A Formula for the I/O Cost of Linear Repair Schemes and Application to Reed-Solomon Codes
Abstract: Node repair is a crucial problem in erasure-code-based distributed storage systems. An important metric for repair efficiency is the I/O cost which equals the total amount of data accessed at helper nodes to repair a failed node. In this work, a general formula for computing the I/O cost of linear repair schemes is derived from a new perspective, i.e., by investigating the Hamming weight of a related linear space. Applying the formula to Reed-Solomon (RS) codes, we obtain lower bounds on the I/O cost for full-length RS codes with two and three parities. Furthermore, we build linear repair schemes for the RS codes with improved I/O cost. For full-length RS codes with two parities, our scheme meets the lower bound on the I/O cost.
- A. G. Dimakis, P. B. Godfrey, Y. Wu, M. J. Wainwright and K. Ramchandran, “Network Coding for Distributed Storage Systems,” IEEE Trans. Inf. Theory, vol. 56, no. 9, pp. 4539-4551, Sep. 2010.
- K. V. Rashmi, N. B. Shah, and P. V. Kumar, “Optimal exact-regenerating codes for distributed storage at the MSR and MBR points via a product-matrix construction,” IEEE Trans. Inf. Theory, vol. 57, no. 8, pp. 5227-5239, Aug. 2011.
- I. Tamo, Z. Wang and J. Bruck, “Zigzag Codes: MDS Array Codes With Optimal Rebuilding,” IEEE Trans. Inf. Theory, vol. 59, no. 3, pp. 1597-1616, Mar. 2013.
- M. Ye, A. Barg, “Explicit constructions of optimal-access MDS codes with nearly optimal sub-packetization,” IEEE Trans. Inf. Theory, vol. 63, no. 10, pp. 6307-6317, Oct. 2017.
- Y. Liu, J. Li and X. Tang, “A Generic Transformation to Generate MDS Array Codes With δ𝛿\deltaitalic_δ-Optimal Access Property,” IEEE Trans. Commun., vol. 70, no. 2, pp. 759-768, Feb. 2022.
- K. Shanmugam, D. S. Papailiopoulos, A. G. Dimakis and G. Caire, “A Repair Framework for Scalar MDS Codes,” IEEE J. Sel. Areas Commun., vol. 32, no. 5, pp. 998-1007, May 2014.
- V. Guruswami and M. Wootters, “Repairing Reed-Solomon Codes,” IEEE Trans. Inf. Theory, vol. 63, no. 9, pp. 5684-5698, Sep. 2017.
- H. Dau and O. Milenkovic, “Optimal repair schemes for some families of Full-Length Reed-Solomon codes,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Jun. 2017, pp. 346–350.
- I. Tamo, M. Ye and A. Barg, “Optimal Repair of Reed-Solomon Codes: Achieving the Cut-Set Bound,” in Proc. IEEE 58th Annu. Symp. Found. Comput. Sci. (FOCS), Oct. 2017, pp. 216-227.
- H. Dau, I. Duursma, and H. Chu, “On the I/O Costs of Some Repair Schemes for Full-Length Reed-Solomon Codes,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Jun. 2018, pp. 1700–1704.
- H. Dau and E. Viterbo, “Repair Schemes with Optimal I/O Costs for Full-Length Reed-Solomon Codes with Two Parities,” in Proc. IEEE Inf. Theory Workshop (ITW), Nov. 2018, pp. 1-5.
- W. Li, H. Dau, Z. Wang, H. Jafarkhani, and E. Viterbo, “On the I/O Costs in Repairing Short-Length Reed-Solomon Codes,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Jul. 2019, pp. 1087–1091.
- Z. Chen, M. Ye and A. Barg, “Enabling Optimal Access and Error Correction for the Repair of Reed–Solomon Codes,” IEEE Trans. Inf. Theory, vol. 66, no. 12, pp. 7439-7456, Dec. 2020.
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