{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:XX2WQSCXZNQHX64RTLH7E7OF3K","short_pith_number":"pith:XX2WQSCX","canonical_record":{"source":{"id":"2507.14060","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.DS","submitted_at":"2025-07-18T16:41:21Z","cross_cats_sorted":[],"title_canon_sha256":"f12ccc8f3a6348ce4667a1021638792d976e95630e674f9f7411858448bf86c3","abstract_canon_sha256":"ff8d7d7e64baf2a47c2adad4d9b2a3eb88b68219fc34af68bc153cd40d0dd830"},"schema_version":"1.0"},"canonical_sha256":"bdf5684857cb607bfb919acff27dc5dab19d37cd978803f842fccd1f45eca26d","source":{"kind":"arxiv","id":"2507.14060","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.14060","created_at":"2026-07-05T11:39:27Z"},{"alias_kind":"arxiv_version","alias_value":"2507.14060v1","created_at":"2026-07-05T11:39:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.14060","created_at":"2026-07-05T11:39:27Z"},{"alias_kind":"pith_short_12","alias_value":"XX2WQSCXZNQH","created_at":"2026-07-05T11:39:27Z"},{"alias_kind":"pith_short_16","alias_value":"XX2WQSCXZNQHX64R","created_at":"2026-07-05T11:39:27Z"},{"alias_kind":"pith_short_8","alias_value":"XX2WQSCX","created_at":"2026-07-05T11:39:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:XX2WQSCXZNQHX64RTLH7E7OF3K","target":"record","payload":{"canonical_record":{"source":{"id":"2507.14060","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.DS","submitted_at":"2025-07-18T16:41:21Z","cross_cats_sorted":[],"title_canon_sha256":"f12ccc8f3a6348ce4667a1021638792d976e95630e674f9f7411858448bf86c3","abstract_canon_sha256":"ff8d7d7e64baf2a47c2adad4d9b2a3eb88b68219fc34af68bc153cd40d0dd830"},"schema_version":"1.0"},"canonical_sha256":"bdf5684857cb607bfb919acff27dc5dab19d37cd978803f842fccd1f45eca26d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:39:27.978857Z","signature_b64":"Ype53d2WDu1j0dP89vMLQAbYzaiBtCdq6cIANhPUjyytbc1M349ck2mlN9kjfRUPGTFWGbnmXBXR5CUI3i4uAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bdf5684857cb607bfb919acff27dc5dab19d37cd978803f842fccd1f45eca26d","last_reissued_at":"2026-07-05T11:39:27.978358Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:39:27.978358Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2507.14060","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T11:39:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Fq+dPAINhWEmFrgJuq03yVbF4wJWs+ZqNcz6yG+piZuzPbO1/YjOlhQBGUqRW5ik3ryz8liXFwH1T23sCzOaCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T11:49:07.307256Z"},"content_sha256":"0e08d0483676df93794dff3ce704d66c64712348bed9bb27ee7e3660b4cf8983","schema_version":"1.0","event_id":"sha256:0e08d0483676df93794dff3ce704d66c64712348bed9bb27ee7e3660b4cf8983"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:XX2WQSCXZNQHX64RTLH7E7OF3K","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sparse Navigable Graphs for Nearest Neighbor Search: Algorithms and Hardness","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Ashwin Padaki, Erik Waingarten, Sanjeev Khanna","submitted_at":"2025-07-18T16:41:21Z","abstract_excerpt":"We initiate the study of approximation algorithms and computational barriers for constructing sparse $\\alpha$-navigable graphs [IX23, DGM+24], a core primitive underlying recent advances in graph-based nearest neighbor search. Given an $n$-point dataset $P$ with an associated metric $\\mathsf{d}$ and a parameter $\\alpha \\geq 1$, the goal is to efficiently build the sparsest graph $G=(P, E)$ that is $\\alpha$-navigable: for every distinct $s, t \\in P$, there exists an edge $(s, u) \\in E$ with $\\mathsf{d}(u, t) < \\mathsf{d}(s, t)/\\alpha$. We consider two natural sparsity objectives: minimizing the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.14060","kind":"arxiv","version":1},"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/2507.14060/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T11:39:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BdIQsgEw+n6djJJx7g1fLddSzI97moZSIAmYlGRCz2oVyt3Y7juijb9IXtK6KLMnMSDASK8ZoEZjBHAbwNVjCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T11:49:07.307973Z"},"content_sha256":"25a3757aef934c83280d065e6faebaff9717e881e595f73b4ecdebabc22ffb8f","schema_version":"1.0","event_id":"sha256:25a3757aef934c83280d065e6faebaff9717e881e595f73b4ecdebabc22ffb8f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XX2WQSCXZNQHX64RTLH7E7OF3K/bundle.json","state_url":"https://pith.science