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Inverted matrix Efficient discovery of frequent items in large datasets in the context of i


? ? T f `x i ? ` Q i ? V t ggH Q I n i ?H I ? d ` p i whhs?g'mS???h?PPSRDsw%Y?Ycqqp T ` Q T VHg i d I I ig ? T V I T V a I ? ? | ? i Q V i d T ` ? T i ? Q ` T ` s8bX%PS?YP?hX8hc?Y?sRx?S??i ? SxXxhi ? 58ch?5? Q V t Q n I T `H Q fg ` I ? i ? T V I Q i I V Q V ? i U d Vg ?g i p i d Q F i d `x Q T iH a cDSv5sPS5PsY?w5T DXgSwYX??hesSc%dPwqSYrwgc?Wrw%wDk lx f I Q ` T i d V i d f Q V d i QHg i t T ? i I ` ?g Q T i d d f a I p t QH d ` Ug 5Y3X5eS?eSSc?wSfP1hSQ PH ?YX` ? Sd ? fSbSY53qhSfSXXPXV U TH TH p ig f T `H Q VH a ` I I o ? I Q i I V Q V ? i t Q d `x I T V a I ? ? | ? i Q V hP5%q3PhSd sPS?wbYY3vSwY??ch?hS?X?rhc?Y?sR!?S??i ? l9SvcsPY1YXqq?hSeX1whw5Dwi ? 5jsPhsY5X??bsPq??DSQ i d d ` i ?H I ? d ` p i p i t Q T ` ? a T i ? T i ? t UH t i I f V a Q t UH p Q V t T `H Q V Q sPS??5f ? XW?PYYX?8sqfPr?hqDi ? 5sq5SPScsPXehPSRPwi p ` a i ?H I I V p T ` ?gH ? V i t ? T i ? I p t QH d ` Ug V U TH Q IH F i t Q n d i ? i ? ` ? p UH ? V d g igg V d d ` g VH Q T i f ? i I V U TH I f ? i I hS??9?r?3?? sP5X?cV ? w%Pc?cV ? XhX%Sbhb?Yh1hPY5?Ys` ? l ` T i i ? i ? V t I T `H Q fg ` I ? T ~ ? I d V i ? ? ix Q I Vg i t Q TH I iH ? f Q cSd ? w'Ws?hrhXPS5PsYqbcV vSc??w?iRX%WhSmP?PDSRI ? T V p x ` I f a `x i t Q T i i ? I V t n I tI V Q Q T V Qd rc?gcqhb?!hSz'jXD?rYX?jbc?YX` ? PqRsq!hS3X` pH Q I ` p i t Q x i T ` I V ? i d i ?H I T ` a n I ig f d T `H Q VH a ` I I V U TH d i ? ` a IH ? ? I ig f d T `H hs?X#?Sw5%Y5swPhS?sPSc%bYYcd5%S9??YPg1PhSqsPSQ l VH a ` I I V x ` ? d i ? ` a IH ? i t Q I U TH TH p V Q V ? TH I t I V Q g V aH T ` T V c%bYYcWX?Yw??YP5?5S?PH 5PhPqd?c5zP?rYc??X?P5shc?a i t Q x ` i T ? ? V Q V ? x ` I T `H Q a igg ` a i U d Vg i I i t Q p ` dx I T d i Q Q g f hS?X?5???c5Gc35sPSwPPshXSc%?YwhS?sS?hS9SYcV ? 5?x l9YhhfPX%SbSX` ? hXGbc?DfhXPY5c1sPSwX?YrwvhS?X?¤hX%SPsYI i I f ?gg VH Q T i Q ? T V Q T V a kH T UH I x ` T `H Q a V d Q F i i t Q d `x I T `H Q fg ` i p `I IiI ` qsYsYX` ? Sd ? n PhsSd ? PhS3YX??S?PSYr?DSm%D89S??YSd p ig ? ` IH t Q i I V i ` Q I iH d Q Q V t Q ?g i k t a d V i I i V n U TH TH p V Q V ? ? ? iH ? f Q I U TH i ? Q f ` t QH ? i t ggH Q I IH V Q V ? x w55%5Pq???hhv?P5DSR?5%w'h5XhSf?3Q ? ??PPSRgP8??h?X`

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ABSTRACT

Inverted Matrix: Ef?cient Discovery of Frequent Items in Large Datasets in the Context of Interactive Mining

Department of Computing Science University of Alberta Edmonton, AB, Canada, T6G 2E8

mohammad@cs.ualberta.ca

Mohammad El-Hajj

1.1 Problem Statement

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? U TH Q Q i I i ?H Q a V d i Q TH ? T V i ?H QH Q i d V TH I f ` i 5hPSYwYhs%S9X?wSbP!hX1?PSfSwi ? SedP35sXU l V Q T V ? ? V ?g d Vg f aH Q d IH t a V ` ? V ? i I V ? l t IH ? I I i a a V p ` ? T V ??rcssdfSc%hPSY?V ? Pd?XsSd h? d?YcD?bYP5WYc?sDc?d d f ? ? I p i QH i f ?H T f x ` d i ? p f T i U d Vg ? d i V t QH ? I i I V ? V Q V ?)qSfm5b5h?chw'j5r!XSc%#Yw?? Sf?ewYXDc?c5? g V T `H Q a V I T V d Q i U d Vg ? d i ? U TH TH p TH ?gg VH a I i i i d u l ? ? p d ` XDXPSwXS5X?YmsS?%)Yw?8hPhPq#P¨fPX%i ? wjS?yGG?SX?x l d Q f ` t a V ` ? V FH d Q ~ ? i Q d i ? T | d f ` Q V t Q g V i ? i d I iH ? f Q Ywi ? s?XsSd h? }fSYcV ?SYw?ry!s8chS3c??S?P5hSRI g V Q T i pH d F E ? ? i ? i i T IH U TH Q T f ` a ? T V T `H Q V d i T i U ? a V ?H X?bqPSwi ? G?w??5dPhPSbhXDc?X%S??whws9Xh%? l T V a p f pH TH p I V I Q T i p i dH f ? i d ? d ` p i p i t Q I i a f ? i d I I i a ` 5c??hePhPqsXiSbqSfhb?SeYXqqWhS?wh?SrYbSd ? U TH TH p i ?H I d f a i d l T ` T ? T V i pH I V n ?gg V TH ? ? I i a T i d d f a a ` l ` hPhPq1?PYS5wS?s5qhXj%g ? PY??bfPXh%??whSYb?Xwa U TH ?H d V p p f I U TH ?gH f ? IH i i d Q Q T i ? T i ? TH gg V p ?g i ?H Q Vg i d hPPSc?q5YmhPP5h?PhSYrb5hi ? DP?Pc?YI f?PSc%S?V n Sf?bhb?S?9c?mc??vDsw¨YwX?sDX??Sf?XYI p i QH Q T i f ? i dx t a V i d `x n ? T ` a i ? ? I I i a a V p ` ? T V d t QH ? T V a gg fx V T V t Q I I ig TH ? i ? iH t a V i ? ?g f ` a I T d i Q Q Q T i f ? i d U T Ph?d?XDS1YPP¨??w%59c?'?%hsw?5SwSYcV ? rwhb?S?x 5PH l ? T k t aH t ? TH n i I V rDDe9Ph??P?sYcDt ? hPhPqr5S3hPS5WYXhX?c5shS'cghP5T U TH TH p i t Q U TH d f ? i I V ? V Q V ? i t Q x ` U TH l T V a I i H Qg f p I Q T i ? i Q V t Q FH d Q ~ ? i Q d i ? T | ? igg V a Q f ` ? V 5c?YqPg ? SfhehSrw?Sd ? ?DS¨fSY?V ¨?SY9?rp?PPX??5Xb%g i I V ? V Q V ? ? i V ` Q TH ? i Q d i ? T ` a I V Q V ? g V T `H Q a V I T V d Q n Q I dH YcDX??h?whT qSbP¨?SY9?rs?PH ?ch?cDX%S9XShc?YG?RSf?? ?X?%?9hT SS#5P5fg h? xb?whw%wqwqSf??Ph?X9P5?? I V i ?H ? i i i d t Q U TH ? ? V ? ? ? a T iH a ? i I QH I i ? iH t a V t aH t nvPSY?V }?SYwsy}?P%c?{hSfSXX%czhPhPqqPhSysPSc%bYYcV FH d Q ~ ? i Q d i ? T | ? igg V a p t QH d ` Ug V U TH TH p ig f d T `H Q VH a ` I I ? i I V ? l t IH ? ? i T V I i I ` d ?YXhybYP5xw5grYX` ? Sd ? 9i ? V ? Phrvw%c?cqDXrq%S?SSQ IH t u ? TH V U V ? T V i pH Q I i d f l a f d Q I V Q V ? i U f t i I i t Q x ` U TH ?gH f ? i t Q ` Q U TH ? V ig ? d V I I i a i w5SYR8??hmXhrjYwhSvX8hP5%Ph5jhS?S!5P5s%#Y?SY5T i d V I p t QH d ` Ug V i p V I i t Q x ` I T f d ? T V p n I d i Q i p V d i T f Q ` S??q5SPScsPX?qcShhS?X?h5S?rc?DSwS9qX?cV ? hhSm?u ?YbSd ? Yw??YP5js5%??5t rcsceSScDe?PSwX?9SrPqDcV IIia ` ? d i ? ` a IH ? i U ? ig ? ` T ? T V x ` i d f Q V T i ?H Q a V d i Q TH ? T i ?H QH Q i d i t Q IH Q T i pH ? s%SfSwi ? S?5S?P3bqP?i ? PhXp??3whSchW?S5XYSd Yc` pH d ` ? V p d i t Q ` T o ? I i a d f ` I i ? d lmwqjP?hsPc?X??bS?P?YbSd ? 5PhPqr?PYS5wS?hS?csbYc` p i p TH I f `H a V d ` ? ` ` Q IH I I i a ` U TH TH p i ?H I d f a i d i t Q d ` n ? d lmwqq%c??%ihhSj?q5YYX?%ifPhhWS5S9hSYRh?c5gPSbX?PXU p i p TH V p TH Q k ` Q ? i p f I I V IH QgH f ? i d f Q a f d Q I V Q V ? aH Q T V UH i5S?w5Sfrvw55hi ? 5dYXqwq#XPh?hS3P?PhXSd ? cR??p t Q d i t QH i e ? a T i ? T i ? ? d ` p i p t UH t i t Q IH p ig ? ` d ` ? V i T ? ? I Q i I V Q V ? g V T `H Q a V I T V d Q i ?H I I V p U TH TH p T i t ? I p ig ? ` 5??SwYX??h?cDX%S9XShc?Y??PYYX??hP5%q!h?rqw%5sSd ? ? T V p p ` dx d i u f I I p t QH d ` Ug V U TH TH p ig f d T `H Q VH a ` I I V U TH Q IH F bX?ysS?wvY¤q5SPScsPXrhPhPq?PhSeXPSc%bYYXW5%SRPGE

1.

i p fg ` ? I f ` ? T i p i d Q V n T ` T i p ` T i IH t Q ` Q i f ? ? t Q ? ` d U i ?H qhPs?¤hsDqwSY3X?s5qs5ht ? %5S?S?5gG?Ss?Scs?PYI l I V p IH t Q ? `gg `x Q ` T ?H ? V Q V ? i t Q U TH ? ?g V T V ? T V U TH Q V fg V ? Rc?gPhS??%PX??ch)%58?c5j5S85%w5fcDXqhX!hPSch%cswi x` di ? c?wr?` ? 5PYYbSd ? hS?%xSs?SXgc` Sc?!qXSd5r}sS9i U TH I I i a ` i t Q TH t Q ? ` d U x i Q V d i p V I i t u ? a Q n ? d V QHgH p n i TH aH ? i p n I I i TH I f ? n i a T iH a I x ` I ?g i k gg V TH V Q V ? U T bYc?fPPqrshPP?qYhPYh5hXhPY¤Xr%D'PX!P???hm5PH l Q V d i T i U TH t Q ? ` d U i ?H I ` F i T V ? i I I i T QH ? i ? V t I ? V ? Q T i a i Yc?9hX!P8Ss?Sc??PYX%g ? w!c!YYhSf????DrhrrwW?

