Coherence-Aware I/O Lower Bounds for Dense Linear Algebra: A Red-Blue-Green Pebble Game Analysis

We introduce a novel theoretical framework for analyzing I/O complexity in the presence of memory coherence constraints, extending the classical Hong-Kung red-blue pebble game with a third "coherence" dimension. Our model captures the fundamental trade-offs between memory capacity, bandwidth, and coherence width in modern computing systems. We establish tight lower bounds for dense linear algebra operations including matrix multiplication (GEMM), QR factorization, Cholesky decomposition, and Krylov subspace methods. Our main result shows that any algorithm for these problems requires Omega(n^3/\sqrt{S} + n^3/C) I/O operations, where S is the fast memory size and C is the coherence width. We present coherence-aware streaming algorithms that achieve these bounds up to logarithmic factors, demonstrating their practical optimality. This work provides the first rigorous treatment of coherence as a fundamental computational resource alongside memory and bandwidth, with implications for quantum-inspired classical algorithms and modern high-performance computing systems.