FPGA Intra-cluster Routing Crossbar Design

Dr. Philip BriskDepartment of Computer Science and Engineering

University of California, Riverside

CS 223

Generating Highly Routable Sparse Crossbars for PLDs

Guy Lemieux, Paul Leventis, David LewisInternational Symposium on FPGAs, 2000

Basic Notation

Fully Populated Crossbar

• Full capacity – can connect as many signals as the number of outputs

• Flexibility – Can connect any input to any output

Full-capacity Minimal Crossbars

• Full capacity• Reduced Flexibility: you lose the ability to

connect any input to any output• p = m(m – n + 1)

switches

Full-capacity Minimal Crossbars

…

• Area savings is minimal if n >> m

Perfect and Sparse Crossbars

• Perfect crossbars– Can disjointly route any m-sized subset of the n inputs

to the m outputs– Both full and full-capacity minimal crossbars are perfect

• Sparse crossbars– Has p < m(m – n + 1) switches– Cannot be perfect

Bipartite Graph Representation

I1 I2 I3 I4 I5 I6

O1

O2

O3

O4

I1

I2

I3

I4

I5

I6

O1

O2

O3

O4

Evaluation Challenge

• How “routable” is a given crossbar?

– Build an FPGA, map 20+ applications, observe results• Slow, highly subject to the application mix

– Monte Carlo Test• Generate random test vectors• Route each test vector on the crossbar (network flow)• Report number of successes as a percentage• A highly routable sparse crossbar has a >= 95% success rate

Hall’s Theorm• Given a bipartite graph G = (V, E)– X, Y are the bipartite independent sets of G

G has a matching of X onto Y if and only if

N(v) is the set of neighbors of vertex v N(S) is the set of neighbors of all vertices in S

• Leverage Hall’s Theorem to generate routable sparse crossbars!

Practical Issues

• Cannot enumerate all subsets of m inputs• N(x) should be approximately equal for all

input vertices x in X– Otherwise, any subset containing a large number

of low-degree vertices is unlikely to be routable• N(y) should be approximately equal for all

output vertices y in Y– Symmetric argument

Hamming Distance and Coding Theory

• Represent N(v) as a bitvector bv– bv[i] = 1 if v fans out to Oi

• Hamming Distance– d(bv1, bv2)

• Strategy– Maximize d(bvi, bvj) for every pair of distinct vertices vi

and vj

Switch Placement Optimizer

• Start with initial switch placement• Generate random swap of switch positions– Accept the swap if there is an improvement– Otherwise, reject the swap

• Stop after a fixed number of swap candidates (e.g., 10K) fails to find an improvement

• Objective is to minimize:

Example

Identical Hamming costs before and after the swap

Before: cannot route {1, 2, 3}After: reduces Hamming costs

168x24 Crossbar, 10K Test Vectors

Altera Flex 8000 HP Plasma Hextant

# Switches vs. Routability

Using Sparse Crossbars within LUT Clusters

Guy Lemieux, David LewisInternational Symposium on FPGAs, 2001

Five Questions1. Will depopulation save area, require greater routing area, or create

unroutable architectures?2. Will depopulation reduce or increase routing delays?3. What amount of depopulation is reasonable?4. How much area or delay reduction can be attained, if any?5. What are the other effects of depopulating the cluster?

Architecture and Parameters

Results

Designing Efficient Input Interconnect Blocks for LUT Clusters Using

Counting and Entropy

Wenyi Feng and Sinan KaptanogluACM Transactions on Reconfigurable Technology and

Systems (TRETS), 1(1): article #6, March, 2008

Note: Paper is from Actel (now Microsemi)

Count Configurations (Details Omitted)

784 Configurations 312 Configurations 256 Configurations

Routing Requirement Vector (RRV)• An ordered list of N subsets

containing K distinct signals

• The ith subset is K distinct signals to route to the ith K-LUT

• Total number of RRVs for the crossbar:

M inputsKN outputs

Entropy of an Intra-cluster Routing Crossbar

• H = lg(# routable RRVs)– Accounts for equivalence of LUT inputs

• Why Entropy?– # routable RRVs is huge– Minimum number of configuration bits to program the crossbar– Inversely correlated with usage of global routing muxes (details

omitted)• If we reduce the routability of the crossbar, we will end up

programming more global routing muxes to compensate for the entropy loss

Conceptual Idea

intra-cluster crossbar

global routing

Theorem

• Let P and L be the number of muxes and switches in a crossbar– The entropy is at most Plg(L/P)– The entropy per switch is at most log(L/P) / (L/P) – These bounds are achieved only when each mux

has size L/P and each configuration realizes a unique RRV

• Proof omitted because I DO NOT HATE YOU!

What are we doing here?

• Lemieux and Lewis– Routability: Monte Carlo simulations– Area: Count switches

• Feng and Kaptanoglu– Routability: Crossbar entropy– Area: Entropy per switch– Caveat: Focus only on crossbars where we can count

routable, non-redundant RRVs!

Type-1 Crossbar

• 1-level– L2 muxes are driven

directly by crossbar input signals

– #routable RRVs depends on L2 crossbar topology

• Not area-efficient due to big L2 muxes

• Xilinx Virtex-style

Type-2 Crossbar

• 2-level– L1 is sparsely

populated– L2 is fully populated

• Fully populated L2 reduces area efficiency

• VPR– Fc,in determines L1

population density

Type-3 Crossbar

• 2-level, Partitioned– L1 partition Pi only drives

L2 partition Oi

– From input m to LUT input n, all paths go through muxes in Pi and Oi exclusively

– #Routable RRVs is the product of #Routable RRVs for each disjoint sub-crossbar

Proposed Type-3 Crossbar and Generation Algorithm

• Each sub-crossbar is Type-2• Can count #routable RRVs (Details omitted)

Entropy vs. # Switches

Entropy vs. Global Routing Mux Usage

The Bottom Line…

• Who cares…– Theoretical properties are cute– Actel/Microsemi did not use these crossbars in

their FPGAs

• Practical observation…– The cheaper you make the intra-cluster routing

crossbar, the more expensive the global routing…

A 65nm flash-based FPGA fabric optimized for low cost and power

Jonathan W. Greene, et al.International Symposium on FPGAs, 2011

Note: Paper is from Microsemi (Feng and Kaptanoglu are co-authors)

Corporate Secrets Divulged• They used a Clos Network– Three parameters: m, n, r

Clos Network Properties

• Used when the physical circuit switching needs to exceed the capacity of the largest feasible single crossbar

• Much cheaper than a fully populated nxn crossbar

Strict-sense Nonblocking Clos Network(m > 2n – 1)

• An unused input on an ingress switch can always be connected to an unused output on an egress switch, without reconfiguration!

Rearrangeably Nonblocking Clos Network(m > n)

• An unused input on an ingress switch can always be connected to an unused output on an egress switch, but reconfiguration may be necessary!

Recursive Clos Network Design• Scalable to any ODD

number of stages– Replace center crossbar with

a 3-stage Clos Network