INTRODUCTION

Department of Computing Science University of Alberta Edmonton, AB, Canada, T6G 2E8

zaiane@cs.ualberta.ca Osmar R. Za¨ane ?

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1.2 Related Work

2. TRANSACTION LAYOUT 2.1 Observations on Super?uous Processing

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T i p V ? T fx IH t Q x ` i I f i t u ? Q T i f ? i dx i ? ` Q i ? V t I Q i I p i QH l ? l ISfGPXw?bhb?S?'8S?9YqSfyb?X??chS85PSc?R)yYwi ? Sd ? QH gg V n Q T i f ? i dx i ? ` Q Q i I p i QH l ? V d `x Q V t Q U TH Q V Q I ? Q d ` i T ` Q ` T ` p V I i I f Q | ? I p t QH d ` Ug V T ? ` T t Q I ` p x ` T `H Q V ? T f ` 5sSchsq8hY5hyjqhSfSXX%c¨??5gRsqsc¨sPS?hD5s?x i5SiqSc?eYPc?hSfSXXPXv?X??D9hPhri??? ?5SfSXsPcv9X?9D?? t Q I p d `x ` Ig V p t QH d ` Ug V ? ? ?? ? ? ? IH t u ?? ? p t QH d ` Ug V ? ? ?? ? ? i5S3PWSwYqSf?b5b?S?m5%S??wqhr?X??hX%S??PPg h? V bc?p t Q IH I Q i I p i QH Q T i f ? i dx U TH Q V d i p f T i d `x I T `H Q V aH ? ? TV TH d Vg %?c%hf ? ` ? RXqi5Si'WS¤qY?9P5?Xq5SPScsPXhw??5Shc` Q I ` p i t Q i ? ` Q I p i i I t aH t ? n I p t QH d ` Ug V ? i t i t Q x i T ? ?? ? n ¨ ?? I ig f d T `H Q VH a ` I I V U TH TH p x ` p ig ? ` i t Q I I i d ? ? 5?i?c?§f?PhS?sPS?wbYYX1hP5%qwc?w%5sSd ? 5SiYSrhXV ` Q i d f Q V d i QHg i t Q TH ? i I ` SmS5Sc?9SPPm5S?P?Ys` ? Sd ? w'j??D1q5SPScsPXvc?w?r? T i i ? i ? V t I p t QH d ` Ug V g V d i ? i

1.3 Motivations and Contributions

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2.2 Horizontal vs. Vertical layout
T10 P T9 T8 T7 T6 T5 T4 T3 T2 L R Q A O K L M N O A H K A A B F A B C A E C H G N (A) B H D C E G I E I A G E I A C M J E T10 P T9 T8 T7 T6 T5 T4 T3 T2 L K L A H K A A B F A B C A E N R (B) B H M C D C E E Q A H G G I E A N A C I M G J I E O O

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2.3 Inverted Matrix Layout

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p i QH T V I V t t a V ` ? V IH t Q TH ? d ` a i d t a V E p i QH i t Q IH ? d ` a i SfmcvXDq?XsSd 5? vPhSjPm?XwSj?XvG? SfshS?Pm?XwSd t a V i x ? i t i t Q t a V ` ? V IH t Q T | ? d i t Q i U ` Q I p i QH i p V I i t 9c?sX` w?m5S#?XsSd h? 1PhSqpm?9hSwXsS?qSfeqXSm5SQ i d V t I Q V t Q I T `H Q a V I T V d Q gg V I i Q Vg i d Q f ` ? Vg g V aH Q d i ? i t u ? I p i Q S?DYvchSihXPSwXS5X?Y5PcrS?wSvsb%hc?PSYw??hrGqSfH ?? x` i ?V p I T HQaVI TVdQ ¨? x` pVI V IQ TiIi id o? id f H §GXrs?ihsP`SwcShc?Y?GXsPig ? XS?iSbYSd ? Sx?hS5XPU?? ?sPhc??cDX%S9XShc?Y?5SgX?dwse5SicjPsS1hS?%g ? sPSwcV ig ? V Q g V T `H Q a V I T V d Q i t Q d `x ? i t i t Q x ` ig ` d i t Q I ? V T `H Q a l I T V d Q i t Q x ` ? | i t Q t a V ` ? V IH t T | ? d i t Q i U ` Q T `H Q a V I T V d Rhc?YhhS?X?p15Sq?XsSd 5? gP5SQ pW?whS9ssSqsPSwcShc?YQ i p V I i t Q T ` I p i QH gg V I i Q Vg i d n ? i I f ?g T ` p p ` a Q I ` p i t Q I qcS?hS?s?qSfrPXmSc%SGv?Yh#fhXqqseRsq?hS?PH t aH t ? n i T ` Q I d k i t u ?g V aH Q d i ? ? T V g V Q T ` ?H d ` t ?g i p V T n I Q f 9Ph?¤shX1RShmhr!?X%SY9??hXrc?bsPSX5qfqXhGS5X` l ? Vg ` ? Q i t Q x ` i T ` i I f I p t QH d ` Ug V U TH Q IH F E ? ? i I f p t QH d ` Ug rery3hS¤Xg5sgY5sqhSfSXX%cqhPSRP?vv?Yh?5SfSXsPcV U TH TH pl ig f d T `H Q VH a ` I I V i t Q x ` ? a T iH a ? i i t Q U TH TH p d i Q i ? T hPhPq?wPhS?sPSc%bYYcj5Sic)whw%wqw?hSqhPhPqSwS95#PH ig ` d Q T V Q d PsS?bX?Yc` ? PqXr%g ? YXhX??hWhSmPr5sPSwXS5X?YhhPScSRI pH T V I ? V i I V ? V Q V ? i t Q TH I T `H Q a V I T V d Q U TH d ` Q xcqc?Sc?8hrxPhcS?s#hPSbYSd ? SqXjS55%SYY??PPXa ` Q V p d `x i t u ? I ig ? VH d V ? U TH Q T i I i i d d ` I i Q f ?H d Q Q V ? igg V I5qhPsa IhX?hsPS?sSwY5s?X1SSc????PPX?1??Sd Gc` T p fg ` H ? T V n I T `H Q V ? d i I ? ` d ` I i d f Q V ix ? igg V a I ? ` F x i ? V p I ig ? V Q t a V E ? I ig ? V Q ? igg V a I ? V d d V g V T `H I T i pH ? l ` ? 5s??PH Phc??Xv?Phc?x?PPX?q?Yc3XhsPYhqPpcryQ ` Q TH V Q V ? U TH d ` Q I x ` I Q IH I T ` a g i ? ` p i I V ? V Q V ? g V T `H Q Vg i d i t SrPq?c58hPSXSRvX3SRPYhsw¤rqeYcDX??h?cDsPS?wS?5ru

?  I V Q T i f ? i dx I V d ` Q T i f ? i dx i d ` p i d V Q V t ¤%XWb5b?S?sX3c?rwhb?S?3SXq1ScW?DSQ I p i QH t QH ? ? ig i ? Vg I i ? ` T ?g T ` I TH V Q T ` a  p i QH Q T i f ? i dx T i ?H qSfjSf???P'c%G5r51P5sG5%c?bse$SfGbhb?S?w?PsU V d `x i i d Ql | ? ? % V Q V t Q IH n i a T i d i uH ? d i t Q ` T o ? T `H Q a i I IH t Q T rc?sSY?y?gsF?chSvPbXhS9vPiwhSch3vwsPSwYvPhSmPH d i Q V ? i TH V F i I V d i Q T f ` a T `H Q H aH Q d 9Sc%g ?hPcg ? w?cgwSb5sqsPS?V ? PSYcV ? !YPX355'?wSbhXa V ` Ig V Q f ? n d i Q T f ` ? a T i f ? i dx V ? T V g i ? Vg p i QH i t Q ?g T ` Q ` T TH V Q T ` a I i ? ` T i t 9h5rS?8#Dc?w'X%?Sfj5S)fhs1ch?%c?bshrhehSQ ? T V ngg i ? I V U TH T T V a Dc??Prgc)hPh5X?YI ? ymXSYX'qhP??PPXqSYehS?PhrhPPg f l p ` Q Q ` ? U TH ? `gg V i i d Q i t Q TH I t TH gXhsPSwSf5d??h8wSY?y?gs% 5S??w?wr??yhXPSwXS5X?YQ V T `H Q a i dH ?H ? i ? V t I i i d Ql | ? ? i t Q n d i ? i ? ` ? ? I T `H Q a V I T V d l ? f I U TH Q T i I i i d I t Q hYg5PSrwYSd ? SG5ScV ? Sf?rY?Ywi ? SY15wSd ? 5SeDcbn SfH t QH ? i I l d i i d Q F k i i t Q ? T V p i Q Q T i f ? i dx i p V I i t Q U TH TH V Q T ` a I i ? ` T x ` T `H I I i a a f I V ` Q U T rwhb?S??qXS?5S#5%5PX?bse5r5?cxXPYY5Yq?S)hPH l Q TH YbPs` ? S9SrPX` ? c?rXPSX5#hXhqSfhb5b?S?q?Sw5?c¨Sf?? I d i Q TH g V Q T ` ?H d ` t ? T V I p i QH Q T i f ? i dx ? i d i ? d ` t QH d i ? V i t V i ? V t ? i t Q V t Q i I T i I i t Q TH g V d i T i U TH i i d Ql ? w5X?h????DwhSQ ?DS#YhY?hS?%?X?9hX?P¨wSY?GG? g V T `H QH ? T ` a i t Q ` Q d VgH pH I i d V Q d ` t I d `x i i d u l | ? ? % d ` n i i d XhsPSf5hs3hSeSs?%%qPYgS?? YX5YWX?WS?yy?gs!chXS?u piQ| Q Ti f ?id ? ia Tidd faa ?l ` Syqb5b?S5?hwSY5?pc?%  P5hzrqSY?%c?Yzhru ?gH f ? i ? I i i d Q gg V p I i t ? ? i TH p I V T ` ` I I V ? i ? d V a IH w?5PqWcqsbY?Xd?5???YP5? i d V I i i d Q i t u ? ?g i ?H I d f a i d I i i d Ql ? f I g V T `H QH ? T ` a U TH ?gH f ? Q f ScswSYW5r?bf?PYS5wSiSY?hYvXhsPSf5DX1hP5%Ph5rs` l t QH ? ? T V T `H Q V d i T i U ? a V ?H ? T V a i t Q U TH ?H pH TH p t QH ? n QgH f Sf?qhX?XPSc?w5s)9Xh%DXjhSq5%w%qP5%q#Sf?v?PP5h? i d V ? i t Q I V T ` ` I I V I i i d Q i t Q x ` i T ` t a V i ?g i Q V d i I i TH p i ScewhS?X8XbYiciwSYrhS'cW5sm?XqfSc??V ? Yr5Pqs?5 ?sYcDX??h?XDXPSwXS5X?YmhS?%?wSP1b5b?S???X??c?1SYQ i I V ? V Q V ? g V T `H Q a V I T V d Q i t Q TH p i QH Q T i f ? i dx t a V i d `x I i i d gPX?Yxf?PSc%Sjrw5Dwi ? 5h%?hP5%Ph55SrvfSYcV SYw?? g V p I ?g i ?H Q Vg i d Q T i ? T i ? TH U TH ?gH f ? T i t Q n FH d Q ~ ? i Q d i l T | i t Q p ` dx ?g Q a i dH ? I p i QH Q T i f ? i dx d `x I T `H Q a V I T V d Ql ? f I U T 5pWhS?sSmfSwSf5iqSf?b5b?S?rc??5sPSwcShX?Y?hY1hPH l ? V i d T ` Q I d k I iHg i d I iH a T i f ? i dx U TH Q rs?SqXrRShrw%PSiPh5b?S?g5%Sf ? XsX?q9c?Sd 5? r5?? p ` a d `x t a V ` ? V d f ?? ? ?? I d ` t Q f V i t Q ? ???P1SchS5chhS?b? ? i Q I i Q Q V I V ig ? Vg V a I ?g d ` ?SRSY?sX1Phc%X?Y8fSXb` ? 'jSm???Sd ? DcshS9SYcV ? bhb? i ? `Q ?i ?` ? TV I TdiQQ Q Ti f l i dx i t Q I V i ? f QH T U V p x ` d i ? d ` i p V I i t Q T ` i d V Q V t Q n I i i d Q l ? wSv5S¤Xi5hSfhsc?DcGw5?ciqcS?5S1XrSc?chSXwSY?GG? gXhsPSf5hs¨%Pc??wSY?hY?chw'j5r8XSc%#Yw?d?P5h? V T `H QH ? T ` a ? igg V a n I i i d Q l ? f I x ` d i ? p f T i U d Vg ? d i ? V I ?gH f Q V t Q i TH Q f ` d i ?H I d f a i d V ? ? ? i a V i d IH T `H Q V d i T i U ? a V ?H ? T V cDS?hPSsS??PYS5Shqbq?c%g ? S?P8sPSc?9hXqwX5%5Dc?a i t Q n? ? t Q ? ` d 0 l ? ? I V t a f I I ? ` t Q i p T | ?? ? T `H Q V d i T i U ? a V hSr?c2 #Ss?S31GG?X8?hYv5rhS9q8yv??? ¨sPS??whXqwch? lfDXdhS3X` w%YmwhY!hSqS855?yPPcDsPY5qP5xsPh?X` H ? T V a i t Q x i ?H I d i i t I i t Q ` Q i f ? ? QHg V T `H I T i pH ? t UH t x I i I V a TH p ig ? ` F i p ` a V i p ` a i ? Q t UH Yc?qP?P5sSd ? wPg ? XqjqX'3rXPqp ? SR?PhS?qhSfSd i Q I IH t Q I p t QH l ` Ug V ? i I V ?l? ? ?? ? ? ? T | ? ? d i ? ` a IH ? ig f d T `H Q VH a ` I I V d `x T `H Q csPX?YXD???X??D9yqbYw??YP?%5SsPSc%bYYcjc?¨XPScV l d ` ?g Q I ` a V ? i d i ?H I T ` a IH I iH a T i f ? i dx x ` T `H Q V d i T i U i t Y9i ? ?fSRsq?S95%YhXqPmPhwhb?S?3XXPSc?w5s!hru

l ? SdqPhX¤dXSX?9'#?5sPSbq)hPYYwbSd ? hXhf ? wi ? Yq5SQ i d ? ig ? V u ? i d `x i ? ? i T `H Q T i p U TH I I i a ` I f ` d f I i t ?H ` ? V i ? n I f t u ? I p i QH Q T i f ? i dx i t ?g T ` TH V Q T ` a Q V t Q I T `H X??j?'hrr?qSf?b5rS?hhSQ f5sqPX?bs3chS?hXPSQ laVI TVdQ i tQ x` IQd wXS5X?Y?hS3XmSY?V ? %%5hSzXa ri?sPS??bP?% 5SQ sSx ?gH f ? i d T V i ? n T `H Q V a `g i t p `d U TH Q d V Q I I p i QH x ` TH V t a i t Q FH d Q ~ ? i Q d i ? T i t Q TH U TH ? ` 5PSYc?RmqSfhcPXh9!hSzfSYcV ??SYwsy| hS?P?hP??Pg lg `x ? ? ?g ` t I i d t Q Q d ? f I d Vg f aH Q d IH t Q d `x Q T V ? ig i d dH i d fs?!i3 w%s5YS5SWYX` 5? Yh?5PSYcV ? PhS?X?3bXXPSYfeS?V I p i QH d i t Q ` i t Q gg o ? Q T i f ? i dx i d V d i Qx V d V ? V Q V t Q I p i Q qSfWw5SXhhSvP3??bhb?S1S?}% wSc?c?i 5? 3chSsqSfH i t ?g T ` n % t QH ? U T `g V n ? i d i ? d ` i d V FH d Q V p i t Q x ` I i THg i t 5SQ fhXv3ESf??hsPc¤v?Sw?XjS?)fSYc??hSiXgwhPPe5SQ i a TH ? ? ? a T i f ? i dx i t Q I V t t aH t ? whPebA w5hb?S?5S?Xh??Ph?y% Sfe5S3SbYSd ? S8§? p i QH i t Q I Q T i I i i d " i TH ? ? " ? i THg T `H Q V a `g p ` dx I I i a ` U TH TH p i t Q Q d V Q I ` Q I i a 5P?i§mhPP)sPSc?bP?XS??YbSd ? 5%5PqghSsY??RjSshk lx f I QH n ? T V t Q d i Q V i d U Q d ? f I t QH ? I T d i Q Q ?Ydf?XhS!wS??SX8Yc` h? Y}Sf?qhS9SYcV ? b5b?S??PcV Q T i f ? i dx gg ? T k ` Q I i ?H a i ? d i I f i t Q xH n i p V F i d ` ? ? ? d V ? d `x Q t UH V d Q I I DhmS?w5%?9YhhhSGf5sPg ? ch9?chsv?csYc?WbsPX?YRWPH V i ?H aH I V ? i t u ? FH d Q V p IH t Q p ` dx ?g Q a i dH ?  I Q i I p i QH Q T i f 5%!PYcD8hr¨wfSY??jPhS?sS??fSwSf5?$SwYqSfjb5b? l i dx d ` I ig f d T `H Q VH a ` I I V U TH TH p d `x p t QH d ` Ug V T V I Q T i I i wSjX? PhS?sPSc%bYYcqhPhPqjX??hSfScsPX?XeSbYSd ? T `H Q a i I Q F i T i t u ?gg V d `x ? T V i a T ` QgH f ? IH QH n i I V ? V Q V XPSwY)rw5?hr??PX?c?}DX?hXqPP5h)PqfeXYXDc?c5? g V T `H Q a V I T V d Q T i ?H U V d ` ? ? i I V ? V Q V ? g V T `H Q a V I T V d Q i t Q x XDXPSwXS5X?Y??Ps?qXh?sYXhX??heXDXPSwXS5X?Y?5Sjc` U TH I I i a ` l i i ? ` Q ? i p f I I V IH FH d Q ~ ? i Q d i ? T | i t Q U TH ?gH f hPYYbSd ? wSd ? '?S3?q5YYX?P?PSY?V h?SYwsy?hSshP5%Phi3

3. CO-OCCURRENCE FREQUENT-ITEMTREES: DESIGN AND CONSTRUCTION
$ " %#! E R?T(¤¤?3 ? §         "   § ?  2   2 6 ?( ? ?  ?  ? § #¨??s¨?Y?¤?!?R? $ #! E ¨¤rhr?D? ?  2 ?? 6?26 ? 6  4 4 ?  ? § S 2 ?  ?  ?§ ? ?§    § ?  2   2 6 ? ( ? ? ? §   ? 2 ?D????%@R?V¤)5?)§q¨??s¨?Y?¤??@??hr

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C E C E C E C E C E E A

(A) A A A A A

C E E A

Frequent Items

(B) A Occurs 5 1

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? i ? ` T Q ` ` d i t Q ` Q ? i Q Vg i d I T d i Q Q Q T i f ? i d srhWXbS3hSjS!Sc%SshSwSY?V ? b5rS?x i t Q gg V i a f ? ` ` Q ? i Q I i Q ? T V ? i Q V d i T i U i ? T V a i i d Q d i t Q ` T hS'PX?hrSd ? Sq?SRSqDcd?S??whws?'8XgSYrw5SX5XV ? T V i pH Q IH t Q Q V ? i ? ` p i d i ? T V a DchqPSs%5S?c8???qSg'dc?% Sy?XhS?y??gs4hru piQ| x ` iid ul| ? ? % i t ?w%?S?V ? PSYcV ? % Sf?chShhSwSY?V ? bhb?S?f5sSf?hhf TH i Q H aH Q d p i QH Q V t Q I T d i Q Q Q T i f ? i dx ?g T ` t QH ? I U TH ? V ig ? i Q QH p ` i d V I T d i Q Q Q T i f ? i dx l T ` T gg V ?gg V TH ? ? e % hPr??Pe?SYfqs?S?i5SwSYcV ? bhb?S?s5hPXmPPcDP?? 3F? QH i t V p ` Q T d i Q Q ? e % ? U TH Q IH F i i t Q ` ? i ? ? V IH i fg V ? I Q f1?c?8S)SwSY?V ? G3F?hPSRPw?hSqSQ ?55sgP?hPXXgSfH ? T V ? i a f ? ` IH ? e % ? T d i Q Q Dc??hrSd ? Phh3F?SwSYcV ? vX5rhyWhSsX?3vX3hPXX? o ? i ? ` T ? i t Q d `x ? x ` i fg V T `H Q H aH Q d sPScV ? PSYcV ? Sf?? ? 3% r e? w95X?rhXjPGSRP59?PGcvRSQ t QH ? e n e ? n t a T V d ? i T ` TH I Q IH F i QH I V Q I i ` Q p i QH Q I Vg i t Q IH p i QH ? i t Q n ?gg V TH ? ? A i p ` a i ? ` Q ? i Q V S?SfrRX%?hSvP?Sf??hS?bfPXhP?vr!qsw'hSj?Sc5? ? f IH ? a T i f ? i dx I QH ?g T ` T i t Q n ? ` Q Ig V f ? i i fg V ? ? a T i f ? i dx V t QH P?w5hb?S?Sf?fhXj5Ss5?S?PcDb?r5%cs?w5hb?S3qSf?? I Q IH F i ? ? V i dg V T d i Q Q SRPw?rs?SfX8SwSY?V ? 3 hPYqDc895X?5?%5S?XS?m?55? % i a TH I ? T V t a T V d ? IH t Q p ` dx ? i a f lcSd ? P?B'33  srhz3 5SrX??wc35%cs!sPScV ? PSYcV ? Sf?? ` IH  ? e % ? i ? ` T i t Q d `x ? x ` i fg V ? T `H Q H aH Q d t QH  A 3% DXqA 33  ?hc?5?hsqPTgSRPIw8?(WsSYWPIhS)PT?SfQH e ? TV e ta TVd ? i T ` H IQ H Fi ? 3? n iidQ H tQ H pi Qrwhb?S?dhswY?5rWrS?cDb?3fh?X?qS!5?5?P1hPXX? T i f ? i dx ? T ` a i I i t u ? A ` Q g V f ? i QH i t V p ` Q ? i ? i i T IH i fg V ISf?hPSc5? ? ?fhXqDXh5?S?PXhb??5%csrYX` h? Y!Sf?rSRPwi QH U TH Q V f ?g T ` ? T V n ? ` Q Ig V f ? i i fg V ? Q d ? f I t QH ? I Q IH F ? ? V i dg V T d i Q Q rs?SfX?SwSY?V ? DXY8hr?s?3!?Dc!s3FdS??Ph?? ? T ` a i I i t u ? ? e % o ? T V ? e % ? i d V t aH t TdiQQ SwSY?V ? FW?hS?sS??S??whX)YPX8Sch5SwSYcV ? 5hrds? % ? o i t Q p ` dx ? i Q V d i T i U ` Ig V i d V I T d i Q Q l ? f ? ?

