Item-Based Collaborative Filtering Recommendation Algorithms

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Item-Based Collaborative Filtering Recommendation Algorithms Item-Based Collaborative Filtering Recommendation Algorithms Badrul Sarwar, George Karypis, Joseph Konstan, and John Riedl f×aÖÛaÖ¸ kaÖÝÔi׸ kÓÒ×ØaÒ¸ ÖiedÐg@c׺ÙÑÒºedÙ GroupLens Research Group/Army HPC Research Center Department of Computer Science and Engineering University of Minnesota, Minneapolis, MN 55455 ÚaiÐabÐe iÒfÓÖÑaØiÓÒ ØÓ ¬Òd ØhaØ Ûhichi× ABSTRACT ØhÖÓÙgh aÐÐ Øhe a ÑÓ×Ø ÚaÐÙabÐe ØÓ Ù׺ ÊecÓÑÑeÒdeÖ ×Ý×ØeÑ× aÔÔÐÝ kÒÓÛÐedge di×cÓÚeÖÝ ØechÒiÕÙe× ÇÒe Óf Øhe ÑÓ×Ø ÔÖÓÑi×iÒg ×ÙchØechÒÓÐÓgie× i× cÓÐ ÐabÓÖa¹ ØÓ Øhe ÔÖÓbÐeÑ Óf ÑakiÒg Ô eÖ×ÓÒaÐiÞed ÖecÓÑÑeÒdaØiÓÒ× fÓÖ ØiÚe ¬ÐØeÖiÒg [½9¸ ¾7¸ ½4¸ ½6]º CÓÐÐab ÓÖaØiÚe ¬ÐØeÖiÒg ÛÓÖk× bÝ iÒfÓÖÑaØiÓÒ¸ ÔÖÓ dÙcØ× ÓÖ ×eÖÚice× dÙÖiÒg a ÐiÚeiÒØeÖacØiÓÒº bÙiÐdiÒg a daØaba×e Óf ÔÖefeÖeÒce× fÓÖ iØeÑ× bÝ Ù×eÖ׺ AÒeÛ Ìhe×e ×Ý×ØeÑ׸ e×Ô eciaÐÐÝ Øhe k¹ÒeaÖe×Ø ÒeighbÓÖ cÓÐÐabÓÖa¹ Ù×eÖ¸ ÆeÓ¸ i× ÑaØched agaiÒ×Ø Øhe daØaba×e ØÓ di×cÓÚeÖ Òeigh¹ ØiÚe ¬ÐØeÖiÒg ba×ed ÓÒe׸ aÖe achieÚiÒg Ûide×ÔÖead ×Ùcce×× ÓÒ bÓÖ׸ Ûhich aÖe ÓØheÖ Ù×eÖ× ÛhÓ haÚe hi×ØÓÖicaÐÐÝ had ×iÑiÐaÖ Øhe Ïebº Ìhe ØÖeÑeÒdÓÙ× gÖÓÛØh iÒ Øhe aÑÓÙÒØÓfaÚaiй Øa×Øe ØÓ ÆeÓº ÁØeÑ× ØhaØ Øhe Òeighb ÓÖ× Ðike aÖe ØheÒ ÖecÓѹ abÐe iÒfÓÖÑaØiÓÒ aÒd Øhe ÒÙÑb eÖ Óf Úi×iØÓÖ× ØÓ Ïeb ×iØe× iÒ ÑeÒded ØÓ ÆeÓ¸ a× he ÛiÐÐ ÔÖÓbabÐÝ aÐ×Ó Ðike ØheѺ CÓÐÐab¹ ÖeceÒØÝeaÖ× Ô Ó×e× ×ÓÑe keÝ chaÐÐeÒge× fÓÖ ÖecÓÑÑeÒdeÖ ×Ý×¹ ÓÖaØiÚe ¬ÐØeÖiÒg ha× beeÒ ÚeÖÝ ×Ùcce××fÙÐ iÒ b ÓØh Öe×eaÖch ØeÑ׺ Ìhe×e aÖe: ÔÖÓ dÙciÒg high ÕÙaÐiØÝ ÖecÓÑÑeÒdaØiÓÒ׸ aÒd ÔÖacØice¸ aÒd iÒ b ÓØh iÒfÓÖÑaØiÓÒ ¬ÐØeÖiÒg aÔÔÐicaØiÓÒ× Ô eÖfÓÖÑiÒg ÑaÒÝ ÖecÓÑÑeÒdaØiÓÒ× Ô eÖ ×ecÓÒd fÓÖ ÑiÐÐiÓÒ× aÒd E¹cÓÑÑeÖce aÔÔÐicaØiÓÒ׺ ÀÓÛeÚeÖ¸ ØheÖe ÖeÑaiÒ iѹ Óf Ù×eÖ× aÒd iØeÑ× aÒd achieÚiÒg high cÓÚeÖage iÒ Øhe face Óf Ô ÓÖØaÒØ Öe×eaÖch ÕÙe×ØiÓÒ× iÒ ÓÚeÖcÓÑiÒg ØÛÓ fÙÒdaÑeÒØaÐ daØa ×ÔaÖ×iØݺ ÁÒ ØÖadiØiÓÒaÐ cÓÐÐab ÓÖaØiÚe ¬ÐØeÖiÒg ×Ý×ØeÑ× chaÐÐeÒge× fÓÖ cÓÐÐab ÓÖaØiÚe ¬ÐØeÖiÒg ÖecÓÑÑeÒdeÖ ×Ý×ØeÑ׺ Øhe aÑÓÙÒØÓf ÛÓÖk iÒcÖea×e× ÛiØh Øhe ÒÙÑbeÖ Óf ÔaÖØici¹ Ìhe ¬Ö×Ø chaÐÐeÒge i× ØÓ iÑÔÖÓÚe Øhe ×caÐabiÐiØÝ Óf Øhe cÓй ÔaÒØ× iÒ Øhe ×Ý×ØeѺ ÆeÛ ÖecÓÑÑeÒdeÖ ×Ý×ØeÑ ØechÒÓÐÓgie× Ðab ÓÖaØiÚe ¬ÐØeÖiÒg aÐgÓÖiØhÑ׺ Ìhe×e aÐgÓÖiØhÑ× aÖe abÐe ØÓ aÖe Òeeded ØhaØ caÒ ÕÙickÐÝ ÔÖÓ dÙce high ÕÙaÐiØÝ ÖecÓѹ ×eaÖch ØeÒ× Óf ØhÓÙ×aÒd× Óf Ô ÓØeÒØiaÐ Òeighb ÓÖ× iÒ ÖeaйØiÑe¸ ÑeÒdaØiÓÒ׸ eÚeÒ fÓÖ ÚeÖÝ ÐaÖge¹×caÐe ÔÖÓbÐeÑ׺ ÌÓ addÖe×× bÙØ Øhe deÑaÒd× Óf ÑÓ deÖÒ ×Ý×ØeÑ× aÖe ØÓ ×eaÖchØeÒ× Óf Øhe×e i××Ùe× Ûe haÚe eÜÔÐÓÖed iØeѹba×ed cÓÐÐab ÓÖaØiÚe ¬Ð¹ ÑiÐÐiÓÒ× Óf Ô ÓØeÒØiaÐ Òeighb ÓÖ׺ FÙÖØheÖ¸ eÜi×ØiÒg