{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:VS53CVCMP4O2OGUOONTLSKOJRV","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":"eaf98e7bc430650f4190c88e57733d5065b403b7517d0658fcf0b5a2a9c2720b","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-08-01T11:51:48Z","title_canon_sha256":"10d5b9186e1d530b2264cf645456fe7a29340309ca9b42853f5d6b84a10e2d89"},"schema_version":"1.0","source":{"id":"1108.0290","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.0290","created_at":"2026-05-18T02:27:50Z"},{"alias_kind":"arxiv_version","alias_value":"1108.0290v2","created_at":"2026-05-18T02:27:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.0290","created_at":"2026-05-18T02:27:50Z"},{"alias_kind":"pith_short_12","alias_value":"VS53CVCMP4O2","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"VS53CVCMP4O2OGUO","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"VS53CVCM","created_at":"2026-05-18T12:26:44Z"}],"graph_snapshots":[{"event_id":"sha256:d34e1c59e41d87e4c566b32c48f224b426340166a61279f27fc85fc1b2b087da","target":"graph","created_at":"2026-05-18T02:27:50Z","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"},"paper":{"abstract_excerpt":"A realisation of a metric $d$ on a finite set $X$ is a weighted graph $(G,w)$ whose vertex set contains $X$ such that the shortest-path distance between elements of $X$ considered as vertices in $G$ is equal to $d$. Such a realisation $(G,w)$ is called optimal if the sum of its edge weights is minimal over all such realisations. Optimal realisations always exist, although it is NP-hard to compute them in general, and they have applications in areas such as phylogenetics, electrical networks and internet tomography. In [Adv. in Math. 53, 1984, 321-402] A.~Dress showed that the optimal realisati","authors_text":"Alice Lesser, Jack Koolen, Sven Herrmann, Taoyang Wu, Vincent Moulton","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-08-01T11:51:48Z","title":"Optimal realisations of two-dimensional, totally-decomposable metrics"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.0290","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":"sha256:52af2154f59390f59d444f97dbd1698ba706817a26b29a52cb8d2a3e6c67139f","target":"record","created_at":"2026-05-18T02:27:50Z","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":"eaf98e7bc430650f4190c88e57733d5065b403b7517d0658fcf0b5a2a9c2720b","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-08-01T11:51:48Z","title_canon_sha256":"10d5b9186e1d530b2264cf645456fe7a29340309ca9b42853f5d6b84a10e2d89"},"schema_version":"1.0","source":{"id":"1108.0290","kind":"arxiv","version":2}},"canonical_sha256":"acbbb1544c7f1da71a8e7366b929c98d481351fcbb14a50eed1c71ff109e5504","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"acbbb1544c7f1da71a8e7366b929c98d481351fcbb14a50eed1c71ff109e5504","first_computed_at":"2026-05-18T02:27:50.188049Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:27:50.188049Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rmhJNX9OS7Nz3K/t1JIuiMenrRak5RVz8d1fdXP+wD198VaCAvlrgJrOsbdd6oxDoQWGLFytF3Tl5z89dvhwBg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:27:50.188480Z","signed_message":"canonical_sha256_bytes"},"source_id":"1108.0290","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:52af2154f59390f59d444f97dbd1698ba706817a26b29a52cb8d2a3e6c67139f","sha256:d34e1c59e41d87e4c566b32c48f224b426340166a61279f27fc85fc1b2b087da"],"state_sha256":"790300a1ea6b9ec44804ddc4c87a3913155d6ec5090e9be4578e19aedabd769c"}