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For each $i$, let $K_i$ be the Minkowski sum of $C_i$ with a disk of radius $r_i$, where each $r_i$ is a random non-negative number drawn independently from the distribution determined by $\\pi$. We show that the expected complexity of the union of $K_1, \\ldots, K_n$ is $O(n^{1+\\varepsilon})$ for any $\\varepsilon > 0$; here the constant of proportionality depends on $\\varepsilon$ and on"},"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":"1310.5647","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2013-10-21T17:08:03Z","cross_cats_sorted":[],"title_canon_sha256":"a2d1a242f2e38d8447287eecf7122165a1e3edd9b8218bfbd785a886af0f3d27","abstract_canon_sha256":"ef6c2ab138bfa04791b6992ba26c07715f86ce4ec086e9b52a8cb86feb47a9a6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:09:34.708668Z","signature_b64":"O4ZT921p0J+6fjjpaEaIjdOxrwTsJefDSTSyxVWbnmlo2wj7R9L/l2BJGzvkGkRmaJDC/clZDE7goQpI34+zCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"68eba29149305e69a9f7ffe5c8dad67636ae9a2257c7c979d44c058975ccb26f","last_reissued_at":"2026-05-18T03:09:34.707965Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:09:34.707965Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Union of Random Minkowski Sums and Network Vulnerability Analysis","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Haim Kaplan, Micha Sharir, Pankaj Agarwal, Sariel Har-Peled","submitted_at":"2013-10-21T17:08:03Z","abstract_excerpt":"Let $\\mathcal{C}=\\{C_1,\\ldots,C_n\\}$ be a set of $n$ pairwise-disjoint convex sets of constant description complexity, and let $\\pi$ be a probability density function (pdf for short) over the non-negative reals. 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