/pith/XX2WQSCXZNQHX64RTLH7E7OF3K/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XX2WQSCXZNQHX64RTLH7E7OF3K/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-08T11:49:07Z","links":{"resolver":"https://pith.science/pith/XX2WQSCXZNQHX64RTLH7E7OF3K","bundle":"https://pith.science/pith/XX2WQSCXZNQHX64RTLH7E7OF3K/bundle.json","state":"https://pith.science/pith/XX2WQSCXZNQHX64RTLH7E7OF3K/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XX2WQSCXZNQHX64RTLH7E7OF3K/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:XX2WQSCXZNQHX64RTLH7E7OF3K","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"ff8d7d7e64baf2a47c2adad4d9b2a3eb88b68219fc34af68bc153cd40d0dd830","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.DS","submitted_at":"2025-07-18T16:41:21Z","title_canon_sha256":"f12ccc8f3a6348ce4667a1021638792d976e95630e674f9f7411858448bf86c3"},"schema_version":"1.0","source":{"id":"2507.14060","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.14060","created_at":"2026-07-05T11:39:27Z"},{"alias_kind":"arxiv_version","alias_value":"2507.14060v1","created_at":"2026-07-05T11:39:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.14060","created_at":"2026-07-05T11:39:27Z"},{"alias_kind":"pith_short_12","alias_value":"XX2WQSCXZNQH","created_at":"2026-07-05T11:39:27Z"},{"alias_kind":"pith_short_16","alias_value":"XX2WQSCXZNQHX64R","created_at":"2026-07-05T11:39:27Z"},{"alias_kind":"pith_short_8","alias_value":"XX2WQSCX","created_at":"2026-07-05T11:39:27Z"}],"graph_snapshots":[{"event_id":"sha256:25a3757aef934c83280d065e6faebaff9717e881e595f73b4ecdebabc22ffb8f","target":"graph","created_at":"2026-07-05T11:39:27Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2507.14060/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We initiate the study of approximation algorithms and computational barriers for constructing sparse $\\alpha$-navigable graphs [IX23, DGM+24], a core primitive underlying recent advances in graph-based nearest neighbor search. Given an $n$-point dataset $P$ with an associated metric $\\mathsf{d}$ and a parameter $\\alpha \\geq 1$, the goal is to efficiently build the sparsest graph $G=(P, E)$ that is $\\alpha$-navigable: for every distinct $s, t \\in P$, there exists an edge $(s, u) \\in E$ with $\\mathsf{d}(u, t) < \\mathsf{d}(s, t)/\\alpha$. We consider two natural sparsity objectives: minimizing the","authors_text":"Ashwin Padaki, Erik Waingarten, Sanjeev Khanna","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.DS","submitted_at":"2025-07-18T16:41:21Z","title":"Sparse Navigable Graphs for Nearest Neighbor Search: Algorithms and Hardness"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.14060","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:0e08d0483676df93794dff3ce704d66c64712348bed9bb27ee7e3660b4cf8983","target":"record","created_at":"2026-07-05T11:39:27Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"ff8d7d7e64baf2a47c2adad4d9b2a3eb88b68219fc34af68bc153cd40d0dd830","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.DS","submitted_at":"2025-07-18T16:41:21Z","title_canon_sha256":"f12ccc8f3a6348ce4667a1021638792d976e95630e674f9f7411858448bf86c3"},"schema_version":"1.0","source":{"id":"2507.14060","kind":"arxiv","version":1}},"canonical_sha256":"bdf5684857cb607bfb919acff27dc5dab19d37cd978803f842fccd1f45eca26d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bdf5684857cb607bfb919acff27dc5dab19d37cd978803f842fccd1f45eca26d","first_computed_at":"2026-07-05T11:39:27.978358Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:39:27.978358Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ype53d2WDu1j0dP89vMLQAbYzaiBtCdq6cIANhPUjyytbc1M349ck2mlN9kjfRUPGTFWGbnmXBXR5CUI3i4uAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T11:39:27.978857Z","signed_message":"canonical_sha256_bytes"},"source_id":"2507.14060","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0e08d0483676df93794dff3ce704d66c64712348bed9bb27ee7e3660b4cf8983","sha256:25a3757aef934c83280d065e6faebaff9717e881e595f73b4ecdebabc22ffb8f"],"state_sha256":"a2e8c5e9616f3e7b18d1ae563ecbddd855903aefd1967bff262ccfe8a319ccc6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"G52pqZaDELbQRVw6tlCb7LT50c0vh3DaGFY7Fo0Mr7b2AQCq72+yFc1bazlLixVLvJwE6z1erm0AfkRIQs2tBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T11:49:07.314137Z","bundle_sha256":"2853c930dbd3a6ca6b00c905ab52fe2f197da2162bc925da37ea42faef80ee1d"}}