topological_order

std.linalg.topological_order · Level L3

A topological order of a finite directed graph by Kahn's algorithm, always taking the smallest available source, so the order is the lexicographically first one. dag is 1 when every vertex was taken, which is exactly when the graph has no cycle. A Scan of topo_step, n steps.

order = Kahn(A), smallest source first

Signature

topological_order(A: f64[n, n]) → f64[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 Kahn's algorithm with the smallest source first in exact rational arithmetic, on all 40 test cases.
All 198 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

Identity

Calls

topo_step

Called by

—

sha256:ddb05706e72d499328aeaa118a935fa7512b5c8fc7cafeee975369a52a637266

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.topological_order · Ω:topological_order
Pins: sha256:bd8b2c38d5e3dd8338c02c53ebd8023cb25328f3f915d8d08bc12fadd663d859this graph and the 1 it reaches through calls
Evidence: sha256:455c1c96ba7e7c0ff2953473b745d675d3f24eb3c3378cc1c0fbd14573e667c4the 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.