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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 $"},"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":"0803.0929","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"cs.DS","submitted_at":"2008-03-06T18:03:06Z","cross_cats_sorted":[],"title_canon_sha256":"0c8c0bd43d17546ad8113f4522ec6ff0795fce0f8add6ea4c900613c271e5a6e","abstract_canon_sha256":"312f6b6b8d271db558150f939109c7d83ff39447391797600510642565924f7e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T17:15:58.002547Z","signature_b64":"VHItAoewYkE3IJV2izOpXqsN0uPhgLoxaCpmgFme6fomDn4ZyA0OvZM+HEIB4pUoUx/xFE8XkX1oBQBTQfkqDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e2d8335687b7071959fa52b5769ba2e45958b6ef56c7cd1ea08b8585877277c8","last_reissued_at":"2026-07-04T17:15:58.002054Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T17:15:58.002054Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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