📐algorithm-prufer

The groupant beside algorithms/src/prufer/mod.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

descriptant

algorithm-prufer: epoch memory

2026-10-30: prospective model selected exact capped algorithms with deterministic canonical outputs. For a labeled undirected tree, encode repeatedly removes the smallest live leaf and emits its neighbor until two remain. Decode counts occurrences, sets degree(v)=1+occ(v), repeatedly joins the smallest degree-one vertex to the next code symbol, then joins final leaves. Sort normalized output edges. Empty codes are disambiguated by explicit n=0,1,2.

On n>=2, degree(v)=1+occ(v) initially in decoding; each step consumes a leaf and one occurrence while maintaining the live-tree degree sum. Encoding validates n-1 distinct loopless edges and acyclicity, which implies connectivity for n>0. Codes have length max(n-2,0). n=0/1 encode/decode empty edges.

O(n log(n+1)) time for ordered duplicate checking, heap operations and output sort; O(n) storage. n<=1000000 checked before allocation. Encoding adjacency scans total O(n): each removed vertex scans its own adjacency once. Rejected wrong lengths avoid unbounded edge processing.

Retained observations: All words for n=0..6 compare both directions with independent adjacency-matrix leaf elimination, connectivity, edge uniqueness, canonical order and degree laws; explicit minimum-leaf example and invalid tree/code boundaries. The compiled production mutation fails canonical_minimum_leaf; exact original bytes precede the four positive runs. No timing comparison or speed improvement was attempted. Models describe intended relations and retain ambiguous reverse fibers; source identity and structural validation do not prove semantics.

Pruefer (1918), Neuer Beweis eines Satzes ueber Permutationen. Original heap implementation and original fixtures.

Model refinement, 2026-10-30: five typed views now distinguish the two baseline relations and three concrete specializations. New validation, observation and oracle constraints retain which claims have native witnesses and which remain source-derived. Rust bytes, partial syntax and native evidence are unchanged.

model

algorithm-prufer

Forward

For a labeled undirected tree, encode repeatedly removes the smallest live leaf and emits its neighbor until two remain. Decode counts occurrences, sets degree(v)=1+occ(v), repeatedly joins the smallest degree-one vertex to the next code symbol, then joins final leaves. Sort normalized output edges. Empty codes are disambiguated by explicit n=0,1,2.

Invariant

On n>=2, degree(v)=1+occ(v) initially in decoding; each step consumes a leaf and one occurrence while maintaining the live-tree degree sum. Encoding validates n-1 distinct loopless edges and acyclicity, which implies connectivity for n>0. Codes have length max(n-2,0). n=0/1 encode/decode empty edges.

Cost

O(n log(n+1)) time for ordered duplicate checking, heap operations and output sort; O(n) storage. n<=1000000 checked before allocation. Encoding adjacency scans total O(n): each removed vertex scans its own adjacency once. Rejected wrong lengths avoid unbounded edge processing.

behavior_complement

U contains labeled edge lists or words, explicit n, and candidate Result values. Wrong lengths/labels, loops, reversed duplicates, or cyclic n-1 edge lists cannot yield Ok. For valid trees, code from a nonminimum current leaf is excluded; decode outputs with wrong degrees, noncanonical edge order, cycles or missing connectivity are excluded.

reverse_reconstruction

For n>=2 fixed n, the valid code identifies one undirected labeled tree; encoding forgets edge order and endpoint orientation, so its concrete edge-list fiber contains all permutations and orientations of that tree. Decode with fixed n has one compatible word. Without n, [] also represents the distinct n=0,1,2 conventions; omitting n loses that distinction.

degree_fiber

For n>=2, fixing just decoded degree vector d admits every word of length n-2 with exactly d(v)-1 occurrences of v; occurrence order is lost by the degree observation. Different such words may encode different trees. Full code retains this order and restores the unique labeled edge set.

Provenance

Pruefer (1918), Neuer Beweis eines Satzes ueber Permutationen. Original heap implementation and original fixtures.

Limitations

Partial syntax is source-bound, not a semantic AST; macros remain opaque. Finite exhaustive small domains and mutation witnesses are not formal proof. No runtime speed claim, approximation cap, automatic substitution or unbounded resource guarantee.

tree_validation_partition

encode first rejects n>MAX_VERTICES, then edges.len()!=n.saturating_sub(1), then processes edges in order: u label, v label, self-loop, normalized duplicate, DSU cycle. A reversed repeat reports RepeatedEdge before the cycle check; a first cycle reports NotTree before inspecting later edges. With n-1 distinct loopless edges, acyclicity implies connectedness for n>0. decode checks cap then length then labels. Thus a wrong-length word with an invalid symbol yields WrongLength. n=0/1 accept only empty edge lists and words.

canonical_tree_identity

For fixed n>=2, decode(encode(n,E)) equals the sorted set of normalized edges {(min(u,v),max(u,v))}; it need not equal E as an ordered oriented list. encode(decode(n,C))=C for every valid word. At each elimination the heap chooses the smallest live degree-one label, so the star/path fixture [(4,0),(4,1),(4,2),(4,3),(3,5)] yields [4,4,4,3]. The degree fiber alone loses this temporal order: for n=4 codes [0,1] and [1,0] have the same degrees [2,2,1,1] but different decoded trees.

evidence

Native checks

Three named tests pass twice debug and twice release with identical normalized names/outcomes. All words for n=0..6 compare both directions with independent adjacency-matrix leaf elimination, connectivity, edge uniqueness, canonical order and degree laws; explicit minimum-leaf example and invalid tree/code boundaries.

Compiled production mutation for prufer fails the named assertion canonical_minimum_leaf; production bytes restored before positive repetitions. Strict all-target algorithms Clippy and narrow rustfmt pass. Six partial syntax projections generated twice identically; full scaffold groupants checked twice. Raw logs live in $MOVA_DATA/runs/octopus-wave16-graphs-2026-10-30; evidence.json binds hashes. No live corpus, parser, teacher, GPU or full-school execution.

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