4. INVERTED MATRIX ALGORITHM

?Id faa Sb` I TdiQQ hSwSY?V ? bhb?ShXxYw??YP5? hS?Ph?P)YXht ? hPhPqp Q T i f ? i dx x ` ? d i ? ` a IH i t Q t aH t ? TH i I V U TH TH i t Q ? T V QgH f ? IH FH d Q ~ ? i Q d i ? T | i t Q t aH t ? TH U TH I I i a ` l i hSjhXsfPh5?PqfSYcV m?SYw?rp?5S??Ph?mP?hPYYbSd ? 9Sd ? i t Q ?g i p V T I I i a ` hSxPwqXD!YbSd ? hPhPqqhS?X8YXht ? ry#5S?hXqXV U TH TH p i t Q x ` I i I V ` ?Q i tQ U T ` p ? i ?H ?H ? i d V I p t QH d ` Ug V i I i t u ? t d ` ? I f ` d f I x ` I Q a i u i i t ?rP5iSc¤qhSfSXXPXrYwhrG'Scrvhs5f ? wi ? Yhc¤Swwvw?hSQ U TH a f ? i d ? T V T `H Q V d i T i U ? a V ?H ? T V a g V pH TH t QH ? I i I V ? V Q V hP55?S?Dc)sPSc?9hX!9Xh%DX'X?PhPqp Sf?sYXhX??h? gXhsPSwcShX?Y8sS?)5PhPq?c?Ys` ? 5f ? 5SSP??qhSfSXXPXV V T `H Q a V I T V d Q i U d Vg U TH TH p x ` i I d i t Q t QH ? I p t QH d ` Ug x ` I Q i I i d V I p t QH d ` Ug V ig f d T `H Q VH a ` I I FH d Q ~ ? i Q d i ? T | i t X3SwY1Scgq5SfSXsPceP5S#X%S?%bYYXV PSY?V )?SYwsy1hru

????

4.1 Building the Inverted Matrix

Q IHg p i QH i t u p i QH t a V i x ` ? a T i f ? i dx i t ? T k ` Q i I V ? V Q V RPP?Sfehre? Sf??XsX)9h5rS?e5SQ DD8SjYXhX??h? ig ` t ? i t Q I T V a I I I Q I d k i t u ? i I V %Xh??hSvhXYrYXV ? RSh?hrGsYXht ? hPYYbSd ? 9Sd ? hShhPH U TH I I i a ` l i i t Q U T lY55jYXhX?c5j5SicgYYcV ? rp)P?hshP)fSYcV q?SY9?rp| d f ? iIV ?VQV ? i tQ x` IiII ` ? Q TH i T ` ? IH FH d Q ~ ? i Q d i ? T i t Q U TH ?gH f 3 ? T `H Q a V I T V d Q i p V I i t T p i QH Q F i T i t Q x ` ? ` hSqhPP5?6?X%S9XShc?YhqXShhSQ PH SfWrwh1hSGc??Sd i t Q TH I I i d ? ? V g V aH I ? i t Q ? T V n T `H Q a V I T V d Q i p V I i t Q TH p i Q hS1P?YS5src?PY5bt ? hSghXbwsPSwcShX?Y?qXSihS1P?SfH Q Fi T i tQ x` Qd 9h?hS?chYcV ? w5hPjhSqPhYS5s?c?PY5bt ? hSW?XPSc?p F i ? TH i t Q TH I I i d ? ? V g V aH I ? i t Q e T `H Q V l d `x TH ? I ?g ` t dH Yc?5P??5%s5rfXV ? 9c??Swh?sSwSb%X` ? hPSbYSd ? SvSfXV ? X` t a V i i d i t ? n I d i Q TH U TH Q T i I i i d I dH x i ? V p IH ? ` d t a V E ? Q d 5X?rP??S89civ?Y?V ? 95DP3hS?%?Sf?hX!Sf?!Sc%bYI F i ? TH i t Q TH p i QH i T ` t QH ? ? i Q VH a ` l I V IH ? ` d t a V t aH t ? TH I ? ` d x ` Q i I V IH ? V d d V g V T `H Q a V I T V d Rc3P??Sd9c?i ?Ph?dPW??S¤XWwYmWPd?Yc?XDXPSwXS5X?YQ i t u ? ? a T i f ? i dx i ?H Q a I i d dH i t Q ? T V I p i QH i t Q I TH V Q T ` F i hr?bw5hb?S?s%S9i ? SWPwhSdDc?qwSP15S3hPc?rXa 95? l TH i t u ? ? V d d V g V T `H Q a V I T V d Q i t Q ? T V F i ? TH i t Q e I Q d P?hrGb??Y?wXDXPSwXS5X?Y?hSmDc?w5hP35S?SYcV ? ?pwX` ` ?Q x i ? V p IH Q V t Q Q f ` ? Vg V Q V ? ? i I V ? l t IH ? V IH FH d Q ~ ? i Q d i ? T | i t 5X??P?chS?5Xb%W??h1?YXh?rYP3GP1fSYcV 1SYw?rpihru

? ? ? i Q T i p i d a TH IH I i ? ` T i I i t Q T ` I i fg V ? T `H Q H aH Q d gg b?SbqSw5Pq%qw5rhdYhSzsq5PXXsPS?V ? PSYcV ? PWo ?¨ Sg%cDb?ehPXX¨X%S?V ? PSYcV ? Sf?DcmqPq95X?h%5S?X` ` Q Ig V f ? i i fg V ? T `H Q H aH Q d I QH ? T V ? IH t a T V d ? IH t Q T o x ? a T i f ? i dx i t Q I V ? e % ? o T d i Q Q i t Q I i Q V d i T i U o I V rc` w5hb?S?5S?Xe?3FW#SwSYcV ? 5SgSc?9hX?hXht Q V t Q t a T V d ? ? T ` a i I i t u ? ? e % ? T V ? e % o i d V t aH t ? T ?DSx?hc?5hswY)hr?5r383 Dcz?36qS?}9P5?PH I i Q H aH Q d S?V ? PSYcV ? % wSP??DSghS9SYcV ? hYi%c8Sc?w5s!S??5T p i QH Q V t Q I T d i Q Q ? f I gg V i Q V d i T i U ` Q ? i i i ? n ? e % 3 o T d i Q Q Q I d k i t Q T | ? T d i Q Q IH t Q x ` ? a T i f ? i d ?¤5?3!?SwSY?V ? RShehSqp??S9SYcV ? P5SrX#w5hb?Sx i t Q IH t aH t ? n ? ? ? ? i Q T i p i d a TH i d V t a T V d ? IH t Q TH I i ? ` T gg 5S3P)?%5?'5!bd?SbqSw5PeS?q95X?5WPhS)P3w5rhGPcV d `x I i fg V ? I T `H Q H aH Q d X??whPXXmhXPScV ? PSY?V ? hr?5)S?PXDb??w5hb?S?!o i t u ? ? ` Q Ig V f ? i ? a T i f ? i dx % 3 TdiQQ SwSYcV ? RS5jhSgw?X???Ph?d¨ cqhPXXsPS?V ? PSYcV ? DcV Q I d k i t Q I i t V p t aH t ? x ` i fg V ? T `H Q H aH Q d ? T ? x ` i fg V ? ? a T i f ? i dx V I V t t a T V d ? Q I d k i t Q TH ?XhhPcs8w5hb?Sj3cD!95X?5WRS5ghSdP?o wSyis5r5T piQ| ? i ? ` i p V I i t Q x ` i fg V ? T `H Q H aH Q d i t Q I f TH p t a T V d ? i t Q TH p i Q qcSrhS5Xr5PXXmsPScV ? PSYcV ? hS¤5Pq?95X??hS?P?SfH Q I d k i t Q x ` ? a T i f ? i dx i t Q IH t a T V d ? t a V i x ` ? a T i f ? i dx i t RS5?5SsX?whwhb?S?q5S?P?hX?9c??X¨whwhb?S?q5ru ?9 ? 3% DXD j'sw??@DXW ? 3GhXvbA 33 D1? S?q9P5?? e ? T V n e ? n ? e o ? T V e % ? T V n e n ? e o i d V t aH t i i d Q % i t Q TH I i t a T V d ? ` ? Q TH I Q IH F i o p i Q | ? o p i QH IH t aH t wSYz!hSmPvh?hc?51rym%vSRPw?Sys??SfiP?9P5?? nsSY?hS?P?wSP1b5b?S?gRXqmhS?sS?hSY??RI ?% Sfsc` i i d Q i t Q TH p i QH Q T i f ? i dx Q I ` p i t Q p ` dx I Q d V Q ? p i QH x i i d Ql| ? ? % ? i t Q U TH TH ~ ? ? T V t Q d i Q V i d U ?g ` t I i d t Q Q d ? f wSY???gs)zhS)5%5Pd?rzcDSj9Sc?Sc?XhYSSeYc` h? YI V U TH p f I I V ? i d i ? ` a IH ? ?hPq5YYXSw??YP5?% 5PhPX?bswmSwYqSfqb5rS?d5SQ U TH TH V Q T ` a I Q i I p i QH Q T i f ? i dx i t I??hY?FS5sP?iwYYbSd ? Sc?595X?5vPXdh?qDwghSs?V ? ` t I  i d f UH ? ? ? i I I i a ` i d V I i t a T V d ? gg V T i t ? ? T i i t Q Q ??5???YP5?ScghXhbhb?S?s5jhrm?RP%8¨P#fPSc?X` ? wSQ i ? d V a IH ? i d V I i T ` Q T i f ? i dx l T ` T i t u ? Q IHg V TH ?gH d V d pi ? i d ` Q I ? T V ? i kH Q T i ?H i d V I T d i Q Q Q T i f ? i dx i Q V ?H ? T V a n Q T f ` SXSRqDc8DfSb5%gS?rhS9SYcV ? rwhb?S?WSc5Dc?r?bhswa T `H Q H aH Q d XPScV ? PSY?V ? hS?hXebhXjYc` h? YqhSQ hPY5?sSYmhS?c` i t Q ? T V Q T f ` a Q d ? f I i t U TH I f n i i d Q i t Q x t a T V d ? t a V i p ` d ? ?  i d f UH ? TH ? i Q V d Q I fggH i d V n i p V F i d ` 95X?5q?X??sS1?S5X%??P#Sc?YRhPPPeS?wsPg ? ch9hX?x p i QH Sf?% hS¨Sz?Sc%Sm5SwSYcV ? bhb?SdhrSd ? SmI ? Sb? i t Q ` Q ? i Q Vg i d I T d i Q Q Q T i f ? i dx i a f ? ` `Q iQ

?

????

????

Item A A,B,C 4 C:8, 4 Participation = 4-0=4 Candidate list B:5, 4 D:3,0 ABC: 4 AC: 4 BC: 4 A:4, 4 A:1, 0

????

