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On input of a SDD $n$-by-$n$ matrix $A$ with $m$ non-zero entries and a vector $b$, our algorithm computes a vector $\\tilde{x}$ such that $\\norm[A]{\\tilde{x} - A^+b} \\leq \\vareps \\cdot \\norm[A]{A^+b}$ in $O(m\\log^{O(1)}{n}\\log{\\frac1\\epsilon})$ work and $O(m^{1/3+\\theta}\\log \\frac1\\epsilon)$ depth for any fixed $\\theta > 0$.\n  The algorithm relies on a parallel algorithm for generating low-stretch spanning trees or spanning subgraphs. To this end, we first"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1111.1750","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2011-11-07T21:17:09Z","cross_cats_sorted":["cs.DC","cs.NA"],"title_canon_sha256":"5ecbb6d544dbbbf7971e6154fc1cf897e4a7c66b832e0e2466d274563c85074b","abstract_canon_sha256":"17d08381712843951e909a706581f4b625ddccebe5ba674fc182c611d5d9eba7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:09:24.809746Z","signature_b64":"ZSLsFz3RqTJyQ1UPFjo9bPW/OYf3ufl6XOYEu2jJnGqq6UggIV8cySrFOMA+rAw0MpNxmHmMi/hVq2BwoIwgCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3daaccd2617544873d17385f536799d0f10f2a9399fabbe18a5522b7cc4be0f2","last_reissued_at":"2026-05-18T04:09:24.809069Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:09:24.809069Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Near Linear-Work Parallel SDD Solvers, Low-Diameter Decomposition, and Low-Stretch Subgraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DC","cs.NA"],"primary_cat":"cs.DS","authors_text":"Anupam Gupta, Gary L. Miller, Guy E. Blelloch, Ioannis Koutis, Kanat Tangwongsan, Richard Peng","submitted_at":"2011-11-07T21:17:09Z","abstract_excerpt":"We present the design and analysis of a near linear-work parallel algorithm for solving symmetric diagonally dominant (SDD) linear systems. On input of a SDD $n$-by-$n$ matrix $A$ with $m$ non-zero entries and a vector $b$, our algorithm computes a vector $\\tilde{x}$ such that $\\norm[A]{\\tilde{x} - A^+b} \\leq \\vareps \\cdot \\norm[A]{A^+b}$ in $O(m\\log^{O(1)}{n}\\log{\\frac1\\epsilon})$ work and $O(m^{1/3+\\theta}\\log \\frac1\\epsilon)$ depth for any fixed $\\theta > 0$.\n  The algorithm relies on a parallel algorithm for generating low-stretch spanning trees or spanning subgraphs. 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