{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:Z6UU42NM7EYHXPUGDIUPMXHQQU","short_pith_number":"pith:Z6UU42NM","canonical_record":{"source":{"id":"1906.09224","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2019-06-21T16:13:17Z","cross_cats_sorted":[],"title_canon_sha256":"6335cac83ffc150e8aecf5068082f80403a7edf8f57136ae586c710e3500ec47","abstract_canon_sha256":"ab2f188e98304be17bfcdc51d8f4024f084873ef93e204c5ed92cce449b18479"},"schema_version":"1.0"},"canonical_sha256":"cfa94e69acf9307bbe861a28f65cf0852e44f46c169b9d1bb1679aba36ada559","source":{"kind":"arxiv","id":"1906.09224","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1906.09224","created_at":"2026-05-17T23:42:44Z"},{"alias_kind":"arxiv_version","alias_value":"1906.09224v1","created_at":"2026-05-17T23:42:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.09224","created_at":"2026-05-17T23:42:44Z"},{"alias_kind":"pith_short_12","alias_value":"Z6UU42NM7EYH","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_16","alias_value":"Z6UU42NM7EYHXPUG","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_8","alias_value":"Z6UU42NM","created_at":"2026-05-18T12:33:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:Z6UU42NM7EYHXPUGDIUPMXHQQU","target":"record","payload":{"canonical_record":{"source":{"id":"1906.09224","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2019-06-21T16:13:17Z","cross_cats_sorted":[],"title_canon_sha256":"6335cac83ffc150e8aecf5068082f80403a7edf8f57136ae586c710e3500ec47","abstract_canon_sha256":"ab2f188e98304be17bfcdc51d8f4024f084873ef93e204c5ed92cce449b18479"},"schema_version":"1.0"},"canonical_sha256":"cfa94e69acf9307bbe861a28f65cf0852e44f46c169b9d1bb1679aba36ada559","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:42:44.093086Z","signature_b64":"MF30xeMiQiT9uHvsu6nQm3qyb0ZvmN2W9lgwiKxtxI9fNK8NLqaKUrwsgXLax6cMxhSSkvbkPNvc9GQ7INBVAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cfa94e69acf9307bbe861a28f65cf0852e44f46c169b9d1bb1679aba36ada559","last_reissued_at":"2026-05-17T23:42:44.092427Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:42:44.092427Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1906.09224","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":"2026-05-17T23:42:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7GCCl8uV4bQCSQTtUEKhvDm+ld9itH7TCWd3JWp9epjbrOyNpi3pqLY/1+lOtpfAuDPckUZ+wS+t7X2XhSCfAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T14:47:51.389328Z"},"content_sha256":"2508d9ba140d6c4fd799a63393f57c556faa0293d18d07deace6ca0e190c6fbf","schema_version":"1.0","event_id":"sha256:2508d9ba140d6c4fd799a63393f57c556faa0293d18d07deace6ca0e190c6fbf"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:Z6UU42NM7EYHXPUGDIUPMXHQQU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Multidimensional Dominance Drawings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"(2) Computer Science Department, CA, Crete, Giacomo Ortali (1), Greece, Heraklion, Inc. Berkeley, Ioannis G. Tollis (2) ((1) University of Perugia, Tom Sawyer Software, University of Crete, U.S.A.)","submitted_at":"2019-06-21T16:13:17Z","abstract_excerpt":"Let $G$ be a DAG with $n$ vertices and $m$ edges. Two vertices $u,v$ are incomparable if $u$ doesn't reach $v$ and vice versa. We denote by \\emph{width} of a DAG $G$, $w_G$, the maximum size of a set of incomparable vertices of $G$. In this paper we present an algorithm that computes a dominance drawing of a DAG G in $k$ dimensions, where $w_G \\le k \\le \\frac{n}{2}$. The time required by the algorithm is $O(kn)$, with a precomputation time of $O(km)$, needed to compute a \\emph{compressed transitive closure} of $G$, and extra $O(n^2w_G)$ or $O(n^3)$ time, if we want $k=w_G$. Our algorithm gives"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.09224","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":""},"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":"2026-05-17T23:42:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0Yg0JPORMLV5z6wplPPqp/nTMLQyczvoA+giZzPyyFNYMtXK0Rg1fDPEH0UGFfIwiVLj4EEXf67ZJpZmPzKOBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T14:47:51.389682Z"},"content_sha256":"b906def661f2e2b648899ce67c4d7859659d5ca9c0da41a6c7d0928c2a16c7fe","schema_version":"1.0","event_id":"sha256:b906def661f2e2b648899ce67c4d7859659d5ca9c0da41a6c7d0928c2a16c7fe"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Z6UU42NM7EYHXPUGDIUPMXHQQU/bundle.json","state