A:4, 4

A

B

D

D:3, 1 C:8, 6 B:5, 5

C:8, 6

D,C 2 Participation = 3-1=2 Candidate list ABC: 4 AC: 5 B:5, 5 BC: 5 A,D,C: 1 D,C: 3 A:4, 4 D:3,1 A:4, 0 B:5, 0 A:1, 1 C:8, 0

Item B B,C 1 Participation = 5-4=1 Candidate list ABC: 4 C:8, 5 AC: 5 BC: 5 A,D,C: 1 D,C: 1 B:5, 4

C:8, 8

D:3,0

D:3,3

A:1, 0

Item D

A:1, 1 A:4, 4 Item A B:5, 0 A:4, 0 C:8, 0 B:5, 4

? i I V ? V Q V ? i t Q TH I i a T i d d f a a ` dH i t Q U TH Q T f ` a Q f ` t QH ? I T d i XYXDc?c5hhS8%W5SY5wbWPwhSe5PSr5sWshSf?shS9SQ l Q Q T i d dH i t Q p ` dx ? i a f ? i ? i ? T V a I T d i Q Q l ? f I d i t Q ` x YcV ? bS?V ? fhS?sS?jh?5W'?X?vhS9SYcV ? hYiwhSc¤c` ? a T i f ? i dx i t u ? i i d Q i t Q x ` i ? ` T Q ` ` d i t Q U TH ? fg a F i ? i TH w5hb?Smhr8swSYjhSscm5r5ecbSjhS8hP5hPw??5Pqp U TH i ? IH Q V t Q i i d Ql| ? ? i t Q TH I i ? ` T x ` d i ? p f i t Q IH 5P'?%??DSdwSY???gs% hSx%?rhhcqw'ehrT hSqPy i d i t ? n i pH Q V Q V I T d i Q Q Sw5?¤sqPSdjcehSwSY?V ? p f pH F V p i Q heP5??mS5f ? s?hX8Sc?9i p `a ? TV iQVd l T i ` Q ? i I f IH t a V ` ? V T ? ` ? ` Q 5wsU S?Yh?%¨9c?Sd 5? ??pl ? S?o w%?S?V ? PSYcV ? wSYQ ? TH I i Q H aH Q d iid i t Q x ` Q ` ` d i t Q T ` p i QH i t Q Q V t Q I T d i Q Q Q i I p i QH l t Q T i f ? i d 5S?c?XbS?hS?s?Sf?hSichS?hSwSY?V ? wYqSf?r?bhb?Sx gPX!hP5hD?cjYX` ? 5f ? hS#Sf?dfSb5hi ? 5hPjSY#9c?1X?x g V U TH ? T k x ` i I d i t Q t QH ? ?g Q T i ? T i ? TH i i d Q t a V i d ` i5s!P8YbSd ? 5%5Pq)hry?fPhh?PH SY???gs9hT 5SQ T ` ? IH I I i a ` U TH TH p i t u ? QgH f ? I i i d Ql | ? ? % Q F i i t i d `x i ? ? i ? d V a IH ? T i t Q n ? i TH p n QgH f ? IH i i d Q t a V E ? d i t Q i U ` Sc?w'???c?YP?5S¤whPqG?fPhh?PqSY??Xv)?w5SwsXSQ ? i Q a f d Q I T ` a Q ` T i d V I p i QH Q T i f ? i dx gg V x ` I i i d Ql | ? ? i t ?SwhSYR5s?ch)S??qSf?rwhb?S??PX?cqSY?y?gs% 5ru

? ?

C:8,5

C Frequent list Support > 2 ABC: 4 AC: 5 BC: 5 D,C: 3
A,D,C 1 C:8, 4 Participation = 1-0=1 Candidate list ABC: 4 D:3, 0 D:3,1 AC: 5 BC: 4 A,D,C: 1 A:1, 1 D,C: 1 A:1, 0

? I Q IH F i QH xH i i d SRPw?f?PhwSYQ i t Q TH p i QH i p V I i t Q x ` T `H Q V a `g Q F i T i t Q ` Q I Q TH Q V t Q i i d 5Sq%?SfWqXS?hSwX8sPScb%irwhWhShSvSrPX` ? chSWwSYQ i t Q TH i ? ` t a V i d `x i ? V p ` Ig V IH t THg o ? i i d u l | ? ? % i t Q T 5SdPh5r5T 9c?3X?35s??YPX3P?hPP?rXS?yy?gs45SdPH p i QH t a V i d `x ? d Q T i Q I d k x ` T `H Q V a `g i t Q TH V Q TH V p ` Q I p i Q Sf??XmX??YYb?RS5WczsPSc?bPq5S¨PX?bPX?)S1qSfH Q T i f ? i dx gg V x ` Q IHg V I i Q f QH Q I T ` a n I i i d u l ? ? t QH i tHg n i i d b5rS?3Pc1X!RP%?dSSfSRhsw?S?yGG?SP?? s%P?swSYQ t a V i TH d i ? V i t i t u ? o ? T V 9c?P?w5X?h)5ri9 hX?3 XhSqrwhb?S?qYP?P8P?cV T V t Q Q T i f ? i dx I I ig IH QH I I i I V a ` ? Q Q I Vg i t Q TH ? i d ` T UH IH  ? e o 3 p ` dx o p i QH t QH Yc?1rp?RX%?hSmPq?SX5sP?P?% b1r8% sS??Sf?Sf?? I i pH Q ? I d f a a ` qPS1?S5b?3 Sf8f%cDP?vs'1??% P?'1?8Ps%!9P5?? p i QH ?gg V TH ? ? ? e o ? TH ? e o ? TH IH t aH t nj#XhSgbhb?SeSXq1Ph?DS?Sf#X¨SP?ehXhS5b?? ? T V t Q Q T i f ? i dx i d ` p IH Q V t Q p i QH T V t QH ? i a T ` I d f a a ` p i QH I V t a T V d ? i T ` x ` i ? V p IH i i d Q ? i t u ? A I i p ` a i ? Sfmcz95X?dhX?c!s?mP!SY?8hr?b&qX'?3 ? TV ? Ii p `ai ? DX¨wqs'?% 5S?DXx?Sc5? ? )S?!S9Sr5sxfhXYI I f t Q ? T V ? i Q V f i d V I d i Q T f ` a ?g T ` ` t a T V d ? U TH Q IH F i T V t QH ? F k i i p V I i t Q I i d V t I ? ? V i dg V 95X?5¨hPSRP59X}Sf?zh9Sd ? qcS)hSqSchYrs?SfX)? e 3 % ?gg V TH ? ? I i p ` a i ? ? ? T V n I i p ` a i ? n ? i Q V f I 38'fPXDP?zr qX'?DX?r7 qsw'?% w?S?h? ? mPH ? a T i f ? i dx i t Q ?g T ` ` I n QgH f ? t a T V d ? w5hb?S)hSfhXY??fPh5?hX?V Xhrs?SfXWe?% I V t ? ? V i dg V ? e ? T d i Q Q V ? ?  ? ? A I i p ` a i ? Q d ? f I I QH I V SwSYcG?9e ? qsw'dYc` h? Y!Sf8X?% rh!Q ? w59i i ? ` T ia F t a T V d ? i t Q d `x ? ? Q d ? f I t QH ? t a T V d ? d i t Q ` T V I i Q V i d a 95X?58hSqc?8e8YX` 5? Y?Sf?z?hc?5q9hSX5X?S??Swd? e o ? % ? t a T V d ? i t Q TH i ? ` T t a V i d `x ? ? Q d ? f I t QH ? t a T V d ?3?Cw95X?5s5SjP?5rhj9c?rc?g?YX` h? YqSf?j95X?? i T ` I p d `x ? e o 3 I V ? i Q V i d a i d V I i ? ` T ? ?g Q T i f ? i I T ` ? T 5sWqSX?m'?r8% Xq?S??Sw1Sc3rhq!PSb5b?YhXa DcV nj)Dc?3 ?qfqcD'qSf? SP?3S5b?% wSyhsw%?y?o ? ? T V n o ?g i p V T n I p i QH t QH ? I d f a a ` p i Q| ? i aH ?Q p i QH t QH ? I d f a a ` QH Q f ? n E p i QH t QH ? I T ? V t U TH t Q i p V SfSf?jSbmf?55??·SfSf?e5i h? D)hP5S8qcSI i5rirA P5X?!P?SrwYSd ? YcDc?ch3hS?Px3 Sf!Sf?qfhX` t u ? ig ? V u TH ? i Q T i I i i I V ? V Q V ? i t Q TH p i QH t QH ? ?g T i aH ? Q I d f a a ? p i QH i I f V a i ? IH IH t Q n ? ? ? a T i f ? i dx t QH ? 3 d ` P?y?S5b` ?SfrYhc?'?P?PhSsbew5hb?SeSf??X?x i ? ` T ?gH t a i T ` ? T V ? U TH TH V Q T ` a i ? ` T Q ` ` d i t Q e I i ? ` T ` ? rhq%%59ehXqhXdhPhPc?rX?5rh1cbS?5S3w5rh!ryQ ?g T ` x ` i ? V p IH n ? i d f UH ? n ? d `x i i d Ql | ? ? i t Q Q V t Q i aH Q ` fhX?c?s?3%wbmSsP?w?hc?1SY?y?gs% hSWcDS?PSX39 ? i p V Tl piQ sqXh?mSfH i p V i t Q I V t Q V t i i d Q i t Q T i ? ` T Q I d k i t Q ` Q d i Q TH qcSI hSmcD??DSQ SYd5SxPH rh?RSh!hS¨SqwSbPs` ? V







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4.2 Mining the Inverted Matrix

