Transcript pptx

ESE535:
Electronic Design Automation
Day 5: February 2, 2015
Clustering
(LUT Mapping, Delay)
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Behavioral
(C, MATLAB, …)
Today
Arch. Select
Schedule
RTL
• How do we map to LUTs?
• What happens when
– IO dominates
– Delay dominates?
• Lessons…
– for non-LUTs
– for delay-oriented partitioning
FSM assign
Two-level,
Multilevel opt.
Covering
Retiming
Gate Netlist
Placement
Routing
Layout
Masks
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LUT Mapping
• Problem: Map logic netlist to LUTs
– minimizing area
– minimizing delay
• Old problem?
– Technology mapping? (Day 3)
– How big is the library for K-input LUT?
• 22K gates in library
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Simplifying Structure
• K-LUT can implement any K-input
function
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Preclass: Cover in 4-LUT?
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Preclass: Cover in 4-LUT?
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Preclass: Cover in 4-LUT?
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Preclass: Cover in 4-LUT?
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Preclass: Cover in 4-LUT?
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Cost Function
• Delay: number of LUTs in critical path
– doesn’t say delay in LUTs or in wires
– does assume uniform interconnect delay
• Area: number of LUTs
– Assumes adequate interconnect to use
LUTs
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LUT Mapping
• NP-Hard in general
• Fanout-free -- can solve optimally given
decomposition
– (but which one?)
• Delay optimal mapping achievable in
Polynomial time
• Area w/ fanout NP-complete
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Preliminaries
• What matters/makes this interesting?
– Area / Delay target
– Decomposition
– Fanout
• replication
• reconvergent
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Costs: Area vs. Delay
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Decomposition
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Decomposition
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Fanout: Replication
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Fanout: Replication
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Fanout: Reconvergence
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Fanout: Reconvergence
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What makes it hard?
• Cost does not
monotonically
increase as cover
more of graph.
• Not clear when to
stop?
• We say cost does
not have a
monotone property
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Preclass Revisited
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Definition
• Cone: set of nodes in the recursive
fanin of a node
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Example Cones
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Delay
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Delay of Preclass Circuit?
• Poll: Delay of preclass circuit
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Dynamic Programming
• Optimal covering of a logic cone is:
– Minimum cost (all possible coverings)
• Evaluate costs of each node based on:
– cover node
– cones covering each fanin to node cover
• Evaluate node costs in topological order
• Key: are calculating optimal solutions to
subproblems
– only have to evaluate covering options at
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Flowmap
• Key Idea:
– LUT holds anything with K inputs
– Use network flow to find cuts
•  logic can pack into LUT including
reconvergence
• …allows replication
– Optimal depth arise from optimal depth
solution to subproblems
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Max-Flow / Min-Cut
• The maximum flow in a network is equal
to the minimum cut
– …the bottleneck
• We can find the mincut by computing
the maxflow.
• Conceptually, how would we determine
the maximum flow?
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Example Flow cut
s
t
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Example Flow cut
Visually, what is min-cut?
s
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MaxFlow
• Set all edge flows to zero
– F[u,v]=0
• While there is a path from s,t
– (breadth-first-search)
– for each edge in path f[u,v]=f[u,v]+1
– f[v,u]=-f[u,v]
– When c[v,u]=f[v,u] remove edge from search
• O(|E|*cutsize)
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Example Flow cut
Find a path?
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Example Flow cut
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Example Flow cut
Find a path?
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Example Flow cut
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Example Flow cut
Find a path?
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Flowmap
• Delay objective:
– minimum height, K-feasible cut
– I.e. cut no more than K edges
– start by bounding fanin  K
Examples are K=4
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• Height of node will be:
– height of predecessors or
– one greater than height of predecessors
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• Check shorter first
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Flowmap
• Construct flow problem
– sink  target node being mapped
– source  start set (primary inputs)
– flow infinite into start set
– flow of one on each link
– to see if height same as predecessors
• collapse all predecessors of maximum height
into sink (single node, cut must be above)
• height +1 case is trivially true
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Example Subgraph
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Target: K=4
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Trivial: Height +1
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Collapse at max height
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Collapse at max height
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Collapsed
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Augmenting Flows
Collapsed
Node
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Augmenting Flows
Collapsed
Node
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Augmenting Flows
Collapsed
Node
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Augmenting Flows
Collapsed
Node
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Augmenting Flows
Collapsed
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Collapse at max height: works for K=4
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Collapsed
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Collapse not work (K still 4)
(different/larger graph)
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Forced to label height+1
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Reconvergent fanout
(yet different graph)
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Can label at height
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Flowmap
• Max-flow Min-cut algorithm to find cut
• Use augmenting paths until discover
max flow > K
• O(K|E|) time to discover K-feasible cut
– (or that does not exist)
• Depth identification: O(KN|E|)
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Mincut may not be unique
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Collapsed
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Flowmap
• Min-cut may not be unique
• To minimize area achieving delay
optimum
– find max volume min-cut
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Compute max flow  find min cut
remove edges consumed by max flow
DFS from source
Compliment set is max volume set
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Collapse at max height: works for K=4
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Collapsed
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BFS from Source
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BFS from Source
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BFS from Source
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BFS from Source
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Collapsed
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Does not find rest.
