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G endstream endobj 70 0 obj 17202 endobj 71 0 obj (Mac OS X 10.13.3 Quartz PDFContext) endobj 72 0 obj (D:20180323140905Z00'00') endobj 1 0 obj << /Producer 71 0 R /CreationDate 72 0 R /ModDate 72 0 R >> endobj obj << /Type /Metadata /SubType /XML /Length 4237 >> stream endstream endobj 73 0 obj << /Encoding /MacRomanEncoding /Preamble (ESannopESslashdocumentclass[10pt]ESleftbrackarticleESrightbrack ESslashusepackage[usenames]ESleftbrackcolorESrightbrack %used for font color ESslashusepackageESleftbrackamssymbESrightbrack %maths ESslashusepackageESleftbrackamsmathESrightbrack %maths ESslashusepackage[utf8]ESleftbrackinputencESrightbrack %useful to type directly diacritic characters ESannopend) /EscapedPreamble (ESannoep%5Cdocumentclass%5B10pt%5D%7Barticle%7D%0A%5Cusepackage%5Busenames%5D%7Bcolor%7D%20%25used%20for%20font%20color%0A%5Cusepackage%7Bamssymb%7D%20%25maths%0A%5Cusepackage%7Bamsmath%7D%20%25maths%0A%5Cusepackage%5Butf8%5D%7Binputenc%7D%20%25useful%20to%20type%20directly%20diacritic%20characters%0AESannoepend) /Subject (ESannotWe present first massively parallel (MPC) algorithms and hardness of approximation results for computing Single-Linkage Clustering of ESdollarnESdollar input ESdollardESdollar-dimensional vectors under Hamming, ESdollarESslashell_1, ESslashell_2ESdollar and ESdollarESslashell_ESslashinftyESdollar distances. All our algorithms run in ESdollarO(ESslashlog n)ESdollar rounds of MPC for any fixed ESdollardESdollar and achieve ESdollar(1+ESslashepsilon)ESdollar-approximation for all distances (except Hamming for which we show an exact algorithm). We also show constant-factor inapproximability results for ESdollaro(ESslashlog n)ESdollar-round algorithms under standard MPC hardness assumptions (for sufficiently large dimension depending on the distance used). Efficiency of implementation of our algorithms in Apache Spark is demonstrated through experiments on the largest available vector datasets from the UCI machine learning repository exhibiting speedups of several orders of magnitude.ESannotend) /EscapedSubject (ESannoesWe%20present%20first%20massively%20parallel%20(MPC)%20algorithms%20and%20hardness%20of%20approximation%20results%20for%20computing%20Single-Linkage%20Clustering%20of%20$n$%20input%20$d$-dimensional%20vectors%20under%20Hamming,%20$%5Cell_1,%20%5Cell_2$%20and%20$%5Cell_%5Cinfty$%20distances.%20All%20our%20algorithms%20run%20in%20$O(%5Clog%20n)$%20rounds%20of%20MPC%20for%20any%20fixed%20$d$%20and%20achieve%20$(1+%5Cepsilon)$-approximation%20for%20all%20distances%20(except%20Hamming%20for%20which%20we%20show%20an%20exact%20algorithm).%20We%20also%20show%20constant-factor%20inapproximability%20results%20for%20$o(%5Clog%20n)$-round%20algorithms%20under%20standard%20MPC%20hardness%20assumptions%20(for%20sufficiently%20large%20dimension%20depending%20on%20the%20distance%20used).%20Efficiency%20of%20implementation%20of%20our%20algorithms%20in%20Apache%20Spark%20is%20demonstrated%20through%20experiments%20on%20the%20largest%20available%20vector%20datasets%20from%20the%20UCI%20machine%20learning%20repository%20exhibiting%20speedups%20of%20several%20orders%20of%20magnitude.ESannoesend) /Type (EEtype2EEtypeend) /Color (EEcol0.000000 0.000000 0.000000 1.000000EEcolend) /BKColor (EEbkc1.000000 1.000000 1.000000 1.000000EEbkcend) /Title (EEtitleEEtitleend) /Magnification (EEmag13.000000EEmagend) /Baseline (EEbas0.000000EEbasend) >> endobj xref 0 74 0000000000 65535 f 0000129402 00000 n 0000003474 00000 n 0000003741 00000 n 0000000022 00000 n 0000003454 00000 n 0000003578 00000 n 0000020650 00000 n 0000102454 00000 n 0000000000 00000 n 0000003874 00000 n 0000012950 00000 n 0000000000 00000 n 0000076398 00000 n 0000000000 00000 n 0000057579 00000 n 0000039551 00000 n 0000111278 00000 n 0000084414 00000 n 0000003824 00000 n 0000004438 00000 n 0000004108 00000 n 0000004418 00000 n 0000004669 00000 n 0000012867 00000 n 0000012888 00000 n 0000012909 00000 n 0000012930 00000 n 0000013121 00000 n 0000013349 00000 n 0000020567 00000 n 0000020588 00000 n 0000020609 00000 n 0000020630 00000 n 0000021265 00000 n 0000021494 00000 n 0000039466 00000 n 0000039488 00000 n 0000039509 00000 n 0000039531 00000 n 0000040043 00000 n 0000040265 00000 n 0000057557 00000 n 0000058113 00000 n 0000057794 00000 n 0000058093 00000 n 0000058341 00000 n 0000076313 00000 n 0000076335 00000 n 0000076356 00000 n 0000076378 00000 n 0000076938 00000 n 0000076619 00000 n 0000076918 00000 n 0000077168 00000 n 0000084331 00000 n 0000084352 00000 n 0000084373 00000 n 0000084394 00000 n 0000084906 00000 n 0000085139 00000 n 0000102432 00000 n 0000102765 00000 n 0000102997 00000 n 0000111195 00000 n 0000111216 00000 n 0000111237 00000 n 0000111258 00000 n 0000111770 00000 n 0000111992 00000 n 0000129285 00000 n 0000129307 00000 n 0000129360 00000 n 0000133793 00000 n trailer << /Size 74 /Root 19 0 R /Info 1 0 R /ID [ ] >> startxref 136934 %%EOF