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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"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"},"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"},"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 $M$. 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