{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/graph-sparsification-by-effective-resistances","title":"Graph Sparsification by Effective Resistances","arxiv_id":"0803.0929","date":"2008-03-06","proceeding":null,"authors":["Daniel A. Spielman","Nikhil Srivastava"],"abstract":"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 $x\\in\\R^V$ $(1-\\epsilon)\\sum_{uv\\in E}(x(u)-x(v))^2w_{uv}\\le \\sum_{uv\\in\\tilde{E}}(x(u)-x(v))^2\\tilde{w}_{uv} \\le (1+\\epsilon)\\sum_{uv\\in E}(x(u)-x(v))^2w_{uv}. (*)$ This improves upon the sparsifiers constructed by Spielman and Teng, which had $O(n\\log^c n)$ edges for some large constant $c$, and upon those of Bencz\\'ur and Karger, which only satisfied (*) for $x\\in\\{0,1\\}^V$. A key ingredient in our algorithm is a subroutine of independent interest: a nearly-linear time algorithm that builds a data structure from which we can query the approximate effective resistance between any two vertices in a graph in $O(\\log n)$ time.","url_abs":"https://arxiv.org/abs/0803.0929v4","url_pdf":"https://arxiv.org/pdf/0803.0929v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"graph-sparsification-by-effective-resistances","repo_url":"https://github.com/as-12/gsparse","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"unanswered"}},{"paper_slug":"graph-sparsification-by-effective-resistances","repo_url":"https://github.com/noamrazin/gnn_interactions","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"ok"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=0803.0929","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}