Skip zero entries in fraction-free row elimination#3
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Store the tableau as Z.t numerators over one positive shared denominator d (cell value = entry/d) and pivot with Edmonds' fraction-free rule (a.(j) <- (a.(j)*p - a.(x)*row.(j)) / dprev, exact Z.divexact). This replaces the Q.t array array tableau whose numerators/denominators blew up and paid a GCD on every arithmetic op. The change is behaviour-preserving: every cell value is exactly Q.make entry d, so the Bland/max pivot sequence, verbose intermediate tableaux, and final rational answers are byte-identical (dune runtest passes with no golden regeneration). Selection compares signs (Z.sign) and ratios (Q.make b pv), both denominator-invariant since d > 0.
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What
Short-circuit zero cells in
ff_eliminate(the fraction-free pivot inner loop).Why
I measured cell-level statistics for the inner loop on the benchmark corpus and found these exact-rational tableaux are 65–80% zero mid-solve — much sparser than the ~50% assumed when
ff_eliminatewas written. Concretely, 55–67% of all inner-loop cell visits have botha.(j)=0and the pivot-row entryrow.(j)=0. For those cells the new valuea.(j)*p - ac*row.(j)is identically zero (andZ.divexact 0 dprev = 0), yet the old code spent two bignum multiplies, a subtraction and an exact division to recompute a zero.How
Add an
a.(j)=0short-circuit to every branch offf_eliminate:a.(j)androw.(j)zero → leave the cell zero, no work at all (the dominant case);a.(j)zero → value is-ac*row.(j)(drop the vanishinga.(j)*pproduct);row.(j)zero → value isa.(j)*p(the existing fast path).Each simplified expression is still the full Bareiss numerator for that cell (the dropped term is provably zero), so the exact
/dprevdivision stays valid and every result is bit-identical.dune runtestpasses with no golden changes.Result
Benchmark harness (
bench/evaluate.py --baseline main --candidate <branch> --rule bland --trials 30), correctness-checked against the GLPK + Gurobi oracle consensus:rand_25x6016.5ms → 14.6ms; best instancedegen_141.31×This lands on top of the fraction-free + pricing + phase-1 work already merged in #2.