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? iidQ Q Fi T i tQ d` sSYsrw5ghSWX?x ? i I f i ? T V a i a I ? d ` p i p I QH ? T V n i U V Q I IH t Q Q V ? i I V ig i d i ?Yh3'!c?gcV ? mYcqqrSf?DX5sXX?RrP5SscdYX?PSW'? T V a i i d u l | ? ? % l % i t u ? e o E % ? T V n e o n e E % i d V t aH t X?eS?yy?gs1i5rjbA ?vr#hX¤rA ?r% rA srSc?9P5?? n % ` Q ? i Q Vg i d I T d i Q Q Q T i f ? i dx gg V i Q V d i T i U ` Q i i d u l| ? ? % l 3?S??S?%Si5SwSYcV ? bhb?SDPXgS??whws1S3Swp??gs1i% i t Q T ` ? iH ? V i d V 5S?s?PPg 5? !Sc)? 5SfSXsPWz%e5r5Swdv?s!SsP??PH p t QH d ` Ug o TH I ? ` t Q i ~ ? o ? i d f UH ? T I V i i d u l | ? ? % l i t ?gH f ? T V a i ? n TH V t a dH i t Q t QH ? XeS?y??gs(i% hSQ %Phh#Xjr'wPXD?hf5S#Sf?y% SfH piQ d `x I iH d Q T i gg V U TH T T V a I ? 3 ? I iH a T i f ? i dx dH i t Q Q T i p i d a TH ?g T X?iPSYb'PX?5%5hc?YqiCP5hb?S?f5SibqS9hPqfhX` ggH ? i ? I V I i ? ` T ? i T i Q V i d a ` Q ? i i ` T i ? ggH ? i d i t Q T i t %P?m??cew5rh?w5?Sc?SwdS??5T hq'i%P?mSw5S¨5SQ n d i ? d ` i p V I i t Q TH ? i Q V i d a T i i ? ? ? V i dg V i ? V t I p i QH i I i t Q d ` ?w5?c?qXSshSmPq?S??Swmw'qX?SfX???DvqSf?YhSiX?x I iH d Q T i i a TH ? ? PSYbg5PD3% SfdSf?? sPSwcShX?Ys9hSX5X1S?DYqYPX?o p i QH t QH T `H Q a V I T V d Q d i t Q ` T V i d V t I ` Ig V ? T V n E I p i QH Q V t Q I T V i p IH t u ? n ? ?  x ` ? d Q T i T V ? T k i DX???qSf?chShhXqePhr)9 ? §RFX#YYbX?DDq??  ? bBR X%`S??bP`Gc?Dcb9 ? bBR8PI¨% SfQDX1?Y?hcDsP`SwcShc?YQ n 7? T HQVa g QV ? TV n n 7? H pi H x` ?VddV gV T HQaVI TVd i t Q x ` ? d Q T ? T ` a i I i t u ? ¨ ` Q Q i I IH T `H Q H aH Q d dH i t Q ? T 5S?X!YYrwi DswY1hrWSWwY3%qX%S?V ? PSYcV ? f5SdDcV ? ` Q Q i I i d V I p i QH i I i t Q gg V x ` ? a T i f ? i dx i t u ? ?SqwY)Sc?qwSPdYhS?PX1czwh5b?S?d5ri?o Sf8X?x p i QH d ` ? d Q T i T V U TH ?g ` t ? i Q V i d a IH E p ` dx i ? ` T ?gH t a d i t Q ` T V ? T YYb#X!hP5%Xhq?Sc?S9gP??sSe5r5#%Ph?g9hSX5X#DcV n?E wSPWc??YYbi XmhP5%s5d?S??SwW%cg5r5d%%59WRS5k p i QH d `x ? d Q T T V U TH ?g ` t ? i Q V i d a IH % x ` i ? ` T ?gH t a Q I d i t Q n IH t Q p ` d ? ? T `H Q a V I T V d Q i p V I i t p i t Q t QH ? I i d V t I n ? 5S'PhS?XS51wsPSwXS5X?Y1qXS?hSQ hSqSf??S?DYw§? T `H Q V a `g Q V o p i QH d i t Q ` i t Q Q V t Q ? T k i ?  ?n 7 ?  T `H Q V a `g Q X%S??bPscy?wSPrwhSc?hS?cDSmhD3??BPrR#sPS??bPsro ??eP??%5?!B}sPS??bPj??Sf85S¨Sf?sPSwXS5X?YmqcSI E IH t aH t ? 7 ? T `H Q V a `g Q V p i QH i t Q t QH ? T `H Q a V I T V d Q i p V i5SiS?DYif??DS?SX55?% Sf?X?e?YcvXhsPSwcShX?Ys5SQ t Q I i d V t I QH Q V t Q I i Q ` T i ? p i QH d `x ? V d d V g V T `H Q a V I T V d Q i t TH Q T i p ig i Q I d k i t u ? FH d Q %WbqP3RShh5r?vPSY?V ~ ?SY9?rp1hSGc?Y?V ? w5h%15SQ ?iQdi ? T| i tQ x` Qd F i ? TH i t TH " ? T `H Q V a `g Q V %e?sPS??bP?c% Sfg'wP%??Sf?rwhb?S?sRS535S!5SQ p i QH i ? ggH ? p i QH Q T i f ? i dx Q I d k i t Q T i t ? T V t Q d i Q V i d U IH ? x| n ig ? V u T | i p V F i I f `H ? i d f ` T #XhSe9Sc?Sch%?s?? %5X?¨pmPg ? X5whhXPwSd ? 5X¨y| ? IH ? i Q V ?H ? T V o p ` dx I T d i Q Q ?QRP?1S?h%5hXr% ?XS?r5SwSYcV ? QbTihfb??iSd?lTs`5Tgi???`qpSdq?d%shYwS5SsYX` 5? YI Xd?Ycr x i ?g ` t I i d t Q Q d ? f T ` ? i I V 3 ? ? `Q` ? 8Sc?0 7 r? i ? ` Q F i Q a ig i I ? o i ? ` T ? 5r5T 9hT ww%wY???5r5j" r?

5.

A

Time in seconds

E

EXPERIMENTAL EVALUATIONS AND PERFORMANCE STUDY
A:5:0 E:5:0 C:5:0 A A:6:0 E:6:0 (A) (B)

1000

1500

2000

2500

3000

500

0

Apriori

1M

5M

FP-Tree

Size in Millions

Inverted Matrix

Time in seconds

  

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Apriori

FP-Tree

Inverted Matrix

6.1 Compressing the size of Inverted Matrix

? i a T V p d `xd s5X?SX?Ywi ? i t Q d i t Q d fx i ? ` pH I f t Q ? T V I ? ? | x ` d i ? p f i t i a f ? i hS?w5SY5?8??Sd ? P?hrS?hX8?sR1Xq9'jhrT 5SQ h?Sd ?gg V aH Q V p V d ? T V a i ? FH d Q V p ? i I I i p ` a t a f I t QH 5 ? ig ? V fPX?PS??X?!X?refSYc???YYSd ? X?95Y?SPCr? Phc?u TH ? i Q a i ? IH o ? i d f UH ? ` Q U TH ? T I i d d ` a FH d Q ~ ? i Q d i ? T P??S9%H ? 51P??dSsP?)S!5P5DX` ? SYXw¨fSYcV #?SY9?rp| ? i I I i p ` a i t u ? ? i I I i p ` a i ? T V a FH d Q ~ ? i Q d i ? T | i t ?YYSd ? sdhrw?YYSd ? sd'xc?fSY?V xSYw?rp!hSQ Q V t Q ? T k T V a i ? ? U `g ` ? ` t Q i p i p V I i t Q U TH I D ? QH U TH ? i a i cDSqDh!c?hr?XXPs5r5Swq?qcSghS?hPY?C?fj5P5?wSd ? p i Q H i t Q x ` I i a T i d d f a a ` i t Q I Q T i I i i d ?? I Q i t a V d ? T i i ? Q i ? d i Sf35S¤X?hwSY5bWhSiSrwYSd ? SiSw??X?5?wrp9'sw'? l p f T ? T V i d i t ? n  ?   ? ?   ? ?  ? ?¨ x ` U T H Q I H I T ` a T ` H Q a mhrqbXjSwh?v9) ?49 ?&9 ?  § ?¤X!hPSRPYhX¨X%S9XV l I T V d Q i T ` I V p i t Q ? iH ? T V a i ? n ?g Q T i f ? i I T ` ? o ? T V n E n R5X?Y?5ssc?5S}wP8XgrbbfSbhb?Y5s?% ?8Dc5?53% IH t aH t ? n F ? f I i p V I i t Q i d V t I I T `H Q a V I T V d Q t Q ` ? o n E n n Pe9Ph?cvm5YiqXSr5S?SchY?hsPSwcShc?YeSXi3 1?sX3% e? n 0 ? T V o n E n n n ? i d V t aH t ? n I T `H Q a V I T V d Q ? i T ` ? Q I i a f ehXrsb3% 33 j?S??9Ph?r5sPSwXS5X?Y?whh?p?55? l ` ? a T i f ? i dx dH i t Q ` Q U TH ? d ` a a V d i ? d ` U TH ? T i a I V ` Q TH g V f I cSd ? wh5b?S?iPwhShS?5P5?XwXrw5?c?5P5DwYX?SbPDXhYhf I V I T `H Q a V I T V d Q t Q ` ? U TH d i ? d ? ? n ? n n E n o ? T V E n ? n n c?5sPSwcShX?Y?SX'3hPSw5?3?e0 es3% ss?jhX#?es3% 33 n1?qSfH PX?bswW5sPSwcShX?YerpsRS5?hSs?DSqDh!Xhr? o I p i Q TH V Q T ` a I T `H Q a V I T V d Q ` ? Q Q I d k i t Q Q V t Q ? T k T V a i n o ? i d f UH TH i p V F i d ` ? ? I T `H Q a V I T V d Ql ? f I i U d i p T i ? i d 1?qS5X%?? %?%g ? X5w3XhrhX%S9XShc?Y?5hYhXSwqdw?3X` n5s?eq55wh1SbPrhX%S9XShc?Yrc%PqPY3XSwqq%hsw35s` I i T ` ? p p f ? ? i T ` Q TH I T `H Q a V I T V d Q d VgH pH I i U d i p ?g f ` a i T nPhS??Ph?X!¤e? hSfScsPX?fSYcV q?SY9?rpj5Sicqwhw%wDk IH t Q i ? iH t a V ` u p t QH d ` Ug V FH d Q ~ ? i Q d i ? T | i t Q x ` ? a T iH a lx i i t Q i ? ` pH ?g f ` a Q V t Q i f I IH Q T V Q d wqhSm??Sd ? %?%5s?chS85YYPebc?YX` ? PXePq??h¨bXV pH T V IH V Q V ? ? T U TH I `g Q f ` t QH ? FH d Q ~ ? i Q d i ? T | i t Q x ` i ?H I i t Q U TH I I i p ` hPYsPW5XhSf?dfSYcV !?SYwsyhhSGc?w%Yh5SmhPYYSd ? s?%

10000

12000

2000

Time in seconds

4000

6000

8000

10000

15000

20000

25000

0

0.01

0.0075 0.005 Support (%)

0.0025

6.2 Reducing the number of I/Os needed

?g i ? ig i I V ? V Q V ? i t Q Q V I i U i p V I i t Q T ` I T `H Q a ?sPhYXDc?c5hhS?c3XXV ? qXShhS8sWhX%S9XV l I T V d Q i p V I i t Q U TH d i Q I fg a ? ? i I V ? V Q V ? i t Q p ` dx ? V i d I i U R5X?YgqcSghSj5PSwSRhP8bhYXhX?c5hhS?XS?8X?S?scV ? xX¤w'j5vhSWhP55?S5X?pfPPhPYYs` ? hS35PSc?PSRsP1fSbSd ` d i ? p f T i t Q U TH a f ? i d x ` ? QHgH ?H I I i t Q U TH Q V UH Q I i ? TH ?g Q T i lYhScg?CsYXhX?c5hhS?XS?gscV ? 5sdcDSgScqqhPP%c?a d f a i d V i 5 ? i I V ? V Q V ? i t Q p ` dx i U i T ` T V t Q i d ` p U THgg V ? ? i T ` ? i ? T V a T `H Q a V I T V d Q i T ` U TH I d i ? V d u ? FH d Q ~ ? i Q d i ? T r?5s53'qX8XPSwXS5X?Y?hsh5PYSw????vvfSYcV q?SY9?rp| i t Q x ` p ` Q Q ` ? i t Q Q V ? i d i Q I fg a i d V I p i QH Q T i f ? i d ? ? ? a T i f ? i d hSDc?sSYc'?5SGcjSwSRhPrS?GqSf?b5b?S5Gb9h5rS?x dH i t Q T ` ? i I V ? I T `H Q a V I T V d Q i t Q f ` d U FH d Q ~ ? i Q d i ? T | i t PwhS?s??YXhshsPSwcShc?YW5SrI ? XSX?fSYcV ?SYw?rp3hru

