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ACM"],"published-print":{"date-parts":[[2026,2,28]]},"abstract":"\n Given a set\n \n \\(S \\subseteq \\mathcal {U}\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n and a function\n \n \\(f:S\\rightarrow \\lbrace 0,1\\rbrace ^r\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n , a static\n retrieval<\/jats:italic>\n data structure for\n f<\/jats:italic>\n supports queries that return\n \n \\(f(x)\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n for\n \n \\(x \\in S\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n and an arbitrary value from\n \n \\(\\lbrace 0,1\\rbrace ^r\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n for\n \n \\(x \\in \\mathcal {U}\\setminus S\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n . Retrieval data structures can be used to implement a static\n approximate membership query (AMQ)<\/jats:italic>\n data structure, i.e., a Bloom filter alternative, with false positive rate\u00a0\n \n \\(2^{-r}\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n . The information-theoretic space lower bound for both tasks is\n \n \\(r|S|\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n bits, and here we aim to use space\n \n \\(r|S|(1+\\varepsilon)\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n bits for a small\n overhead<\/jats:italic>\n \n \\(\\varepsilon\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n , including\n succinct<\/jats:italic>\n constructions with\n \n \\(\\varepsilon = o(1)\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n .\n <\/jats:p>\n \n A well-known approach to this task associates each key\n \n \\(x \\in S\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n with a row vector\n \n \\(\\smash{\\vec{h}}(x) \\in \\lbrace 0,1\\rbrace ^{m}\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n and stores a matrix\n \n \\(Z\\in \\lbrace 0,1\\rbrace ^{m\\times r}\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n such that\n \n \\(\\smash{\\vec{h}}(x)\\cdot Z = f(x)\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n for every\n \n \\(x \\in S\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n . We propose a new variant where\n \n \\(\\smash{\\vec{h}}(x)\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n contains a short block of random bits at a random position\n \n \\(s(x)\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n , and is otherwise zero. Sorting the row vectors by\n \n \\(s(x)\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n gives a matrix\n \n \\(A \\in \\lbrace 0,1\\rbrace ^{n \\times m}\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n with non-zero entries concentrated in a \u201cribbon\u201d along a generalized diagonal. This makes a variant of Gaussian elimination particularly efficient at computing\n Z<\/jats:italic>\n . We thus obtain simple data structures called\n Standard Ribbon Retrieval<\/jats:italic>\n and\n Homogeneous Ribbon Filter<\/jats:italic>\n . We then refine the construction using\n bumping<\/jats:italic>\n (a variant of backyarding) and\n overloading<\/jats:italic>\n (using\n \n \\(m \\lt n\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n ) to obtain\n bumped ribbon retrieval<\/jats:italic>\n (\u201cBuRR\u201d), with overhead\n \n \\(\\mathcal {O}\\!(\\frac{\\log w}{rw^2})\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n , query time\n \n \\(\\mathcal {O}\\!(1+\\frac{rw}{\\log n})\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n , and expected construction time\n \n \\(\\mathcal {O}\\!\\left(nw\\right)\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n , for a tuning parameter\n \n \\(w=\\mathcal {O}\\!\\left(\\log n\\right)\\)<\/jats:tex-math>\n <\/jats:inline-formula>\n that opens a trade-off between space and running time.\n <\/jats:p>\n Our experiments reveal our implementations to be the first to simultaneously achieve small overheads and fast running times in practice, with BuRR achieving overheads well below 1\u00a0% while being faster than most competitors, which have larger space overheads. This efficiency, including favorable constants, stems from a combination of simplicity, word parallelism, and high locality.<\/jats:p>\n We offer a unified theoretical perspective on these three ribbon-based data structures, including a nontrivial rigorous analysis of their running times and memory consumption.<\/jats:p>","DOI":"10.1145\/3785417","type":"journal-article","created":{"date-parts":[[2026,1,3]],"date-time":"2026-01-03T20:18:27Z","timestamp":1767471507000},"page":"1-45","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Ribbon: Fast Succinct Static Retrieval and Approximate Membership"],"prefix":"10.1145","volume":"73","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5484-3474","authenticated-orcid":false,"given":"Martin","family":"Dietzfelbinger","sequence":"first","affiliation":[{"name":"Technische Universit\u00e4t Ilmenau","place":["Ilmenau, Germany"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-2662-2607","authenticated-orcid":false,"given":"Peter C.","family":"Dillinger","sequence":"additional","affiliation":[{"name":"Meta Platforms Inc","place":["Seattle, United States"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4315-3264","authenticated-orcid":false,"given":"Lorenz","family":"H\u00fcbschle","sequence":"additional","affiliation":[{"name":"Karlsruhe Institute of Technology","place":["Karlsruhe, Germany"]},{"name":"Firebolt Analytics","place":["Munich, Germany"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3330-9349","authenticated-orcid":false,"given":"Peter","family":"Sanders","sequence":"additional","affiliation":[{"name":"Karlsruhe Institute of Technology","place":["Karlsruhe, Germany"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6477-0106","authenticated-orcid":false,"given":"Stefan","family":"Walzer","sequence":"additional","affiliation":[{"name":"Karlsruhe Institute of Technology","place":["Karlsruhe, Germany"]}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2026,2,13]]},"reference":[{"key":"e_1_3_2_2_2","doi-asserted-by":"publisher","DOI":"10.1109\/FOCS.2010.80"},{"key":"e_1_3_2_3_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-04128-0_66"},{"key":"e_1_3_2_4_2","doi-asserted-by":"crossref","unstructured":"Michael Axtmann Daniel Ferizovic Peter Sanders and Sascha Witt. 2022. 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