{"paper":{"title":"Approaching optimality for solving SDD systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Gary L. Miller, Ioannis Koutis, Richard Peng","submitted_at":"2010-03-15T16:37:51Z","abstract_excerpt":"We present an algorithm that on input of an $n$-vertex $m$-edge weighted graph $G$ and a value $k$, produces an {\\em incremental sparsifier} $\\hat{G}$ with $n-1 + m/k$ edges, such that the condition number of $G$ with $\\hat{G}$ is bounded above by $\\tilde{O}(k\\log^2 n)$, with probability $1-p$. The algorithm runs in time\n  $$\\tilde{O}((m \\log{n} + n\\log^2{n})\\log(1/p)).$$\n  As a result, we obtain an algorithm that on input of an $n\\times n$ symmetric diagonally dominant matrix $A$ with $m$ non-zero entries and a vector $b$, computes a vector ${x}$ satisfying $||{x}-A^{+}b||_A<\\epsilon ||A^{+}b"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.2958","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}