_url":"https://pith.science/pith/Z6UU42NM7EYHXPUGDIUPMXHQQU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Z6UU42NM7EYHXPUGDIUPMXHQQU/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":"2026-05-28T14:47:51Z","links":{"resolver":"https://pith.science/pith/Z6UU42NM7EYHXPUGDIUPMXHQQU","bundle":"https://pith.science/pith/Z6UU42NM7EYHXPUGDIUPMXHQQU/bundle.json","state":"https://pith.science/pith/Z6UU42NM7EYHXPUGDIUPMXHQQU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Z6UU42NM7EYHXPUGDIUPMXHQQU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:Z6UU42NM7EYHXPUGDIUPMXHQQU","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":"ab2f188e98304be17bfcdc51d8f4024f084873ef93e204c5ed92cce449b18479","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2019-06-21T16:13:17Z","title_canon_sha256":"6335cac83ffc150e8aecf5068082f80403a7edf8f57136ae586c710e3500ec47"},"schema_version":"1.0","source":{"id":"1906.09224","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1906.09224","created_at":"2026-05-17T23:42:44Z"},{"alias_kind":"arxiv_version","alias_value":"1906.09224v1","created_at":"2026-05-17T23:42:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.09224","created_at":"2026-05-17T23:42:44Z"},{"alias_kind":"pith_short_12","alias_value":"Z6UU42NM7EYH","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_16","alias_value":"Z6UU42NM7EYHXPUG","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_8","alias_value":"Z6UU42NM","created_at":"2026-05-18T12:33:33Z"}],"graph_snapshots":[{"event_id":"sha256:b906def661f2e2b648899ce67c4d7859659d5ca9c0da41a6c7d0928c2a16c7fe","target":"graph","created_at":"2026-05-17T23:42:44Z","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":"Let $G$ be a DAG with $n$ vertices and $m$ edges. Two vertices $u,v$ are incomparable if $u$ doesn't reach $v$ and vice versa. We denote by \\emph{width} of a DAG $G$, $w_G$, the maximum size of a set of incomparable vertices of $G$. In this paper we present an algorithm that computes a dominance drawing of a DAG G in $k$ dimensions, where $w_G \\le k \\le \\frac{n}{2}$. The time required by the algorithm is $O(kn)$, with a precomputation time of $O(km)$, needed to compute a \\emph{compressed transitive closure} of $G$, and extra $O(n^2w_G)$ or $O(n^3)$ time, if we want $k=w_G$. Our algorithm gives","authors_text":"(2) Computer Science Department, CA, Crete, Giacomo Ortali (1), Greece, Heraklion, Inc. Berkeley, Ioannis G. Tollis (2) ((1) University of Perugia, Tom Sawyer Software, University of Crete, U.S.A.)","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2019-06-21T16:13:17Z","title":"Multidimensional Dominance Drawings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.09224","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":"sha256:2508d9ba140d6c4fd799a63393f57c556faa0293d18d07deace6ca0e190c6fbf","target":"record","created_at":"2026-05-17T23:42:44Z","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":"ab2f188e98304be17bfcdc51d8f4024f084873ef93e204c5ed92cce449b18479","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2019-06-21T16:13:17Z","title_canon_sha256":"6335cac83ffc150e8aecf5068082f80403a7edf8f57136ae586c710e3500ec47"},"schema_version":"1.0","source":{"id":"1906.09224","kind":"arxiv","version":1}},"canonical_sha256":"cfa94e69acf9307bbe861a28f65cf0852e44f46c169b9d1bb1679aba36ada559","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cfa94e69acf9307bbe861a28f65cf0852e44f46c169b9d1bb1679aba36ada559","first_computed_at":"2026-05-17T23:42:44.092427Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:42:44.092427Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MF30xeMiQiT9uHvsu6nQm3qyb0ZvmN2W9lgwiKxtxI9fNK8NLqaKUrwsgXLax6cMxhSSkvbkPNvc9GQ7INBVAA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:42:44.093086Z","signed_message":"canonical_sha256_bytes"},"source_id":"1906.09224","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2508d9ba140d6c4fd799a63393f57c556faa0293d18d07deace6ca0e190c6fbf","sha256:b906def661f2e2b648899ce67c4d7859659d5ca9c0da41a6c7d0928c2a16c7fe"],"state_sha256":"92e2be1d31d634f59abcfd4c0422fb5b2487b70d644a346ccd622924d938ed77"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PSZs5oiHcBjUDX69TdyCJEfIZ1FkqJVXDj5ISmNtfYgb2MthxHUUa4a9hjgP1IcjQnEQ9BYjEattHYH+5IZCBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T14:47:51.391579Z","bundle_sha256":"5235357761a27a0b1afdce641a8a6f067f5606fea91ebf7bf3edc9a8670a9cb5"}}