aÐgÓÖiØhÑ× ØeÖiÒg ØechÒiÕÙe׺ ÁØeѹba×ed ØechÒiÕÙe× ¬Ö×Ø aÒaÐÝÞe Øhe haÚe Ô eÖfÓÖÑaÒce ÔÖÓbÐeÑ× ÛiØh iÒdiÚidÙaÐ Ù×eÖ× fÓÖ ÛhÓÑ Ù×eÖ¹iØeÑ ÑaØÖiÜ ØÓ ideÒØifÝ ÖeÐaØiÓÒ×hiÔ× b eØÛeeÒ di«eÖeÒØ Øhe ×iØe ha× ÐaÖge aÑÓÙÒØ× Óf iÒfÓÖÑaØiÓÒº FÓÖ iÒ×ØaÒce¸ iØeÑ׸ aÒd ØheÒ Ù×e Øhe×e ÖeÐaØiÓÒ×hiÔ× ØÓ iÒdiÖecØÐÝ cÓÑÔÙØe if a ×iØe i× Ù×iÒg bÖÓÛ×iÒg ÔaØØeÖÒ× a× iÒdicaØiÓÒ× Óf cÓÒ¹ ÖecÓÑÑeÒdaØiÓÒ× fÓÖ Ù×eÖ׺ ØeÒØ ÔÖefeÖeÒce¸ iØ ÑaÝhaÚe ØhÓÙ×aÒd× Óf daØa Ô ÓiÒØ× fÓÖ iØ× ÁÒ Øhi× ÔaÔ eÖ Ûe aÒaÐÝÞe di«eÖeÒØ iØeѹba×ed ÖecÓÑÑeÒ¹ ÑÓ×Ø fÖeÕÙeÒØ Úi×iØÓÖ׺ Ìhe×e \ÐÓÒg Ù×eÖ ÖÓÛ×" ×ÐÓÛ dÓÛÒ daØiÓÒ geÒeÖaØiÓÒ aÐgÓÖiØhÑ׺ ÏeÐÓÓkiÒØÓ di«eÖeÒØØech¹ Øhe ÒÙÑbeÖ Óf Òeighb ÓÖ× ØhaØ caÒ b e ×eaÖched Ô eÖ ×ecÓÒd¸ ÒiÕÙe× fÓÖ cÓÑÔÙØiÒg iØeѹiØeÑ ×iÑiÐaÖiØie× ´eºgº¸ iØeѹiØeÑ fÙÖØheÖ ÖedÙciÒg ×caÐabiÐiØݺ cÓÖÖeÐaØiÓÒ Ú׺ cÓ×iÒe ×iÑiÐaÖiØie× b eØÛeeÒ iØeÑ ÚecØÓÖ×µ aÒd Ìhe ×ecÓÒd chaÐÐeÒge i× ØÓ iÑÔÖÓÚe Øhe ÕÙaÐiØÝ Óf Øhe Öec¹ di«eÖeÒØØechÒiÕÙe× fÓÖ ÓbØaiÒiÒg ÖecÓÑÑeÒdaØiÓÒ× fÖÓÑ ØheÑ ÓÑÑeÒdaØiÓÒ× fÓÖ Øhe Ù×eÖ׺ Í×eÖ× Òeed ÖecÓÑÑeÒdaØiÓÒ× ´eºgº¸ ÛeighØed ×ÙÑ Ú׺ ÖegÖe××iÓÒ ÑÓ deеº FiÒaÐÐݸ Ûeeܹ ØheÝ caÒ ØÖÙ×Ø ØÓ heÐÔ ØheÑ ¬Òd iØeÑ× ØheÝ ÛiÐÐ Ðikeº Í×eÖ× Ô eÖiÑeÒØaÐÐÝ eÚaÐÙaØe ÓÙÖ Öe×ÙÐØ× aÒd cÓÑÔaÖe ØheÑ ØÓ Øhe ÛiÐÐ "ÚÓØe ÛiØh ØheiÖ feeØ" bÝ ÖefÙ×iÒg ØÓ Ù×e ÖecÓÑÑeÒdeÖ ba×ic k¹ÒeaÖe×Ø ÒeighbÓÖ aÔÔÖÓachº ÇÙÖ eÜÔ eÖiÑeÒØ× ×Ùg¹ ×Ý×ØeÑ× ØhaØ aÖe ÒÓØ cÓÒ×i×ØeÒØÐÝ accÙÖaØe fÓÖ ØheѺ ge×Ø ØhaØ iØeѹba×ed aÐgÓÖiØhÑ× ÔÖÓÚide dÖaÑaØicaÐÐÝ b eØØeÖ ÁÒ ×ÓÑe ÛaÝ× Øhe×e ØÛÓchaÐÐeÒge× aÖe iÒ cÓÒ­icظ ×iÒce Øhe Ô eÖfÓÖÑaÒce ØhaÒ Ù×eÖ¹ba×ed aÐgÓÖiØhÑ׸ ÛhiÐe aØ Øhe ×aÑe Ðe×× ØiÑe aÒ aÐgÓÖiØhÑ ×Ô eÒd× ×eaÖchiÒg fÓÖ Òeighb ÓÖ׸ Øhe ØiÑe ÔÖÓÚidiÒg b eØØeÖ ÕÙaÐiØÝ ØhaÒ Øhe b e×Ø aÚaiÐabÐe Ù×eÖ¹ ÑÓÖe ×caÐabÐe iØ ÛiÐÐ b e¸ aÒd Øhe ÛÓÖ×e iØ× ÕÙaÐiØݺ FÓÖ Øhi× ba×ed aÐgÓÖiØhÑ׺ Öea×ÓÒ¸ iØ i× iÑÔ ÓÖØaÒØ ØÓ ØÖeaØ Øhe ØÛÓ chaÐÐeÒge× ×iÑÙй ØaÒeÓÙ×ÐÝ ×Ó Øhe ×ÓÐÙØiÓÒ× di×cÓÚeÖed aÖe b ÓØh Ù×efÙÐ aÒd 1. INTRODUCTION ÔÖacØicaк ÁÒ Øhi× ÔaÔ eÖ¸ Ûe addÖe×× Øhe×e i××Ùe× Óf ÖecÓÑÑeÒdeÖ Ìhe aÑÓÙÒØ Óf iÒfÓÖÑaØiÓÒ iÒ Øhe ÛÓÖÐd i× iÒcÖea×iÒg faÖ ×Ý×ØeÑ× bÝ aÔÔÐÝiÒg a di«eÖeÒØ aÔÔÖÓachßiØeѹba×ed aÐgÓ¹ ÑÓÖe ÕÙickÐÝ ØhaÒ ÓÙÖ abiÐiØÝ ØÓ ÔÖÓ ce×× iغ AÐÐ Óf Ù× haÚe ÖiØhѺ Ìhe b ÓØØÐeÒeckiÒcÓÒÚeÒØiÓÒaÐ cÓÐÐab ÓÖaØiÚe ¬ÐØeÖ¹ kÒÓÛÒ Øhe feeÐiÒg Óf b eiÒg ÓÚeÖÛheÐÑed bÝ Øhe ÒÙÑbeÖ Óf iÒg aÐgÓÖiØhÑ× i× Øhe ×eaÖch fÓÖ Òeighb ÓÖ× aÑÓÒg a ÐaÖge ÒeÛ b Ó Ók׸ jÓÙÖÒaÐ aÖØicÐe׸ aÒd cÓÒfeÖeÒce ÔÖÓ ceediÒg× cÓѹ Ù×eÖ Ô ÓÔÙÐaØiÓÒ Óf Ô ÓØeÒØiaÐ Òeighb ÓÖ× [½¾]º ÁØeѹba×ed aй iÒg ÓÙØ eachÝeaÖº ÌechÒÓÐÓgÝ ha× dÖaÑaØicaÐÐÝ ÖedÙced Øhe gÓÖiØhÑ× aÚÓid Øhi× b ÓØØÐeÒeckbÝ eÜÔÐÓÖiÒg Øhe ÖeÐaØiÓÒ×hiÔ× baÖÖieÖ× ØÓ ÔÙbÐi×hiÒg aÒd di×ØÖibÙØiÒg iÒfÓÖÑaØiÓÒº ÆÓÛ beØÛeeÒ iØeÑ× ¬Ö×ظ ÖaØheÖ ØhaÒ Øhe ÖeÐaØiÓÒ×hiÔ× beØÛeeÒ iØ i× ØiÑe ØÓ cÖeaØe Øhe ØechÒÓÐÓgie× ØhaØ caÒ heÐÔ Ù× ×ifØ Ù×eÖ׺ ÊecÓÑÑeÒdaØiÓÒ× fÓÖ Ù×eÖ× aÖe cÓÑÔÙØed bÝ ¬ÒdiÒg edº Be¹ Copyright is held by the author/owner. iØeÑ× ØhaØ aÖe ×iÑiÐaÖ ØÓ ÓØheÖ iØeÑ× Øhe Ù×eÖ ha× Ðik ÛeeÒ iØeÑ× aÖe ÖeÐaØiÚeÐÝ ×ØaØic¸ WWW10, May 1-5, 2001, Hong Kong. caÙ×e Øhe ÖeÐaØiÓÒ×hiÔ× b eØ ACM 1-58113-348-0/01/0005. 285 ÁÒ ÓÒe ×ØÙdÝ Ù×iÒg ×ÝÒØheØic daØa¸ ÀÓÖØiÒg ÔÖÓ dÙced b eØØeÖ iØeѹba×ed aÐgÓÖiØhÑ× ÑaÝ b e abÐe ØÓ ÔÖÓÚide Øhe ×aÑe ÕÙaй ÔÖedicØiÓÒ× ØhaÒ a ÒeaÖe×Ø Òeighb ÓÖ aÐgÓÖiØhÑ [½]º iØÝ a× Øhe Ù×eÖ¹ba×ed aÐgÓÖiØhÑ× ÛiØh Ðe×× ÓÒÐiÒe cÓÑÔÙØa¹ ËchafeÖ eØ aк¸ [¾6] ÔÖe×eÒØ a deØaiÐed ØaÜÓÒÓÑÝ aÒd eÜaѹ ØiÓÒº ÔÐe× Óf ÖecÓÑÑeÒdeÖ ×Ý×ØeÑ× Ù×ed iÒ E¹cÓÑÑeÖce aÒd hÓÛ Úide ÓÒe¹ØÓ¹ÓÒe Ô eÖ×ÓÒaÐiÞaØiÓÒ aÒd aØ Øhe ×aÑe 1.1 Related Work ØheÝ caÒ ÔÖÓ caÒ caÔØÙÖe cÙ×ØÓÑeÖ ÐÓÝaÐØݺ AÐØhÓÙgh Øhe×e ×Ý×ØeÑ× haÚe ÁÒ Øhi× ×ecØiÓÒ Ûe bÖie­Ý ÔÖe×eÒØ ×ÓÑe Óf Øhe Öe×eaÖchÐiع b eeÒ ×Ùcce××fÙÐ iÒ Øhe Ôa×ظ ØheiÖ Ûide×ÔÖead Ù×e ha× eÜÔ Ó×ed eÖaØÙÖe ÖeÐaØed ØÓ cÓÐÐab ÓÖaØiÚe ¬ÐØeÖiÒg¸ ÖecÓÑÑeÒdeÖ ×Ý×¹ ×ÓÑe Óf ØheiÖ ÐiÑiØaØiÓÒ× ×Ùch a× Øhe ÔÖÓbÐeÑ× Óf ×ÔaÖ×iØÝiÒ ØeÑ׸ daØa ÑiÒiÒg aÒd Ô eÖ×ÓÒaÐiÞaØiÓÒº Øhe daØa ×eظ ÔÖÓbÐeÑ× a××Ó ciaØed ÛiØh high diÑeÒ×iÓÒaÐiØÝ ÌaÔ e×ØÖÝ [½¼] i× ÓÒe Óf Øhe eaÖÐie×Ø iÑÔÐeÑeÒØaØiÓÒ× Óf cÓй aÒd ×Ó ÓÒº ËÔaÖ×iØÝ ÔÖÓbÐeÑ iÒ ÖecÓÑÑeÒdeÖ ×Ý×ØeÑ ha× Ðab ÓÖaØiÚe ¬ÐØeÖiÒg¹ba×ed ÖecÓÑÑeÒdeÖ ×Ý×ØeÑ׺ Ìhi× ×Ý×¹ b eeÒ addÖe××ed iÒ [¾¿¸ ½½]º Ìhe ÔÖÓbÐeÑ× a××Ó ciaØed ÛiØh ØeÑ ÖeÐied ÓÒ Øhe eÜÔÐiciØ ÓÔiÒiÓÒ× Óf Ô eÓÔÐe fÖÓÑ a cÐÓ×e¹kÒiØ high diÑeÒ×iÓÒaÐiØÝ iÒ ÖecÓÑÑeÒdeÖ ×Ý×ØeÑ× haÚebeeÒdi×¹ cÓÑÑÙÒiØݸ×Ùch a× aÒ ÓÆce ÛÓÖkgÖÓÙÔº ÀÓÛeÚeÖ¸ ÖecÓѹ cÙ××ed iÒ [4]¸ aÒd aÔÔÐicaØiÓÒ Óf diÑeÒ×iÓÒaÐiØÝ ÖedÙcØiÓÒ ÑeÒdeÖ ×Ý×ØeÑ fÓÖ ÐaÖge cÓÑÑÙÒiØie× caÒÒÓØ deÔ eÒd ÓÒ each ØechÒiÕÙe× ØÓ addÖe×× Øhe×e i××Ùe× ha× b eeÒ iÒÚe×ØigaØed iÒ Ô eÖ×ÓÒ kÒÓÛiÒg Øhe ÓØheÖ׺ ÄaØeÖ¸ ×eÚeÖaÐ ÖaØiÒg×¹ba×ed aÙ¹ [¾4]º ØÓÑaØed ÖecÓÑÑeÒdeÖ ×Ý×ØeÑ× ÛeÖe deÚeÐÓÔ edº Ìhe GÖÓÙ¹ ÇÙÖ ÛÓÖk eÜÔÐÓÖe× Øhe eÜØeÒØØÓÛhich iØeѹba×ed ÖecÓѹ ÔÄeÒ× Öe×eaÖch ×Ý×ØeÑ [½9¸ ½6] ÔÖÓÚide× a Ô×eÙdÓÒÝÑÓÙ× ÑeÒdeÖ׸ a ÒeÛ cÐa×× Óf ÖecÓÑÑeÒdeÖ aÐgÓÖiØhÑ׸ aÖe abÐe cÓÐÐab ÓÖaØiÚe ¬ÐØeÖiÒg ×ÓÐÙØiÓÒ fÓÖ Í×eÒeØ ÒeÛ× aÒd ÑÓÚie׺ ØÓ ×ÓÐÚe Øhe×e ÔÖÓbÐeÑ׺ ÊiÒgÓ [¾7] aÒd ÎideÓ ÊecÓÑÑeÒdeÖ [½4] aÖe eÑaiÐ aÒd Ûeb¹ Ù×ic aÒd ba×ed ×Ý×ØeÑ× ØhaØ geÒeÖaØe ÖecÓÑÑeÒdaØiÓÒ× ÓÒ Ñ 1.2 Contributions ÑÓÚie׸ Öe×Ô ecØiÚeÐݺ A ×Ô eciaÐ i××Ùe Óf CÓÑÑÙÒicaØiÓÒ× Óf Ìhi× ÔaÔ eÖ ha× ØhÖee ÔÖiÑaÖÝ Öe×eaÖchcÓÒØÖibÙØiÓÒ×: Øhe ACÅ [¾¼] ÔÖe×eÒØ× a ÒÙÑb eÖ Óf di«eÖeÒØ ÖecÓÑÑeÒdeÖ ×Ý×ØeÑ׺ ½º AÒaÐÝ×i× Óf Øhe iØeѹba×ed ÔÖedicØiÓÒ aÐgÓÖiØhÑ× aÒd ÇØheÖ ØechÒÓÐÓgie× haÚe aÐ×Ó b eeÒ aÔÔÐied ØÓ ÖecÓÑÑeÒdeÖ ideