is_bipartite

std.linalg.is_bipartite · Level L2

Whether a finite undirected graph is bipartite, as 1 or 0: it has no closed walk of odd length, which is tr(Aᵏ) = 0 for every odd k ≤ n. The odd traces are read off without a parity test, as (tr(Aᵏ) − tr((−A)ᵏ))/2. The reference two-colours the graph by breadth-first search instead.

bipartite ⇔ tr(Aᵏ) = 0 for all odd k ≤ n

Signature

is_bipartite(A: f64[n, n]) → f64[]

Structure

The function as NOVA stores it: one box per input, operation and output, and arrows that carry values. A double border marks another library function this one runs — called once, or by Scan once per element; select it to open that function.

input
operation
constant
call
output

Verification

Signature proven by NOVA’s shape solver, for every size.
Equal to the reference two-colouring by breadth-first search in exact rational arithmetic, on all 40 test cases.
All 40 float64 results are exact: the error is zero.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 40 of them exact, with zero error.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 100%
bit-equal to the NumPy formula in float64: 100%
largest error, in units in the last place: 0

Note

Exact while the closed-walk counts stay below 2⁵³: (n − 1)ⁿ, so every n up to about 15.

Identity

Calls

matpow_step

Called by

—

sha256:4602ca548bc7b1c41fb6a6e05da8314806d30859bf3788b85233babffe4051c9

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.

Control handle

Symbol: Ω:std.linalg.is_bipartite · Ω:is_bipartite
Pins: sha256:0f1131073f5cc83f8cc56029875ada53a68e1db9cae158abd1d683c7269ed1d3this graph and the 1 it reaches through calls
Evidence: sha256:93122317f98d529e50dc603b0d08922c1d7d9e80996b46be92b3b02759269cd4the hash of its verification record
Needs: no capability: a pure function

Through NOVA’s control layer, the symbol launches this function only while the program still matches what it pins: a change to this graph, or to any graph it reaches, needs a migration first.