
Metal layer 3
Via
Note: Colors used in this slide are not standard
Metal layer 2
Metal layer 1
©Bazargan

5. 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]

6. Taxonomy of VLSI Routers
Global
Detailed
Specialized
Graph Search
Power/Gnd
General Purpose
Restricted
Steiner
Clock
River
Maze
Iterative
Switchbox
Line Probe
Maze
Channel
Line Expansion
Hierarchical
Greedy
Left-Edge
[©Keutzer]

7. Global Routing Stages Routing region definition
[©Sarrafzadeh]
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 M2 M1 M3

8. 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 t1 t2 t3 t4 t1 t3 t4 t2
The weight on the edges in the middle graph indicate edge cost, not capacity.
The right graph is called “channel intersection” graph. It is more popular than the other two. t1 t2 t3 t4 1 2
[©Sherwani]

9. 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

10. (4-side routing, requires Steiner routing, NP-hard)
(2-side routing, solvable optimally in linear time w/o vertical constraints)
switch-boxes (maroon) in rest if the areas
channels (blue) covering adjacent overlapping module boundary pairs
(4-side routing, requires Steiner routing, NP-hard)
©Dutt for channel & sw-box definition in the left figure

15. 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

19. Optimality of the Left Edge Algorithm
Case 2b:
Closest non-ov
net to e does not
cross L
Case 2a:
Closest non-ov
net to e crosses L
s(e)
e(e)
s(e’) e’
e: Most recently
routed net
e: Most recently
routed net e’
s(e’)
S(L) L L’ L
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
©Dutt

20. 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)

22. Acyclic VCG
Cyclic VCG 1a a 2 b 1b

23. 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