Rewiring – Review, Quantitative Analysis and Applications

Rewiring – Review, Quantitative Analysis and Applications Matthew Tang Wai Chung CUHK CSE MPhil 10/11/2003

Agenda • Review • Introduction • RAMBO, REWIRE • RAMFIRE • GBAW • Quantitative Analysis • Rewiring power • Runtime • Coverage • Applications • Circuit Partitioning • FPGA LUT mapping

Introduction – What is Rewiring? • Replacing a target wire by an alternative wire without changing the functionality of the circuit. • An alternative wire is a redundant wire to the circuit. • Its addition would make the target wire become redundant hence removable. • Let’s check out the classic example.

Classic Example

Classic Example – Resultant Circuit # gates: 9  6; # interconnect: 15  13

How can we find AWs? • RAMBO / REWIREATPG: Mandatory assignments to find candidate nodes for AWs. • RAMFIREFIRE: Redundancy identification.ATPG: connection type. • GBAWGraph-based pattern matching.

Automatic Test Pattern Generation (ATPG) • We consider stuck-at-fault model.A wire is called stuck-at-1/0 (s-t-1/0) if the value at the wire is always 1/0. • A wire is called redundant if its addition or removal to the circuit does not affect the functions at output. • In ATPG, some stuck-at-faults for some wires are not testable  these wires are redundant.

0 0 0 0 0 s-t-1 0 1 0 0/1 0 1 0/1 0/1 1 1 1 ATPG by Example 0/1 g6  g7 s-t-1 fault testable  g6  g7 not redundant

RAMBO – Basic Notion • Consider TW (ts, td) & AW (as, ad) • RAMBO considers absolute dominators td of as candidates for ad. • Create conflicts in M.A. by choosing suitable as.e.g. if dominator is a AND gate, choose with node M.A. = 0.

RAMBO - Algorithm foreach wire w in the network N { perform stuck-at-fault test on w; build candidate set C for w; Verify each wire in C by redundancy test; } • C can be very large  a lot of redundancy test needed.

REWIRE • Observation that some MA is forced, which means conflicts these would result in untestable stuck-at-fault. (observability MA) • Some fast filters are proposed to prune the candidate set to avoid unnecessary time-costy implication. • The flow of REWIRE is similar to that of RAMBO.

REWIRE - Algorithm foreach node nd in the network N { find OMA of nd; construct array of wires with nd; foreach wire wtof the array { do stuck-at-fault test on wt; build candidate set C; apply filters to C; Verify each wire in C; } }

RAMFIRE • Inefficiency in finding redundancy is a big problem with RAMBO and REWIRE. • FIRE: a fast redundancy identification algorithm. • No brute-force testing is needed, instead apply FIRE to get a set of alternative wires directly.

0 0 1 1 1 1 1 1 1 FIRE - Example * Uncontrollability Unobservability * * * * *

RAMFIRE - Algorithm foreach wire w in the network N { perform stuck-at-fault test on w (get MA); perform FIRE (1’ & 0’) on w; Build a set of redundant wire R for w; Form AW in R by creating conflicts in MA; } • Only three logic implications are used for each wire.

GBAW • Based on pre-configured graph patterns. • These patterns are extracted from result of RAMBO. • There are around 20 – 25 patterns in the library. • Each pattern correspond to a known TW / AW pair. • Notion: Pattern matching on the network to find AWs.

GBAW – Pattern Example • Local 13: a champion pattern • Actual logic transformation: DeMorgan’s law Forward Backward

Analysis Basis • Operation 1: Given a target wire, find a set of alternative wires. • Operation 2: Given a redundant wire to be added, find a set of wires becoming redundant after this addition. We compare the algorithms in handling operation 1. The algorithms of RAMBO and REWIRE are modified accordingly. All the experimental results are obtained on the same PC running Redhat Linux 8.0. (Duron 1.3 GHz, 256 MB)

Comparing Rewiring Power

Comparing Runtime (Theoretical) • Consider a circuit n nodes and w wires. • Let c and c’ be the sizes of the candidate set in RAMBO & REWIRE respectively. • Recursive Learning (rmax = 1): O(w2)

Comparing Runtime (Experimental)

Comparing GBAW & RAMBO (coverage) Forward Pattern: PI side  higher depth  lower depth  PO side Backward Pattern:

Comparing GBAW & RAMBO (coverage) • RAMBO is able to locate all forward patterns by GBAW.Why? Forward patterns are actually extracted from results of RAMBO. • RAMBO can’t find any backward patterns by GBAW.Why? The candidate destination node of the AW must be an absolute dominator in RAMBO.

Comments • GBAW & RAMFIRE: fast, strong in locating backward wires • RAMBO & REWIRE: much stronger rewiring power • Tradeoff: CPU runtime  Rewiring Power • GBAW can be an efficient pre-process filter to skip time-costy redundancy test.

Circuit Partitioning • Rewiring can be used in reducing the cut size. • Consider the wires on the cut as target wires, we try to find alternative wires which can be placed within clusters  smaller cut size. • Experimental Results:9% reduction in cut size with small penalty on the # of literals. (5-way partition)

FPGA LUT Mapping • Given a Boolean network, map the nodes into fixed size lookup tables (LUT). • The number of inputs to a LUT is fixed, say k. A k-LUT can implement any Boolean function with k variables. • Objective: • Minimize the total number of LUT used. • Minimize the depth of the mapping solution.

LUT-based mapping Example • k = 3

Motivation • Most LUT-based mapping algorithms do not consider modifying the Boolean network during mapping. • We would like to explore the possibility to apply logic transformation during mapping. • Logic transformation: our favourite – rewiring • We have picked Flowmap to work with due to its solution’s depth optimality.

Intuition – Utilizing LUTs • We found that quite a lot LUTs are not fully utilized, i.e. input used <= k.e.g. pcler8: k = 5; 12 / 47 nodes has 5 inputs. • Using rewiring, we can find backward wire to fill the unused input in LUTs. • When some LUT is utilized further, we may save some LUTs in the whole network.

Example

Example (Cont’) • GBAW: Local 111g6  g7 replaced by g6  g3

Future Work • Verify the intuition through more examples or experiments. • Observe and find out more intuitions. • Purpose a more complete scheme. THANK YOU

Rewiring – Review, Quantitative Analysis and Applications