11 numerical data e.g alphabetic order names w grades allowed - PDF document

Space efficient quantile selection Where U has order C an C U stream Input 9 000 11 numerical data e.g alphabetic order names w grades allowed multiple passes median return the Goal minimum w 6 space passes a quantile queries generally more k element select rank kth largest

Input I pass exikhd.dk n fT

i i kh

Passe Spacey select and sort 1 Och quick select random pivot OCI Ollogn rib Munro Paterson 1981 p Vis 2 quantile summaries

Approximations and KEEN rank E given 0 param element k rank return En w f K Sampling median for 18 0 elements sample f return medias of sample dn k rank for l sample the out de return of rank sample Deterministic

invariant consecutive two Key any 2En have width intervals E E APX quantile summary

Q get 94 ai 3 intervals 9mi w E APX need show to To show Q is 2Eb width property in Q's Take intervals two consecutive E g y Two cases from diff elements sets elements from sets same

Recap we can E APX quantile summaries combine to get E APX quantile summary of whole thing E APX quantile summary sparsify to w Kpoints tat quantile summary APX to address Remains all how make at to one

Space D D D D D D D D D D D D D D D root only keep summaries

theorem 1 pass OC log4n E space deterministic stream quantile over E APX t dyadic intervals trick idea mergability slightly better

Theoremett 1 pass 0C logic Eh 1E space deterministic stream quantile over E APX better Even 2001T Greenwald Khanna space loafers more sophisticated quantile summary merging trick interval