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Max-Volume Mincut
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Flowmap
• Covering from labeling is straightforward
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process in reverse topological order
allocate identified K-feasible cut to LUT
remove node
postprocess to minimize LUT count
• Notes:
– replication implicit (covered multiple places)
– nodes purely internal to one or more covers may
not get their own LUTs
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Flowmap Roundup
• Label
–
–
–
–
Work from inputs to outputs
Find max label of predecessors
Collapse new node with all predecessors at this label
Can find flow cut  K?
• Yes: mark with label (find max-volume cut extent)
• No: mark with label+1
• Cover
– Work from outputs to inputs
– Allocate LUT for identified cluster/cover
– Recurse covering selection on inputs to identified LUT
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Area
Changing Cost Functions Now
(previous was delay)
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DF-Map
• Duplication Free Mapping
– can find optimal area under this constraint
– (but optimal area may not be duplication
free)
[Cong+Ding, IEEE TR VLSI Sys. V2n2p137]
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Maximum Fanout Free Cones
MFFC: bit more general than trees
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MFFC
• Follow cone backward
• end at node that fans out (has output)
outside the cone
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MFFC example
Identify FFC
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MFFC example
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DF-Map
• Partition into graph into MFFCs
• Optimally map each MFFC
• In dynamic programming
– for each node
• examine each K-feasible cut
– note: this is very different than flowmap
where only had to examine a single cut
– Example to follow
• pick cut to minimize cost
– 1 +  cones for fanins
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DF-Map Example
Cones?
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DF-Map Example
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DF-Map Example
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DF-Map Example
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DF-Map Example
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DF-Map Example
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DF-Map Example
Start
mapping
cone
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DF-Map Example
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DF-Map Example
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DF-Map Example
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DF-Map Example
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DF-Map Example
Similar
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Composing
• Don’t need minimum delay off the
critical path
• Don’t always want/need minimum delay
• Composite:
– map with flowmap
– Greedy decomposition of “most promising”
non-critical nodes
– DF-map these nodes
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Variations on a Theme
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Applicability to Non-LUTs?
• E.g. LUT Cascade
– can handle some functions of K inputs
• How apply?
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Adaptable to Non-LUTs
• Sketch:
– Initial decomposition to nodes that will fit
– Find max volume, min-height K-feasible cut
– ask if logic block will cover
• yes  done
• no  exclude one (or more) nodes from block and
repeat
– exclude == collapse into start set nodes
– this makes heuristic
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Partitioning?
• Effectively partitioning logic into clusters
– LUT cluster
• unlimited internal “gate” capacity
• limited I/O (K)
• simple delay cost model
– 1 cross between clusters
– 0 inside cluster
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Partitioning
• Clustering
– if strongly I/O limited, same basic idea
works for partitioning to components
• typically: partitioning onto multiple FPGAs
• assumption: inter-FPGA delay >> intra-FPGA
delay
– w/ area constraints
• similar to non-LUT case
– make min-cut
– will it fit?
– Exclude some LUTs and repeat
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Clustering for Delay
• W/ no IO constraint
• area is monotone property
• DP-label forward with delays
– grab up largest labels (greatest delays)
until fill cluster size
• Work backward from outputs creating
clusters as needed
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Area and IO?
• Real problem:
– FPGA/chip partitioning
• Doing both optimally is NP-hard
• Heuristic around IO cut first should do
well
– (e.g. non-LUT slide)
– [Yang and Wong, FPGA’94]
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Partitioning
• To date:
– primarily used for 2-level hierarchy
• I.e. intra-FPGA, inter-FPGA
• Open/promising
– adapt to multi-level for delay-optimized
partitioning/placement on fixed-wire
schedule
• localize critical paths to smallest subtree
possible?
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Summary
• Optimal LUT mapping NP-hard in
general
– fanout, replication, ….
• K-LUTs makes delay optimal feasible
– single constraint: IO capacity
– technique: max-flow/min-cut
• Heuristic adaptations of basic idea to
capacity constrained problem
– promising area for interconnect delay
optimization
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Today’s Big Ideas:
• IO may be a dominant cost
– limiting capacity, delay
• Exploit structure: K-LUTs
• Mixing dominant modes
– multiple objectives
• Define optimally solvable subproblem
– duplication free mapping
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Admin
• Reading Wednesday on web
• Assignment 3 due Thursday
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