An Efficient Parallel Solver for SDD Linear Systems

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Presents an algorithm that solves SDD (graph Laplacian) systems with nearly-linear work and polylog depth by building an “approximate inverse chain”: repeatedly square the random-walk matrix, apply randomized spectral sparsifiers at each level, and propagate symmetric factorizations back up to approximate the inverse. The chain has about log κ levels (κ = condition number), each level uses sparse spectral specifiers so intermediate matrices stay sparse, and the method generalizes to p‑powers and parallel Gaussian sampling — it achieves the claimed parallel depth and practical sparsity.

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" chapters (25)

1. 0:00Intro
2. 0:58LARGE GRAPHS
3. 1:41GRAPH LAPLACIAN
4. 2:59THE LAPLACIAN PARADIGM
5. 4:22SOLVERS
6. 6:46PARALLEL SPEEDUPS Common architectures multicore MapReduce
7. 10:31OUTLINE
8. 12:43PREVIOUS FAST ALGORITHMS
9. 16:16POLYNOMIAL APPROXIMATIONS
10. 18:53LOWER BOUND FOR LOWER BOUND
11. 21:12SUMMARY
12. 23:53REDUCTION TO (I - A)
13. 25:24INTERPRETATION
14. 27:10SIMILAR TO
15. 29:21WHAT IS AN ALGORITHM
16. 31:32SQUARING
17. 33:03APPROXIMATE INVERSE CHAIN
18. 34:15ISSUE 1
19. 35:46ISSUE 2
20. 40:08PSEUDOCODE
21. 40:46TOTAL COST
22. 43:55SOLUTION 2
23. 45:46SOLUTION 3 ([CCLPT 14])
24. 48:14OPEN QUESTIONS
25. 51:23REPRESENTATION OF (I - A)