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Concretely, for all vectors $x$ and any $\\epsilon>0$, $\\tilde{G}$ satisfies $$ (1-\\epsilon) x^T L_G x \\leq x^T L_{\\tilde{G}} x \\leq (1+\\epsilon) x^T L_G x, $$ where $L_G$ and $L_{\\tilde{G}}$ are the Laplacians of $G$ and $\\tilde{G}$ respectively. We show that the fastest known algorithm for computing a sparsifier with $O(n\\log n/\\epsilon^2)$ edges can actually run in $\\tilde{O}(m\\log^2 n)$ time"},"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":"1209.5821","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2012-09-26T03:15:16Z","cross_cats_sorted":[],"title_canon_sha256":"52ad41329551eaa9159c27beb7513510b70d5d3edc3fe88d522423cd12f652e2","abstract_canon_sha256":"325158e268a11f6f5e96156a5103fabc6ac19675b72eb8c5a0a44df3dab78910"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:07:02.134706Z","signature_b64":"7QEWhHZ2EVtN0I0iT/kHmgC2AUDOHw3oJGvkyV1mp2A9968CYA/BJiVLithql5YPenM46dHVffqEY5RwIJU6Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7364e908412ae3f7e4c092123c1a60f8e1b1af5ebc9e811d0a477971cd0265b1","last_reissued_at":"2026-05-18T03:07:02.133929Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:07:02.133929Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Faster spectral sparsification and numerical algorithms for SDD matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Alex Levin, Ioannis Koutis, Richard Peng","submitted_at":"2012-09-26T03:15:16Z","abstract_excerpt":"We study algorithms for spectral graph sparsification. The input is a graph $G$ with $n$ vertices and $m$ edges, and the output is a sparse graph $\\tilde{G}$ that approximates $G$ in an algebraic sense. Concretely, for all vectors $x$ and any $\\epsilon>0$, $\\tilde{G}$ satisfies $$ (1-\\epsilon) x^T L_G x \\leq x^T L_{\\tilde{G}} x \\leq (1+\\epsilon) x^T L_G x, $$ where $L_G$ and $L_{\\tilde{G}}$ are the Laplacians of $G$ and $\\tilde{G}$ respectively. We show that the fastest known algorithm for computing a sparsifier with $O(n\\log n/\\epsilon^2)$ edges can actually run in $\\tilde{O}(m\\log^2 n)$ time"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.5821","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1209.5821","created_at":"2026-05-18T03:07:02.134061+00:00"},{"alias_kind":"arxiv_version","alias_value":"1209.5821v3","created_at":"2026-05-18T03:07:02.134061+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.5821","created_at":"2026-05-18T03:07:02.134061+00:00"},{"alias_kind":"pith_short_12","alias_value":"ONSOSCCBFLR7","created_at":"2026-05-18T12:27:16.716162+00:00"},{"alias_kind":"pith_short_16","alias_value":"ONSOSCCBFLR7PZGA","created_at":"2026-05-18T12:27:16.716162+00:00"},{"alias_kind":"pith_short_8","alias_value":"ONSOSCCB","created_at":"2026-05-18T12:27:16.716162+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ONSOSCCBFLR7PZGASIJDYGTA7D","json":"https://pith.science/pith/ONSOSCCBFLR7PZGASIJDYGTA7D.json","graph_json":"https://pith.science/api/pith-number/ONSOSCCBFLR7PZGASIJDYGTA7D/graph.json","events_json":"https://pith.science/api/pith-number/ONSOSCCBFLR7PZGASIJDYGTA7D/events.json","paper":"https://pith.science/paper/ONSOSCCB"},"agent_actions":{"view_html":"https://pith.science/pith/ONSOSCCBFLR7PZGASIJDYGTA7D","download_json":"https://pith.science/pith/ONSOSCCBFLR7PZGASIJDYGTA7D.json","view_paper":"https://pith.science/paper/ONSOSCCB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1209.5821&json=true","fetch_graph":"https://pith.science/api/pith-number/ONSOSCCBFLR7PZGASIJDYGTA7D/graph.json","fetch_events":"https://pith.science/api/pith-number/ONSOSCCBFLR7PZGASIJDYGTA7D/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ONSOSCCBFLR7PZGASIJDYGTA7D/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ONSOSCCBFLR7PZGASIJDYGTA7D/action/storage_attestation","attest_author":"https://pith.science/pith/ONSOSCCBFLR7PZGASIJDYGTA7D/action/author_attestation","sign_citation":"https://pith.science/pith/ONSOSCCBFLR7PZGASIJDYGTA7D/action/citation_signature","submit_replication":"https://pith.science/pith/ONSOSCCBFLR7PZGASIJDYGTA7D/action/replication_record"}},"created_at":"2026-05-18T03:07:02.134061+00:00","updated_at":"2026-05-18T03:07:02.134061+00:00"}