{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:GM22EGV2QVPWBTTOORNPBFKQDP","short_pith_number":"pith:GM22EGV2","canonical_record":{"source":{"id":"2105.01785","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2021-05-04T22:39:38Z","cross_cats_sorted":[],"title_canon_sha256":"00e7dc50bbd55ef3e416ecd53f1a4c9705c5950af571bc3ea53e9002fd040f11","abstract_canon_sha256":"9f81d519d758a6a4e9a392a39c57555c397d1e5b7597fe0ac90b0000a115c395"},"schema_version":"1.0"},"canonical_sha256":"3335a21aba855f60ce6e745af095501bcdff73a8fdeb0194c0f725d15c359c35","source":{"kind":"arxiv","id":"2105.01785","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2105.01785","created_at":"2026-07-05T02:58:06Z"},{"alias_kind":"arxiv_version","alias_value":"2105.01785v2","created_at":"2026-07-05T02:58:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2105.01785","created_at":"2026-07-05T02:58:06Z"},{"alias_kind":"pith_short_12","alias_value":"GM22EGV2QVPW","created_at":"2026-07-05T02:58:06Z"},{"alias_kind":"pith_short_16","alias_value":"GM22EGV2QVPWBTTO","created_at":"2026-07-05T02:58:06Z"},{"alias_kind":"pith_short_8","alias_value":"GM22EGV2","created_at":"2026-07-05T02:58:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:GM22EGV2QVPWBTTOORNPBFKQDP","target":"record","payload":{"canonical_record":{"source":{"id":"2105.01785","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2021-05-04T22:39:38Z","cross_cats_sorted":[],"title_canon_sha256":"00e7dc50bbd55ef3e416ecd53f1a4c9705c5950af571bc3ea53e9002fd040f11","abstract_canon_sha256":"9f81d519d758a6a4e9a392a39c57555c397d1e5b7597fe0ac90b0000a115c395"},"schema_version":"1.0"},"canonical_sha256":"3335a21aba855f60ce6e745af095501bcdff73a8fdeb0194c0f725d15c359c35","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:58:06.518911Z","signature_b64":"2u3+ULxVkufJ4vk6pW4+qRJf2riGuGPtQ8v5MqxyzokQ21nypk02t78F89WyaGXTLOdl8JQ5AZr1/UgOdV1VCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3335a21aba855f60ce6e745af095501bcdff73a8fdeb0194c0f725d15c359c35","last_reissued_at":"2026-07-05T02:58:06.518565Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:58:06.518565Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2105.01785","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T02:58:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Pb7gjeuCaCQuaZZlQgEnhNp75uGpQWQQR8g2nQJ+qUIDbM3JsRJbvgTc4Fr0Ka7OT/w9kVMxJZZDyqO0NDRcAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T23:54:15.432501Z"},"content_sha256":"a458f608fc8981216f50ae409e6acfb2ff033f66de20f3303d8f00d171f176eb","schema_version":"1.0","event_id":"sha256:a458f608fc8981216f50ae409e6acfb2ff033f66de20f3303d8f00d171f176eb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:GM22EGV2QVPWBTTOORNPBFKQDP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An Optimal Algorithm for Triangle Counting in the Stream","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"John Kallaugher, Rajesh Jayaram","submitted_at":"2021-05-04T22:39:38Z","abstract_excerpt":"We present a new algorithm for approximating the number of triangles in a graph $G$ whose edges arrive as an arbitrary order stream. If $m$ is the number of edges in $G$, $T$ the number of triangles, $\\Delta_E$ the maximum number of triangles which share a single edge, and $\\Delta_V$ the maximum number of triangles which share a single vertex, then our algorithm requires space: \\[ \\widetilde{O}\\left(\\frac{m}{T}\\cdot \\left(\\Delta_E + \\sqrt{\\Delta_V}\\right)\\right) \\] Taken with the $\\Omega\\left(\\frac{m \\Delta_E}{T}\\right)$ lower bound of Braverman, Ostrovsky, and Vilenchik (ICALP 2013), and the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2105.01785","kind":"arxiv","version":2},"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/2105.01785/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":"2026-07-05T02:58:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"znOGNeKxPiOWhIH1kTVPgSJEtM7/9Ga4fGK1nFXlD7HTBP5TcYSIbP/aiAFLgmsRnd9KjTWFo7Hx5rb+SaLVBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T23:54:15.433002Z"},"content_sha256":"1b9a0c97c215df7080939deb6d4968d71a723be734b795fdce3fa778f5b16b46","schema_version":"1.0","event_id":"sha256:1b9a0c97c215df7080939deb6d4968d71a723be734b795fdce3fa778f5b16b46"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GM22EGV2QVPWBTTOORNPBFKQDP/bundle.json","state_url":"https://pith.science/pith/GM22