{"paper":{"title":"Graph Sparsification by Effective Resistances","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Daniel A. Spielman, Nikhil Srivastava","submitted_at":"2008-03-06T18:03:06Z","abstract_excerpt":"We present a nearly-linear time algorithm that produces high-quality sparsifiers of weighted graphs. Given as input a weighted graph $G=(V,E,w)$ and a parameter $\\epsilon>0$, we produce a weighted subgraph $H=(V,\\tilde{E},\\tilde{w})$ of $G$ such that $|\\tilde{E}|=O(n\\log n/\\epsilon^2)$ and for all vectors $x\\in\\R^V$ $(1-\\epsilon)\\sum_{uv\\in E}(x(u)-x(v))^2w_{uv}\\le \\sum_{uv\\in\\tilde{E}}(x(u)-x(v))^2\\tilde{w}_{uv} \\le (1+\\epsilon)\\sum_{uv\\in E}(x(u)-x(v))^2w_{uv}. (*)$\n  This improves upon the sparsifiers constructed by Spielman and Teng, which had $O(n\\log^c n)$ edges for some large constant $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0803.0929","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/0803.0929/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}