5000

0

Apriori

Algorithm

FP-Tree

Inverted Matrix

6.3 Updateable Inverted Matrix

? i Q d i ? T | i t Q x ` I i U V Q T V ? ? V TH V p i t Q x ` i T ? ? t a d V i I i d d f ?SY9?rpqhSscewsX?bcssPX?q5S?Xqh?)??Sc?YwSes` TH i f I IH Q T V Q d PqhYYPhbX?Yc` ? P?ceP#fSYcV #?SYwsymPhc?Sc5? ? D wsPSQ pH T V IH FH d Q ~ ? i Q d i ? T | ig ? V i Q V ? T `H l V d ` F i p ` a V I T `H Q a V I T V d Q ? i T x ` T `H QH ? ? V i t Q U TH t V ??wi ? qwPg ? s8e5sPSwcShX?Y?9hrc¨X%SfhsmhS?hPrX?p nwsPSchPhjXes5hvc?PSYw?schwX?bsPSchsw5SfhsP8fSYcV ~ T `H Q V TH ? p ` a V Q f ? g V aH Q d i ? d ` T g V Q T ` ?H d ` t d i t QH i T IH FH d Q ? i Q d i ? T | i t u ? T `H Q a V I T V d Q i t Q TH I p i QH gg V x ` I iH d Q T i i I V ? V Q V ?SY9?rpvhr?wsPSwXS5X?Y?hShP¤qSfbPXhX¤PSYrw?YXhX??h? i t Q TH I i Q V hSdPsS?h? ? f PWSf5YSd sPSwcShX?Y??55s)?X5?sb%¤c?a TH I Qg f I i T `H Q a V I T V d Q ? i ? ? V t a V i n Q f ` ? Vg g V lH Q d i ? V t QH 5 ? i I V ? V Q V ? i t Q ` Q I T `H Q a V I T V d Q i I ` t Q U TH ? T fSYwsd?Sf}sYcDX??hq5SdS1hsPSwcShc?YmYshS8hP5hi ? l ? !fg ? PYW%WhXPSwXS5X?Yj5%hXD?sb%¤c?rXPSX5j)Sf}5 V ? pH I IH I T `H Q a V I T V d Q U TH ? ? V n Q f ` ? Vg g V Q T ` ?H d ` t V t QH

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  S  R('V? ?D??¤? w(?r?¨ ¤d¤Dy? ? 6  4 4 ? ?   6  §  6 ?  ? R?8??X¨cw?  §  ?    § ?  2   2 6  § ?  ?       ?§ ?   ? 2 ? ? ?  ¤R¨??¤r¤ ????5?qG?2 ? ??¤D¤§ ? § ? 6 ? ?

6.

?S?y??gs% ?PPX??S5Sw5SYR'PX?Y8f?PSc%S?hPq?f?PSwcV I i i d u l | ? ? ? igg V a I i d f Q a f d Q I gg V p I ?g i ?H Q Vg i d i TH p ?g i ?H Q a l d i Q TH ` Q ? d ` p i p i t Q I i I f ` Ig V Q | ? FH d Q ~ ? i Q d i ? T | ? igg V YwSbPqS?Ycqq!5SeY5?YPcqy¨vPSY?V z?SYw?ry??PPXa Q f ` ? Vg g VH a I V TH I T `H Q a V I T V d Q i t Q i d ` Q I ` Q t IH i t Q I i I sb%?X%i ? qPq5sPSwXS5X?Y?hS8SXSR#S¨YP5? 5SjY5f Q V t Q I p t QH d ` Ug V x ` Q i I ? i T V i I ` i ? ng V ` U IH t Q t a V i d ` ?DSsqhSfSXXPX?c?wYywh?1Ys` ? Sd ? r5?c?XWP5S!?X?Sj?u ? t a d V i I i d d f ` x ` g V ` U TH V p i t Q IH I i I V ? V Q V ? i U d V ?g i p i d Q F ??Sc?YwS?5X?cGXss)PX?1hSWPgYXhX??hmXSc%g fqwSY9i TH U TH TH p ig f d T `H Q VH a ` I I V d `x I p t QH d ` Ug V ig ? Vg V a I U TH ? TH %#hP5%q!PhSX%S?%bYYX8c?mqhSfScsPXqPhc%X?Y#hP5hP??

DISCUSSION AND FUTURE WORK

? 2? di ? piai r2 2 §gw'eg? nbA r& P9 ??7 Xr?5?qf?X??c?j ?5X5YY¨sw%?sRXWQX? ? ? ?e ? n ? ?? ??  ?? ? ? ? ? ? ? ?? ? ? X ? ?  ? ??? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ?  ?  ? ? ? ? ? ? ? ? ?  A A A ? ? ? i ? d f I o e U TH TH hS??'s?wY5sCD9S?rw'Xv?§§B'b9?S5Yyv5hP5%qp T `H Q VH a ` I I V ? i Q f ?H d Q IH ? T V g igg V d V ? ?H t V 7 ? W ? ~ ? " ? sPSc%bYYcd?ShPSYRP5? hX?w%Pc?cGh?rcw85¨?e?f?



? r¨ ¨ ¨ dYchS5h? ? ?dV fd ?i n o n ` U iH ? T V ? n ? & ? ? I i U n y x x x v A ? 1r% sPg!X5?r¨ 7 b?r" WscV ? ??bw au¤ ? t ? ?? ? ?? ?? ? ? A ?? ? ? ? ? ¤ ? ? ?  ¤? ? ? T| ? Q T i p i T k i r??9Sv?Xbs4?X1)c?'?Xv$#h?)pr?bq5DwSd T `H Q fg ` I i d i ?H I I i d U ` sPS5PXYSh?PYYSXXSd ? Sf?h?qPSfhjd%sqwSPH t QH ? VH ? i pH Qg f p TH I p i Q Q T i d d f a i d U TH TH ~ ? f t 7 ? ? ? T V n T V ? ? n i T V9 V 7 ? ? ? ? ? ? ? bSYSj5PhPdh?rvse8DcDwXg55W XhX%@)wr1heecf?

? ¨ ¨ ¨ 15hsW ? i T f n ? & ? e  ?  ? n ? ? ??? ? ? ?? ? e A ? ? R ?    ? T ` IH d 5c" ??A 9BR?'ShX??cYXP'9§d?g§ c? wsYPScV ? sa p ` ? T V ? i ? d f I g V d i T i U V l U TH TH p ig f d T `H Q VH a ` I I V d ` DX!wsS5YwX?w5sj?e5PhPqgPhSqsPSc%bYYc3X?x I p t QH d ` Ug o ? t i ? V ?H i V t V ? ? T V n d i ? Q T f ? D ? H ? ? ? ¨ ? qhSfSXX%WDw5X?PXXX?9 e0 DcD?wwSr5?0 ??n h? 3h5W ?f? ? ¨ ¨ ¨ 'X%PXh? ? n I Vgg V n ? !  ?    T | ? T `H Q V d i T i U i Q V ?H ? T V a Q f ` t QH hv"C  ? pDwsPSc?9hXhSch%DX3s5SP?? I T d i Q Q Q T i f ? i dx U TH TH ~ ? TH ` ? ` ? T V nH i ? ? W n T V ? ? ? 5SwSYcV ? bhb?SmhPhPdh?PgbjaDc'??h5¨wX?h5W ?2 ?



? ¨? i T f l ? r¨ ¨ ?1h5sW r?V ~ n ? & e  ? ? n ? ?? ? ? ? ?? ? ? ? ?? ? ? ? ? ? ? ? ? ?? T ? ? R ? rXA  %7   9  ??¤Xb?99S'?Xr9d?X?8'X?b?Y%?'1SX? ? ? ??? ? ? ? ? ? ? ? ? ig f d T `H Q VH a ` I I V d `x U TH TH p V Q V ? g igg V d hc??w?9hXRhisPhS)sPSc%bYYcgc??hPhPqj??h?w%Pc?cV ? ig ? Vg V a ? ? d V p f ? P ? T V n I ? d V X ? n T V ? ? ? ? E ? Phc%X?r5?c?hhX ?QDcDPH ? Y?1e0 ?cg5e?l ???7 ?

? ? sr¨ ¨ ? n o n ` a IH a T V d T V ? n T V px f V X T V U d ` ~ ? ? ? ? ?? ? X ? ? 1r% 5wY%whX?? X'wX??5X1c?ScdvS9hw?hrYYv? ? ? ? ?? ?? ? ? ?  U ? ?? ??  ?? ? ? ? d i ? p V X ? ~ ? T V T V ? ? ? 'cj?hy9DXvWVb?h?q6?c?D?w'jc1?dDc)X?h5W ?? ?

? ? QH d U i Q T byPScSbPH I Q U TH ?H ? d ` i ? d ` ?g i Q i p ` a QH ?gH f ? i d ` Q U TH ? V t Q f ` t QH SfH hPP?cV ? Rsc#PwSwPg ? swhf?%PhhwS!S!5P?h15s5Sf?? FH d Q ~ ? i Q d i ? T | i t Q i Q V f ` Q I ? V ? Q T iH a ? i U TH Q V UH Q I i ? T fSYcV z?SYwsy!5SqSch? ? ?Se5rjbPqw?hPScsPSR?rPH i d V i 5 ? TH V U V QH U TH ?gH f ? l i d x ` ? i i T i t Q ` Q U TH ? V ig n FH d Q S?e¨!wPX?chfqhPP5h?wS?X)?51hSmS?hP5X?PvwfSY?V ~ ? i Q d i ? T | i t Q I i U T V t a I T `H Q a V I T V d Q ? i T U TH Q ig i ? d ` U TH ? ? V ? ?SYw?ryshS?s5XD?vhsPSwcShc?Yxw5hhPSwPrcghP55sjb? d i t QH i i I V ? V Q V ? i t Q U TH U T V t % ? TH V U V i I V ? V Q V ? i t Q U TH T T V a w5Sf1YXDc?c5hhSj5PshcD?8?PXsX1YcDX??hh5SjhP5hXYI l i d T V i p Q ` T I i ` ? g i ? ig Q d ? f I i t Q U TH U T V t a Q V t Q IH FH d Q wSmc?qrchrb'?w%?YX` 5? YWhS1hPs5XD?icDSi%?fSY?V ~



 

? 2 ? ? ~ n V T ` ?H d o n T ` I a f r7 2 §#V 'DXPSWh?XYh¤u n ? " ? & ? Ii U n ?? ? ? ?? ? ?? ? ? ? ? ?  ¤? ? ?  ¤? ? X?%A A 3XXV ? X?X1?p?h9q b?'X#I'?Xv$#h?? ?iHC  ? F?§??Yr?yhr?ch3wsYXDrw?Sc?p !  ?    G ¨ ¨ ? ¤ ??? ? T| ? VQ V ? Qi tI V ? Qi td V dc??IPhSddX%SQ??PPg ? P!hXq5%Sb5s?wYqwSP3PqXD?? `x ig f T `H V aH pH ? T V U TH Q T f ` a Q i I p i QH aH p V T ? ? fI ?d5Yru ?'? DTcDwX?PP??jh5?hcsyXdh?5?PSi3 ve?" ? ? V n T V pgg D ? ? ? W nH T V ? Q ` ~ ? ? n TH d ? ? ? ?g p ? H V ?H d `H V ? Qg i U d ` ? E ? i ? U d f ? i ? U V ?qSQbt ?HSdc`PSd ? cSXPSd ? X9fsSc'?F?5? 5Sh?X?p lH T ? I ? ? ? ? fx ? Q Q fh5f wa b9h??s e? ? Yrt ? T `H Q V Q T i p i pH H d `H o ? Qg i U d ` ? % ? ?X%S??bqPg ? PwSXPSd ? v?fsSc?3 3??A ? ? ? ¨? di ? piai ? n ? &2 rs¨ ?Ww'jwg'r" r?? I i U n ? ?? ??  ?? ? ? ? ? ? ? ? ?? ? ? ? ?  ? ? ? ??? ? ? ? ?? ? XXV ? Xb?h?qD?X1dXj?'9Y9??cv$s'c??X'?9yh?? A A A? | ? T HQ Ha `IIV pdiQ ? ? T HQV Hd`UiQV §CB)pTv?XP`Sc%VbYYX?SwS?rqX%`S??P?SXXSc?a Q T i p f a ` ? Q F i u ? i T V9 V 7 ? ? ? ? ? T V iH T ` Q T o ? ? ? ~ ? bq5b5Wrw?vshc%@ )v81e)DchPhXSrWD?yl ?ec ??

