{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:6ZKBM26MCXZTD2XKIEXFO3KODU","short_pith_number":"pith:6ZKBM26M","canonical_record":{"source":{"id":"1107.2312","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2011-07-12T14:56:31Z","cross_cats_sorted":["cs.DS","cs.SC"],"title_canon_sha256":"e41813f36c39d3304c6fd02faf6afddc15e7838bda0ba819ecc8952fc129e8fb","abstract_canon_sha256":"534101da24cb797402f6398feb2b9d67f404ab61d2b8273c9f8e3ccfe18f0809"},"schema_version":"1.0"},"canonical_sha256":"f654166bcc15f331eaea412e576d4e1d0470b7865da13cec320903f0516496b5","source":{"kind":"arxiv","id":"1107.2312","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.2312","created_at":"2026-05-18T04:18:23Z"},{"alias_kind":"arxiv_version","alias_value":"1107.2312v2","created_at":"2026-05-18T04:18:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.2312","created_at":"2026-05-18T04:18:23Z"},{"alias_kind":"pith_short_12","alias_value":"6ZKBM26MCXZT","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"6ZKBM26MCXZTD2XK","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"6ZKBM26M","created_at":"2026-05-18T12:26:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:6ZKBM26MCXZTD2XKIEXFO3KODU","target":"record","payload":{"canonical_record":{"source":{"id":"1107.2312","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2011-07-12T14:56:31Z","cross_cats_sorted":["cs.DS","cs.SC"],"title_canon_sha256":"e41813f36c39d3304c6fd02faf6afddc15e7838bda0ba819ecc8952fc129e8fb","abstract_canon_sha256":"534101da24cb797402f6398feb2b9d67f404ab61d2b8273c9f8e3ccfe18f0809"},"schema_version":"1.0"},"canonical_sha256":"f654166bcc15f331eaea412e576d4e1d0470b7865da13cec320903f0516496b5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:18:23.079754Z","signature_b64":"8KvcQHcwmoSpL7yhTPXisVn2srwN2lwSxHa3wpe7ZnyxhgN07SxqAlH/2zfRSTmYfqRQnb3d54P+rT1dxiHFDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f654166bcc15f331eaea412e576d4e1d0470b7865da13cec320903f0516496b5","last_reissued_at":"2026-05-18T04:18:23.079076Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:18:23.079076Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1107.2312","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":"2026-05-18T04:18:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mZFKgAmTYQVR6NvgA/DrIRBxMvR1m4ZByvqI6UDzcphqTt2VVqbzvMqgxfIuwskN4a5HAxLkw1EHLaZe7t6gBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T16:18:50.509873Z"},"content_sha256":"321d3dd7de91601027e4bbcc18beae333b609bd500be3e7168d4d45069227423","schema_version":"1.0","event_id":"sha256:321d3dd7de91601027e4bbcc18beae333b609bd500be3e7168d4d45069227423"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:6ZKBM26MCXZTD2XKIEXFO3KODU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Computing the Distance between Piecewise-Linear Bivariate Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","cs.SC"],"primary_cat":"cs.CG","authors_text":"Boris Aronov (NYU-Poly), Guillaume Moroz (INRIA Lorraine - LORIA)","submitted_at":"2011-07-12T14:56:31Z","abstract_excerpt":"We consider the problem of computing the distance between two piecewise-linear bivariate functions $f$ and $g$ defined over a common domain $M$. We focus on the distance induced by the $L_2$-norm, that is $\\|f-g\\|_2=\\sqrt{\\iint_M (f-g)^2}$. If $f$ is defined by linear interpolation over a triangulation of $M$ with $n$ triangles, while $g$ is defined over another such triangulation, the obvious na\\\"ive algorithm requires $\\Theta(n^2)$ arithmetic operations to compute this distance. We show that it is possible to compute it in $\\O(n\\log^4 n)$ arithmetic operations, by reducing the problem to mul"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.2312","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":""},"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":"2026-05-18T04:18:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VhTkZggv8q6RssoYEho+B8/8D6fnHfs42+c/LLlZ8caZZNFKB72GGgAsA9PBYQGEscKxyXYq31PTik+Ty66KCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T16:18:50.510217Z"},"content_sha256":"a7aec8bbfb1d780e1a2495a00d252dc4fb83ef85a0d3b359482ab3e37d64de6d","schema_version":"1.0","event_id":"sha256:a7aec8bbfb1d780e1a2495a00d252dc4fb83ef85a0d3b359482ab3e37d64de6d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6ZKBM26MCXZTD2XKIEXFO3KODU/bundle.json","state_url":"https://pith.science/pith/6ZKBM26MCXZTD2XKIEXFO3KODU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6ZKBM26MCXZTD2XKIEXFO3KODU/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":"2026-06-27T16:18:50Z","links":{"resolver":"https://pith.science/pith/6ZKBM26MCXZTD2XKIEXFO3KODU","bundle":"https://pith.science/pith/6ZKBM26MCXZTD2XKIEXFO3KODU/bundle.json","state":"https://pith.science/pith/6ZKBM26MCXZTD2XKIEXFO3KODU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6ZKBM26MCXZTD2XKIEXFO3KODU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:6ZKBM26MCXZTD2XKIEXFO3KODU","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":"534101da24cb797402f6398feb2b9d67f404ab61d2b8273c9f8e3ccfe18f0809","cross_cats_sorted":["cs.DS","cs.SC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2011-07-12T14:56:31Z","title_canon_sha256":"e41813f36c39d3304c6fd02faf6afddc15e7838bda0ba819ecc8952fc129e8fb"},"schema_version":"1.0","source":{"id":"1107.2312","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.2312","created_at":"2026-05-18T04:18:23Z"},{"alias_kind":"arxiv_version","alias_value":"1107.2312v2","created_at":"2026-05-18T04:18:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.2312","created_at":"2026-05-18T04:18:23Z"},{"alias_kind":"pith_short_12","alias_value":"6ZKBM26MCXZT","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"6ZKBM26MCXZTD2XK","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"6ZKBM26M","created_at":"2026-05-18T12:26:22Z"}],"graph_snapshots":[{"event_id":"sha256:a7aec8bbfb1d780e1a2495a00d252dc4fb83ef85a0d3b359482ab3e37d64de6d","target":"graph","created_at":"2026-05-18T04:18:23Z","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":"We consider the problem of computing the distance between two piecewise-linear bivariate functions $f$ and $g$ defined over a common domain $M$. We focus on the distance induced by the $L_2$-norm, that is $\\|f-g\\|_2=\\sqrt{\\iint_M (f-g)^2}$. If $f$ is defined by linear interpolation over a triangulation of $M$ with $n$ triangles, while $g$ is defined over another such triangulation, the obvious na\\\"ive algorithm requires $\\Theta(n^2)$ arithmetic operations to compute this distance. We show that it is possible to compute it in $\\O(n\\log^4 n)$ arithmetic operations, by reducing the problem to mul","authors_text":"Boris Aronov (NYU-Poly), Guillaume Moroz (INRIA Lorraine - LORIA)","cross_cats":["cs.DS","cs.SC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2011-07-12T14:56:31Z","title":"Computing the Distance between Piecewise-Linear Bivariate Functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.2312","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":"sha256:321d3dd7de91601027e4bbcc18beae333b609bd500be3e7168d4d45069227423","target":"record","created_at":"2026-05-18T04:18:23Z","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":"534101da24cb797402f6398feb2b9d67f404ab61d2b8273c9f8e3ccfe18f0809","cross_cats_sorted":["cs.DS","cs.SC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2011-07-12T14:56:31Z","title_canon_sha256":"e41813f36c39d3304c6fd02faf6afddc15e7838bda0ba819ecc8952fc129e8fb"},"schema_version":"1.0","source":{"id":"1107.2312","kind":"arxiv","version":2}},"canonical_sha256":"f654166bcc15f331eaea412e576d4e1d0470b7865da13cec320903f0516496b5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f654166bcc15f331eaea412e576d4e1d0470b7865da13cec320903f0516496b5","first_computed_at":"2026-05-18T04:18:23.079076Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:18:23.079076Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8KvcQHcwmoSpL7yhTPXisVn2srwN2lwSxHa3wpe7ZnyxhgN07SxqAlH/2zfRSTmYfqRQnb3d54P+rT1dxiHFDg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:18:23.079754Z","signed_message":"canonical_sha256_bytes"},"source_id":"1107.2312","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:321d3dd7de91601027e4bbcc18beae333b609bd500be3e7168d4d45069227423","sha256:a7aec8bbfb1d780e1a2495a00d252dc4fb83ef85a0d3b359482ab3e37d64de6d"],"state_sha256":"b279ddf0fd684ceaebac5c975e35c9f18f08855d685e738e9f57f00385cb6749"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kHGh4Rgqe97/cyNseQ4late55bNDZmiBLk2+HB0560oKWM8od1HpJe8f3P/q8ZqvP9q1w9+Omc5Hai6Hz3x6Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T16:18:50.512182Z","bundle_sha256":"d87557f48f9e53444f98d4832aad75f73bbff95fc2429bf94519c4a23f727c7e"}}