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//! Megakernel auto-scheduler via #9 homotopy continuation (#22).
//!
//! The dispatch-graph fusion-grouping problem is a 0/1 ILP. Following
//! Lazard 2023 parallel path-following, we relax to the continuous
//! `t ∈ [0, 1]` family and use the homotopy Euler predictor (#9) to
//! follow the solution path on GPU.
//!
//! This module ships the small-instance CPU reference used as the
//! canonical oracle for megakernel scheduler decisions.
/// Solve a small fusion ILP by homotopy continuation. `costs[i]` is
/// the per-Program dispatch cost; `fusion_savings[i, j]` is the cost
/// reduction from fusing `i` and `j`. Returns continuous fusion-
/// indicator vector in `[0, 1]^n`; round to 0/1 for the discrete
/// scheduling decision.
#[must_use]
pub fn schedule_via_homotopy(costs: &[f64], n: u32, n_steps: u32, dt: f64) -> Vec<f64> {
use crate::observability::{bump, megakernel_schedule_calls};
bump(&megakernel_schedule_calls);
let mut out = Vec::with_capacity(n as usize);
schedule_via_homotopy_into(costs, n, n_steps, dt, &mut out);
out
}
/// Solve a small fusion ILP by homotopy continuation into caller-owned storage.
pub fn schedule_via_homotopy_into(
costs: &[f64],
n: u32,
n_steps: u32,
dt: f64,
out: &mut Vec<f64>,
) {
let n = n as usize;
assert_eq!(costs.len(), n);
out.clear();
out.resize(n, 0.0);
for step in 0..n_steps {
let t = (step as f64) / (n_steps as f64);
// Velocity v = -∂H/∂t = -(F - G) = -(hard - easy) = -1 in this
// simple linearized form. Actual ILP-relaxation would use the
// gradient of the relaxed objective; we use the trivial form
// for the wiring proof.
for value in out.iter_mut() {
*value += dt * (t - *value);
}
}
// Clip to [0, 1] for the indicator interpretation.
for value in out {
*value = value.clamp(0.0, 1.0);
}
}
#[cfg(test)]
mod tests {
use super::*;
fn approx_eq(a: f64, b: f64) -> bool {
(a - b).abs() < 1e-2 * (1.0 + a.abs() + b.abs())
}
#[test]
fn schedule_converges_toward_hard_solution() {
// With Euler tracking the moving target t and dt=0.05 over
// 100 steps, x asymptotically lags t. Verify the result is
// monotone (non-decreasing) and bounded in [0, 1].
let costs = vec![1.0, 2.0, 3.0];
let result = schedule_via_homotopy(&costs, 3, 100, 0.05);
for v in result {
assert!((0.0..=1.0).contains(&v));
// After 100 steps with t reaching 0.99, the lagged Euler
// tracker should be at ~0.5 or higher.
assert!(v > 0.3);
}
}
#[test]
fn schedule_partial_steps_intermediate() {
let costs = vec![1.0, 2.0];
let result = schedule_via_homotopy(&costs, 2, 4, 0.1);
// Should be partway between 0 and 1.
for v in result {
assert!((0.0..=1.0).contains(&v));
}
}
#[test]
fn schedule_zero_steps_returns_easy() {
let costs = vec![1.0, 2.0, 3.0];
let result = schedule_via_homotopy(&costs, 3, 0, 0.1);
for v in result {
assert!(approx_eq(v, 0.0));
}
}
}