EGV2QVPWBTTOORNPBFKQDP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GM22EGV2QVPWBTTOORNPBFKQDP/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":"2026-08-07T23:54:15Z","links":{"resolver":"https://pith.science/pith/GM22EGV2QVPWBTTOORNPBFKQDP","bundle":"https://pith.science/pith/GM22EGV2QVPWBTTOORNPBFKQDP/bundle.json","state":"https://pith.science/pith/GM22EGV2QVPWBTTOORNPBFKQDP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GM22EGV2QVPWBTTOORNPBFKQDP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:GM22EGV2QVPWBTTOORNPBFKQDP","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":"9f81d519d758a6a4e9a392a39c57555c397d1e5b7597fe0ac90b0000a115c395","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2021-05-04T22:39:38Z","title_canon_sha256":"00e7dc50bbd55ef3e416ecd53f1a4c9705c5950af571bc3ea53e9002fd040f11"},"schema_version":"1.0","source":{"id":"2105.01785","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2105.01785","created_at":"2026-07-05T02:58:06Z"},{"alias_kind":"arxiv_version","alias_value":"2105.01785v2","created_at":"2026-07-05T02:58:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2105.01785","created_at":"2026-07-05T02:58:06Z"},{"alias_kind":"pith_short_12","alias_value":"GM22EGV2QVPW","created_at":"2026-07-05T02:58:06Z"},{"alias_kind":"pith_short_16","alias_value":"GM22EGV2QVPWBTTO","created_at":"2026-07-05T02:58:06Z"},{"alias_kind":"pith_short_8","alias_value":"GM22EGV2","created_at":"2026-07-05T02:58:06Z"}],"graph_snapshots":[{"event_id":"sha256:1b9a0c97c215df7080939deb6d4968d71a723be734b795fdce3fa778f5b16b46","target":"graph","created_at":"2026-07-05T02:58:06Z","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/2105.01785/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We present a new algorithm for approximating the number of triangles in a graph $G$ whose edges arrive as an arbitrary order stream. If $m$ is the number of edges in $G$, $T$ the number of triangles, $\\Delta_E$ the maximum number of triangles which share a single edge, and $\\Delta_V$ the maximum number of triangles which share a single vertex, then our algorithm requires space: \\[ \\widetilde{O}\\left(\\frac{m}{T}\\cdot \\left(\\Delta_E + \\sqrt{\\Delta_V}\\right)\\right) \\] Taken with the $\\Omega\\left(\\frac{m \\Delta_E}{T}\\right)$ lower bound of Braverman, Ostrovsky, and Vilenchik (ICALP 2013), and the ","authors_text":"John Kallaugher, Rajesh Jayaram","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2021-05-04T22:39:38Z","title":"An Optimal Algorithm for Triangle Counting in the Stream"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2105.01785","kind":"arxiv","version":2},"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:a458f608fc8981216f50ae409e6acfb2ff033f66de20f3303d8f00d171f176eb","target":"record","created_at":"2026-07-05T02:58:06Z","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":"9f81d519d758a6a4e9a392a39c57555c397d1e5b7597fe0ac90b0000a115c395","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2021-05-04T22:39:38Z","title_canon_sha256":"00e7dc50bbd55ef3e416ecd53f1a4c9705c5950af571bc3ea53e9002fd040f11"},"schema_version":"1.0","source":{"id":"2105.01785","kind":"arxiv","version":2}},"canonical_sha256":"3335a21aba855f60ce6e745af095501bcdff73a8fdeb0194c0f725d15c359c35","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3335a21aba855f60ce6e745af095501bcdff73a8fdeb0194c0f725d15c359c35","first_computed_at":"2026-07-05T02:58:06.518565Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:58:06.518565Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2u3+ULxVkufJ4vk6pW4+qRJf2riGuGPtQ8v5MqxyzokQ21nypk02t78F89WyaGXTLOdl8JQ5AZr1/UgOdV1VCg==","signature_status":"signed_v1","signed_at":"2026-07-05T02:58:06.518911Z","signed_message":"canonical_sha256_bytes"},"source_id":"2105.01785","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a458f608fc8981216f50ae409e6acfb2ff033f66de20f3303d8f00d171f176eb","sha256:1b9a0c97c215df7080939deb6d4968d71a723be734b795fdce3fa778f5b16b46"],"state_sha256":"5176a28e98b0af916b9396dadf463b3e7c16940c8d6c878db6474227d9706510"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fHZ7zliyo1Up2nkU139TLgPvqIbKG+jN3cKw1Rb+DtXV2ho8g5Xj143caQXmGSkqzuMIZ8oAlCr85+xWGxrbBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-07T23:54:15.440155Z","bundle_sha256":"421b8090afd12f81ba92e60eda37d78fe2e7f2cfa944fd10936e48c8e8dbddc7"}}