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Shortest-Path FFT: Optimal SIMD Instruction Scheduling via Graph Search

Mohamed Amine Bergach

Abstract

An $N$-point FFT admits many valid implementations that differ in radix choice, stage ordering, and register-blocking strategy. These alternatives use different SIMD instruction mixes with different latencies, yet produce the same mathematical result. We show that finding the fastest implementation is a shortest-path problem on a directed acyclic graph. We formalize two variants of this graph. In the \emph{context-free} model, nodes represent computation stages and edge weights are independently measured instruction costs. In the \emph{context-aware} model, nodes are expanded to encode the \emph{predecessor edge type}, so that edge weights capture inter-operation correlations such as cache warming -- the cost of operation~B depends on which operation~A preceded it. This addresses a limitation identified but deliberately bypassed by FFTW \citep{FrigoJohnson1998}: that optimal-substructure assumptions break down ``because of the different states of the cache.'' Applied to Apple M1 NEON, the context-free Dijkstra finds an arrangement at 22.1~GFLOPS (74\% of optimal). The context-aware Dijkstra discovers $\text{R4} \to \text{R2} \to \text{R4} \to \text{R4} \to \text{Fused-8}$ at 29.8~GFLOPS -- a $5.2\times$ improvement over pure radix-2 and 34\% faster than the context-free result. This arrangement includes a radix-2 pass \emph{sandwiched between} radix-4 passes, exploiting cache residuals that only exist in context. No context-free search can discover this.

Shortest-Path FFT: Optimal SIMD Instruction Scheduling via Graph Search

Abstract

An -point FFT admits many valid implementations that differ in radix choice, stage ordering, and register-blocking strategy. These alternatives use different SIMD instruction mixes with different latencies, yet produce the same mathematical result. We show that finding the fastest implementation is a shortest-path problem on a directed acyclic graph. We formalize two variants of this graph. In the \emph{context-free} model, nodes represent computation stages and edge weights are independently measured instruction costs. In the \emph{context-aware} model, nodes are expanded to encode the \emph{predecessor edge type}, so that edge weights capture inter-operation correlations such as cache warming -- the cost of operation~B depends on which operation~A preceded it. This addresses a limitation identified but deliberately bypassed by FFTW \citep{FrigoJohnson1998}: that optimal-substructure assumptions break down ``because of the different states of the cache.'' Applied to Apple M1 NEON, the context-free Dijkstra finds an arrangement at 22.1~GFLOPS (74\% of optimal). The context-aware Dijkstra discovers at 29.8~GFLOPS -- a improvement over pure radix-2 and 34\% faster than the context-free result. This arrangement includes a radix-2 pass \emph{sandwiched between} radix-4 passes, exploiting cache residuals that only exist in context. No context-free search can discover this.

Paper Structure

This paper contains 21 sections, 2 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Contributions
  3. The Shortest-Path FFT Framework
  4. Context-Free Model
  5. Edge Types
  6. Context-Aware Model
  7. Why Context Matters
  8. Search Complexity
  9. Implementation on Apple M1 NEON
  10. Butterfly Core
  11. Fused Register Blocks
  12. Results
  13. Setup
  14. Algorithm Comparison
  15. Key Findings
  16. ...and 6 more sections

Figures (3)

  • Figure 1: Context-free computation graph for $N = 1024$ ($L = 10$). Edges: radix-2 (blue), radix-4 (orange), radix-8 (red), fused register blocks (green). A path from 0 to 10 is a complete FFT; the shortest path is the fastest. Subset of 30+ edges shown.
  • Figure 2: Context-aware graph (partial view). Each node is expanded by predecessor type: $(3, \text{R2})$ means "stage 3, reached via R2." The edge from $(2, \text{R4})$ to $(3, \text{R2})$ captures R2's cost after R4's cache residual. Optimal path (red): R4 $\to$ R2 $\to$ R4 $\to$ R4 $\to$ F8.
  • Figure 3: Three decompositions for $N = 1024$. Top: pure radix-2 (10 passes, 5.7 GF). Middle: context-free Dijkstra (R4+F8+F32, 22.1 GF). Bottom: context-aware Dijkstra (R4$\to$R2$\to$R4$\to$R4$\to$F8, 29.8 GF). Dashed box: the radix-2 pass exploiting cache residuals from the preceding R4.