On the Optimal Recovery Threshold of Coded Matrix Multiplication

• Published in: IEEE transactions on information theory Vol. 66; no. 1; pp. 278 - 301
• Main Authors: Dutta, Sanghamitra, Fahim, Mohammad, Haddadpour, Farzin, Jeong, Haewon, Cadambe, Viveck, Grover, Pulkit
• Format: Journal Article
• Language: English
• Published: New York IEEE 01-01-2020; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Coded computing; Codes; Coding; Computation; Computational complexity; Computational efficiency; Computational modeling; distributed computing; Encoding; Fault tolerance; matrix multiplication; Multiplication; Polynomials; Recovery; recovery threshold; stragglers; Systematics; Workers
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2019.2929328

We provide novel coded computation strategies for distributed matrix-matrix products that outperform the recent "Polynomial code" constructions in recovery threshold, i.e., the required number of successful workers. When a fixed <inline-formula> <tex-math notation="LaTeX">1/m </tex-math></inline-formula> fraction of each matrix can be stored at each worker node, Polynomial codes require <inline-formula> <tex-math notation="LaTeX">m^{2} </tex-math></inline-formula> successful workers, while our MatDot codes only require <inline-formula> <tex-math notation="LaTeX">2m-1 </tex-math></inline-formula> successful workers. However, MatDot codes have higher computation cost per worker and higher communication cost from each worker to the fusion node. We also provide a systematic construction of MatDot codes. Furthermore, we propose "PolyDot" coding that interpolates between Polynomial codes and MatDot codes to trade off computation/communication costs and recovery thresholds. Finally, we demonstrate a novel coding technique for multiplying <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> matrices (<inline-formula> <tex-math notation="LaTeX">n \geq 3 </tex-math></inline-formula>) using ideas from MatDot and PolyDot codes.