Buffered Routing Tree Construction Under Buffer Placement Blockage

Wei Chen & Massoud PedramUSC

Premal BuchMagma Design Automation

VLSID 2002

Outline

Motivation Preliminaries BRBP Algorithm Experiment Result Conclusion

Motivation (1/3)

Buffering has been a common technique to improve circuit speed

No previous works construct buffered multi-pin routing tree with consideration to blockages

Motivation (2/3)

Motivation (3/3)

Proposed algorithm has three characteristics:Buffer locations are not pre-determinedMulti-pin nets are easily handledUse line-search algorithm instead of maze

route

Preliminaries (1/3)

Problem definition:Given placement blockages and locations of

sources and sinks for all nets

To simultaneously build the net topology and insert sized buffer (inverter)

Preliminaries (2/3) The delay model of buffer/inverter is a re-

formulation of conventional RC model of CMOS gate delay[5]

The delay of a buffer:

d = τ(p + gh)

τ:scaling parameter p : parasitic delay of gateg : logical effort of the gate

h : electrical effor(gain) Cl / Cin

Preliminaries (3/3)

Interconnect delay: Elmore delay model

BRBP Algorithm

BRBP algorithm flow Hanan graph Data structure and base solution Solution propagation Edge buffering Pruning Algorithm flow and time complexity

BRBP － Algorithm FlowHanan graph

Initialize base solutions

Solution queue is empty?

Yes

Implement best solution & map buffer/invters to real library

Update timing

Pop a solution &find its escape nodes

For each escape node

Expand solution to escape node

Merge and prune

New solution been dominated?

No

Insert back to queue

BRBP － Hanan graph

BRBP － Data Structure Each solution has five labels:

root : a pointer to the root of node tree formed by now

cap : capacitive load seen from rootreq : required arrival time at rootreachable_set: a set of the nodes that are

reachable from rootrepeater : the type(buffer, inverter, null) and

size inserted at root

BRBP － Base solution (1/2)

BRBP － Base solution (2/2)

All base solutions are pushed into a priority queue priority_sols

Each time returns the solution with largest req (less critical sinks or partial solutions)

BRBP － Solution Propagation (1/4)

Find escape nodes:

BRBP － Solution Propagation (2/4)

BRBP － Solution Propagation (3/4)

BRBP － Solution Propagation (4/4)

All of the new higher-level solutions are pushed back to priority_sols

After a popped-out solution expands to all its escape nodes, another solution is popped

Repeat the process until the priority_sols is empty or a solution reaches all the sinks

BRBP － Edge Buffering

If a edge between two roots are too long, we should also insert buffer/inverter on it

The maximum length that needs no buffer can be easily determined[10]

Generate new solutions at end of each segment

BRBP － Pruning (1/2)

Consider the set of solutions with the same root and driving the same sink set

For any two solutions u and v, if capu > capv and requ <= reqv, then u is dominated by v and will be dropped from priority_sols

BRBP － Pruning (2/2)

Pruning does not affect optimality

Pruning is performed in two cases:When a solution merges with the other rooted

at its escape node

Edge buffering

Example

BRBP － Time complexity

If we have N M Hanan grids, at most K solutions ‧are kept after each pruning, and n is the number of sinks:

Worst-case space complexity: O(N M 2‧ ‧ n K)‧

Time complexity: O(N2 M‧ 2 2‧ (2n-m) K‧ 2)

Experiment Result

Conclusion

A dynamic-programming based algorithm is presented to perform buffered routing with placement blockages

Experiment result shows average improvement of 7.9% on longest delay and 2.3% on median port slack