?g p Q ? V Q V ? T ? I ? Q I i f ? ? I a ? p ` p ? ? T i ? V pg ? ? ? ? ? Q Q ?qSbt ?c5DRc9RwhbX??8s ?a 5%H w5X?%cV hss?s e? ? Yrt ? i ? ` a T `H Q V d i T i U V Q V ? aH Q i t Q T ? I Q I i X5r)X%S??whwse??hhPSw5Sb5RWR5f Y ?s?P3h91? ? ? T i ? V pg o ? | ?

 2   ¨ ?? n ¨  "  BP?§R  n??  A  BP?rR  n7? ?  n?? BR P?R    BP?§R  n"?  ? P?rBR R BP?§R ?  n 7? ?  n"?  ?  ? 5R R BP?5wR ? n ?? ?  n?? B?R   nrR BR P?R R BP?rR ? ?  n"? ?  nA? ? nA? ? n ¨ BR  ? rBR R  ¨ ?? ? n?? ?  n?? BR  ? ssR R BP?sXR ?  n?? ?  n¨? BR P?R R BP?§R ?  n R BP?2  R  ? r?R ? n??  ? BP?rR ?  n A? ? n R  ? 2  BR  ? r?  ? n ? n? R  ? §R ? n BR  ? r"  ?  n BR BP??? ?  n BR BP?bA  ? n¨? R  ? §R  ? ? ? V d d o g V T `H Q a V I T V d ?Y?GXhsPSwXS5X??u  ?6  7   6 4 ?  5wcr'S  x5&(???'?3

??c5?? §6? 2

6.4 Parallelizing the Inverted Matrix

? ? rs¨ ¨ ? dwi'jwgvr"  " ???6scV ? Xr?h?qf?X1dXj?'9Y9??Xv ? piai ? n & 2 ? " I i U n ? ?? ??  ?? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ??? ? ? ? ? ? ? ? A A A ? T | ? I i I V ? V Q V ? x T V a I i T ` t QH s'c??X'?9yh?C§BqyDYcDX??h?c` XYhhsdSP?? IH I ?g V T V T `H Q VH a ` I I o ? i fg i ? ? ? ? T V n f ? n U T V f ? ? ? ? ? ? PYfXDc#sPSc%bYYWDs5P31!hX5?¨5 ? hXh?e19sf?

? ? ¨? QI fU f b?¨ ?3RhXh3o nr?Yw'P3?XSbsq5vw?  r?2 XgwsXV ? Xr?5?qf?X?j'X? V Q d i ?g o n T ` Q T ` p ? E n ? & ? ? I i U n ? ?? ??  ?? ? ? ? ?  9XS9?ggb?Y%?D1SXG%?Xvg'X??X'?9yh??3gC  ? ? h ? ? ?? ? ? ? ? ?? T ? ? R ? ? ¤ ? ? ?  ? ? ??? ? ? ? ?? ? ? ? ? R ?   ? X ? A | ? T HQai?` HQ H T fQ ? ` ? ? IQiI pi ? r?%?r?pTh?X%`S9R?Sd ? PaSRPIh5SX` h? 8r?SwY?wSPQH Q T i f ? i dx U TH TH ~ ? T V ? ? W ? T V n U T V 5 ? X n T V ? ? ` n fH ? ? ? ? ? bhb?SjhP5%dD?cg550hX555X?m?XGhjb?PwD5W ??f?

? 2? ? ~ n o niI ` TV ? n ?? & 7? IiU rA 2 §#V D?r% XYssW X5?r" §D%A BhXXV ? n ?? ? ? ?? ? ?? ? ? ? ? ?  ¤ ? ? ?  ¤? ?? ? !  ?    X?X13R?h9q b?'c?q%?Xv$#5??i"C rp ? i¨ ¨? ¤??? ? | ?I g fd T HQ Ha `IIV U H H p d x p t Hd` g ??§??Yr?pT5PihSqsP`S?VwbYYXm5PThPTqWX?`?5SPQScsPUXV ? i I V ?l t I V t i ?H Q a i u i T o ? f ` ? ? ? T V n T i t ? ~ n t d V ? ? ?  ? ?YXh?5YXhh?PSwvw!3w?b?dDcDwh?% ?''S?GD5W ?f?

 



?g V pH TH p IH I i ? ` T T i i ? Q i ? T `H Q V aH T f p p ` a i d i t ?X?PhPqWPWw5rh!?pw'!sPSc?P5hjqXhSw5?? n?X%YS9?5c%Srwhb?YshS'c?YXV ? hswYrhSgSrX?PSbiRsqPcV T `H I d i ? g VH Q T i f ? i I i t Q x ` I I ? T ` a i I i t Q ` Q g V aH Q T i ?H Q I ` pg IH i ? ` T g V a `g t a V i d `x I I P?rhwX?bPd?X?X?rYcV ? DswY?5rvSYX` h? Y?c?bPgY5SQ ? T ` a i I i t u ? I Q d ? f I g V a `g i I i t U TH d i Q Q V a I d ` U TH Q I V a ? V ` d ? ? ? d i t QH i i T ` ? i ? ?g f ` a p i QH t a V 5PSwSYc?YvcghPSRX5ssS5?b?whSf?hss'j%hX?Sfm?Xi x ` ? a T i f ? i dx g V ? `g U i t Q U TH Q V d i T i 0 p i QH t a V i d `x ? a T i f ? i d cqw5hb?SvcDX%X35S?5PSc?w5?C? Sf89c?sX?mw5hb?Sx g V a `g i t Q i Q V d i T i U ` Q T V a I d ` I I Q I d k i t u ? V Q V ? g V a `g i t X?bP?hSeS??whws8S#c?YgcgYXV ? RSh1hr?r?chvc?bPe5SQ di ?` IiII wsc?YYcV ? rp?hPYh?hsh'mYPX!%hs?sPSYcV !?SYw?ry| ` ? Q U TH I f i T ` ? i ? ` Ig V ?g f ` ? I i aH d Q ~ ? i Q d i ? T ? i Q f ?H d Q IH ? i I i t Q U TH ?gH f ? I T `H Q a V I T V d Q gg V TH I p i QH i t Q x ?S55%SYRPmY5S!hPP5?3 5sPSwcShX?YGPX?PgqSfj5S?c` ? a T i f ? i dx g V ? `g U i t Q Q a i d ?g f ` ? I i aH d Q ~ ? i Q d i ? T | gg V p w5hb?S?XhsPsq5S1wi ? S¨5s??w%SY?V ??SY9?rp?Pc?YI i t Q x ` Q d F i ? TH i t u ? d i Q I fg a i t Q TH i ? ` T I QH T ` I T `H Q a V I T V d 5ShcGYcV ? w5h%?hr??wSR5P?5S?PrrhGSfeX?hsPSwcShc?YQ gg V I Q a i d Q V t Q FH d Q ~ ? i Q d i ? T | T ? ` I QH ?gH f ? ?g f ` a d ` I I i PX1Swi ? Sg?DSqfSYcV #?SYw?ry¨??gSf#%Ph5?%hX?XYYwa l` XSd ? ?Xv#?w%Pc?cV ? %?fPh5q'?5s?fSYcV ¨?SYw?rp?5ru t a V E ?g igg V d TH QgH f ? i ? ?g f ` a FH d Q ~ ? i Q d i ? T | i t

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7.

? V ?V TV n % ? E ? rhXDXr% 36?5g9 XS??bX?30 ?ScYI p ` dx Q T V d t a d V i lwW¨xbx?SYc` h? YfPX%SYcV ? P}?Sc?YwS?P5rySrwq%S9i ? i ? V ? ? ? i Q d ? f I ?gg VH Q d IH t a d V i I i d IH t u ? I Q T i pH d l F i d f TH ? i I p V d U ` rwW5X` P!?Y5f X?XXSd ? Ss?S31GGhhSGch5rhPhc?5w59i t Q ? ` d 0 l ? ? i t Q x ` i ? ` a ig ? V Q f a i F i t Q t QH ? I f U TH ?H ? ` d `x H i ? T VH t T V t Q i t Q i tH ?g f ` ? i 5SdSf?shm5%%?Sd ? c??w?qX%sW hcDSghSgs%Pg %hXrg¨5

8.

REFERENCES

? 2 ? n ? ?? ?? r7 2 vXr?h?q ?? ? ? ? ? ? ? ? h ? ? ?? ? ? ? ? ?? T ? ? R ? ? ¤ ? ? ?  ¤?? ? ? ? ?  T ?X?j'cd 9XS9?hgb?Y%?'1Qc?'?Xv$??h?jYX?p| ? I ig f d T `H Q VH a ` I I V x ` ? d i ? ` a IH ? Q I Vx d `x I p t QH d ` Ug V ? i PhS)X%S?%bYYX?cdY9??YP53RX?3c?sqhSfSXXPX?w?9 ?H ? ? 5 ? T V n V d V tH U ? ? ~ n ? t Q V d V I V t Q d V ? ? ? nH t V 7 ? ~ ? A ? ?w5?Dc'r?cDPX?'?DbbSc?cScDSY?GD''?rXv8?e?Bf?

? ? 2? di ? pi i X2 gw'eSQ ? ? n igH t n ` U VH Q T V ? n ? & ? I i U n ? ? ? ? 5 ?? ? X%P5?% 5XX%SbX?2 2 b'7 ? ?XXV ? S9c36?c?? ? ?? ? 3 ? ? 1 ¤ ? ? ?  ¤ ? ? ( ¨ ¨ ? ¤ ? ? ? ? T | ? I ig f d T `H Q VH a ` I I b?X?4 9h2%?cv0#5?? )?§??Yr?phPhS)sPS?wbYYXV UhPHhPqWX?rqhSfSXX%c3RXD?bXXPSrh1!Dc?cs?XWh1h? ?? T TH p d `x I p t QH d ` Ug V Q I V ? ? Q T V tH d ? ? ? ? T V g V ? V d U o ? ? ? ? 2 ? ? ~ n? ?  2 #V vP3% j? n T ` Q U TH t I V n " ? ? & ¨ ? I i U n ?? ? ? ? ? ? ? ? ? ? ? ? ? wsSX5PhYX?5 §r'7 ??wsXV ? X?c?3R?59? b?DX? ¤%??Xv$#¤h??i"§  ??§??Yr?pThYXhX??h? ?  ? ?? ? !  ?   ? ¨ ¨ ? ¤??? ? | ?IiIV ?VQV i U d V TH I p i QH x ` I Q i I T i i ? Q i ? I ig f d T `H Q VH a ` I I XSc%g PsqSf?X?SwYdryw'?PhS)sPS?wbYYXV U TH TH ~ ?H p V ? ? ? o ? T V nH t I THg iH p | ? u ng V ? V d U o ? ? ? ? hPhPdh?qXsb'1dhXh?rY5%PPqyh?D?cs?XWD1h9P?



 

ACKNOWLEDGMENTS


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