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Third, given a query pair of nodes $(u,v)$, return the distance from $u$ to $v$. In the easier problem called sensitive reachability oracle problem, we only ask if there exists a directed path from $u$ to $v$.\n  Our first result is a sensitive distance oracle with $\\tilde{O}(Wn^{\\omega+(3-\\omega)\\mu})$ preprocessing time, $\\tilde{O}(Wn^{2"},"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":"1907.07982","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2019-07-18T10:52:13Z","cross_cats_sorted":[],"title_canon_sha256":"e61f6f1d4f5fed448f171c9e1e050c54f33916c46c5eb09823eaf08e036ae2ca","abstract_canon_sha256":"f9aab4bfaa2b80ff64b633b09fcb392f22d497a8ee064b584fc35962fb841034"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:03.105873Z","signature_b64":"oCYo/W5wI27VCPhm0mN7ebiiDGm4DuCPsrBI4o/lsH7rVc0Y3lF3DwieG7vYLfLHA1v2ByoX0M1fnFAokuRpDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3a18ac3f8c8cee1f77c63dd03868f4fa3c668fcff148c759100092d18dcec04d","last_reissued_at":"2026-05-17T23:40:03.105137Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:03.105137Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sensitive Distance and Reachability Oracles for Large Batch Updates","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Jan van den Brand, Thatchaphol Saranurak","submitted_at":"2019-07-18T10:52:13Z","abstract_excerpt":"In the sensitive distance oracle problem, there are three phases. 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