CS137: Simplifying Structure Electronic Design Automation K-LUT - PDF document

CS137: Simplifying Structure Electronic Design Automation • K-LUT can implement any K-input function Day 3: September 29, 2005 Clustering (LUT Mapping, Delay) 1 4 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon Today Cost Function • How do we map to LUTs? • Delay: number of LUTs in critical path – doesn’t say delay in LUTs or in wires • What happens when delay dominates? – does assume uniform interconnect delay • Lessons… – for non-LUTs • Area: number of LUTs – for delay-oriented partitioning – Assumes adequate interconnect to use LUTs 2 5 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon LUT Mapping LUT Mapping • Problem : Map logic netlist to LUTs • NP-Hard in general – minimizing area • Fanout-free -- can solve optimally given – minimizing delay decomposition – (but which one?) • Old problem? • Delay optimal mapping achievable in – Technology mapping? (last week) Polynomial time – Library approach require 2 2K gates in library • Area w/ fanout NP-complete 3 6 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon 1

Decomposition Preliminaries • What matters/makes this interesting? – Area / Delay target – Decomposition – Fanout • replication • reconvergent 7 10 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon Area vs. Delay Fanout: Replication 8 11 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon Decomposition Fanout: Replication 9 12 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon 2

Fanout: Reconvergence Delay 13 16 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon Dynamic Programming Fanout: Reconvergence • 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 each node 14 17 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon Monotone Property Flowmap • Does cost function increase • Key Idea : monotonicly as more of the graph is – LUT holds anything with K inputs included? (do all subsets have property) – Use network flow to find cuts • ≡ logic can pack into LUT including – gate count? reconvergence – I/o? • …allows replication • Important? – Optimal depth arise from optimal depth – How far back do we need to search? solution to subproblems 15 18 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon 3

Trivial: Height +1 Flowmap • Delay objective: 1 1 1 1 – minimum height, K-feasible cut 1 1 – I.e. cut no more than K edges – start by bounding fanin ≤ K 2 1 1 • Height of node will be: 2 – height of predecessors or 2 – one greater than height of predecessors 3 • Check shorter first 19 22 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon Collapse at max height Flowmap • Construct flow problem – sink ← target node being mapped 1 1 1 1 – source ← start set (primary inputs) 2 – flow infinite into start set – flow of one on each link – to see if height same as predecessors 2 • collapse all predecessors of maximum height into sink (single node, cut must be above) • height +1 case is trivially true 20 23 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon Collapse not work Example Subgraph (different/larger graph) 1 1 1 1 1 1 1 1 2 2 Target: K=4 2 2 Forced to label height+1 2 21 24 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon 4

Reconvergent fanout (different/larger graph) Two 3-cuts 1 1 1 1 1 2 Can label at height 2 25 28 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon Flowmap Flowmap • Max-flow Min-cut algorithm to find cut • Min-cut may not be unique • Use augmenting paths to until discover • To minimize area achieving delay max flow > K optimum – find max volume min-cut • O(K|e|) time to discover K-feasible cut • Compute max flow ⇒ find min cut – (or that does not exist) • remove edges consumed by max flow • Depth identification: O(KN|e|) • DFS from source • Compliment set is max volume set 26 29 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon Graph Flowmap • Min-cut may not be unique 27 30 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon 5

Graph: maxflow (K=3) Flowmap • Covering from labeling is straightforward – 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 31 34 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon Graph: BFS source Flowmap Roundup • Label Reachable – 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 32 35 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon Max Volume min-cut Area 33 36 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon 6

DF-Map MFFC example • 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] 37 40 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon Maximum Fanout Free Cones MFFC example MFFC: bit more general than trees 38 41 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon MFFC DF-Map • Partition into graph into MFFCs • Follow cone backward • Optimally map each MFFC • end at node that fans out (has output) • In dynamic programming outside the code – for each node • examine each K-feasible cut – note: this is very different than flowmap where only had to examine a single cut • pick cut to minimize cost – 1 + Σ MFFCs for fanins 39 42 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon 7

DF-Map Example DF-Map Example Cones? 43 46 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon DF-Map Example DF-Map Example 44 47 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon DF-Map Example DF-Map Example 45 48 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon 8

DF-Map Example DF-Map Example Start mapping 1 1 cone 1 ? 1 49 52 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon DF-Map Example DF-Map Example 1 1 1 1 1 1 1 ? 1 50 53 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon DF-Map Example DF-Map Example 1 1 1 1 1 ? 1 1 1 1 51 54 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon 9

DF-Map Example DF-Map Example Similar 1 1 1 1 to previous 1 1 1 1 1 1 1 1 1 1 3 ? 55 58 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon DF-Map Example DF-Map Example 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ? ? 56 59 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon DF-Map Example DF-Map Example 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ? ? 57 60 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon 10

DF-Map Example DF-Map Example 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ? 2 ? 61 64 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon DF-Map Example DF-Map Example 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ? 2 ? 62 65 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon DF-Map Example DF-Map Example 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 2 2 ? 63 66 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon 11

DF-Map Example DF-Map Example 3 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 3 3 2 ? 2 ? 67 70 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon DF-Map Example DF-Map Example 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 3 2 ? 2 3 68 71 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon DF-Map Example DF-Map Example 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 3 2 ? 2 3 ? 69 72 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon 12

DF-Map Example DF-Map Example 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 4 2 3 ? 2 3 ? 73 76 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon DF-Map Example DF-Map Example 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 4 4 2 3 ? 2 3 ? 74 77 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon DF-Map Example DF-Map Example 3 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 4 4 2 3 ? 2 3 ? 75 78 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon 13