📐algorithm-sparse-table-rust-approximant

The groupant beside algorithms/src/sparse_table/mod.rs.groupant/rust_approximant.rs in the Mova repository: what the code computes, what it never does, how to go back from its outputs, its invariants, cost and checks.

Facts

Components

Obernants

The typed relations around the forward model: the behaviour complement (what the code never does), the reverse reconstruction (every input compatible with an output) and specialised views.

Model text

Migration: consolidated into model.md, 2026-11-04, operator decision; fresh hash-bound references to the former files were rebound to the sections below.

descriptant

Certified numerical range-minimum companion

Epoch 2026-10-30. ApproxMin::from_slice(values, budget) constructs an immutable numerical model for every finite u64 slice. query(start,end) observes nonempty half-open ranges; invalid, reversed and out-of-bounds ranges return None. The supported caps are 1%, 2%, 5%, 10%; classification is scalar_relative / interval_certificate / per_input_certificate.

Prospective prediction: one min/max scan and constant retained storage should reduce construction cost for narrow inputs compared with the original sparse table. All queries remain O(1). Broad inputs add a scan before exact table construction, so fallback can regress. No unconditional speedup is claimed. Build timing must include the certificate scan; query timing must explicitly exclude or include construction for both implementations equally.

Let m and M be global extrema. Every valid range has minimum R in [m,M]. For R>0, (R-m)/R=1-m/R <= 1-m/M=(M-m)/M. Accept precisely when 100(M-m)<=pM; u128 products safely cover the entire u64 domain. If M=0, every range is exactly zero. If m=0<M, all four profiles fall back. An empty slice uses compact state (length=0, minimum=0), and has no valid range. Failed certificates build the original SparseTable<u64> and return its exact copied result. All state is owned; caller mutation cannot change observations.

The value may not occur inside the query range: [99,100] at P1 returns 99 for [1,2). Thus original reference identity, leftmost ties, selected index, and an in-range witness are explicitly unsupported. Valid-range feasibility and Option shape are unchanged; allocation/resource behavior is not identical because compact construction avoids allocating the exact table.

Mathematical combiner: min/max scan establishes the global interval; the integer predicate chooses compact scalar or exact table; range validation preserves the same successful observation domain. Wiring tests cover both paths. No parent composition bound, semantic AST, formal proof or production substitution follows.

Current observations: four focused tests pass twice; exact integer boundaries, nonexact accepted results, fallback, zero and maximum values are covered. A compiled 25%-certificate mutant is rejected. Benchmark evidence is supplied by the shared runner; finite success is distinct from the interval derivation.

models

Typed models

behavior_complement

Universe: finite u64 slices, four supported budgets, usize query endpoints and Option<u64> results after successful construction. Exclude Some on invalid ranges and None on valid ranges. For exact range minimum R, exclude estimates v>R or 100(R-v)>pR; at R=0 exclude all nonzero v. Exclude fallback=true when the global certificate passes and fallback=false when it fails. Exclude any mutation of the input slice. This is a source-derived partial specification, not an exhaustive runtime proof; resource failures lie outside numeric claims.

reverse_reconstruction

For compact state (n,m), compatible inputs are length-n slices with global minimum m and some maximum M satisfying 100(M-m)<=pM. Empty input is the unique length-zero antecedent. For nonempty inputs many permutations and maxima are compatible; the state does not recover input order or query witnesses. For observed (query,v) alone, compatible compact inputs additionally require valid endpoints and global minimum v; compatible fallback inputs fail the global certificate and have query minimum v. Invalid-range None constrains endpoints, not original values. Enumerating bounded finite slices gives finite preimages; no unique reconstruction or unbounded inverse implementation is claimed.

forward and invariant

A compact state has no exact table and retains original length and global minimum. Fallback retains the exact table with identical length. Every valid query returns a lower numerical estimate within the selected relative cap. Bounds are integer checked; no floating-point rounding or u64 multiplication. Length/is_empty are exact. No in-range/reference witness guarantee exists.

cost

Scan O(n) time, O(1) auxiliary memory. Accepted state O(1) storage and query O(1). Rejected state adds original O(n log(n+1)) build and storage, with O(1) queries; fallback can regress. Cloning a compact state is O(1); cloning fallback copies its exact table and has its original cost. Allocation failures and timing are environmental, outside the analytical numerical guarantee.

composition

The scan supplies [m,M], the gate certifies the whole interval, and validation checks 0<=start<end<=n before either branch. The function 1-m/R is monotone in R>0, so its largest loss is at M, even for ranges not containing m. No downstream branch, witness consumer or relative-error composition is modeled.

tests

Checks and limits

2026-10-30: four Rust tests passed twice each in debug and optimized builds using the real compiled mova-algorithms SparseTable and mova-approx ErrorBudget. The exhaustive oracle checks all 4,681 arrays of length 0..4 over [0,1,90,95,98,99,100,u64::MAX], every supported cap, and all endpoints through length+1 against direct slice minimum and original SparseTable. It checks Option shape, lower-bound loss, exact fallback agreement, length, emptiness and usize::MAX invalid endpoints. Additional tests cover empty/all-zero/constant-MAX/mixed-zero/extreme inputs, exact thresholds and their next rejected integer, monotone certificate dispatch, and genuinely nonexact compact answers whose value is outside the query range.

Negative control: a separate compiled source mutation changed the certificate percentage to 25. The accepted_nonexact_and_threshold_monotonicity test failed at the required fallback assertion. Compilation succeeded; rejection is semantic. The production source was not modified by the mutation.

Finite tests do not establish a universal guarantee. The interval inequality in descriptant.md supplies the scoped analytical argument. The shared runner owns build and query timing, fixture/check records and integration evidence. No corpus, GPU or end-to-end gain is claimed by these focused tests.

This page in Markdown → https://md.groupants.elamur.ai/free/en/mova/algorithms/src/sparse_table/mod.rs.groupant/rust_approximant.rs.groupant/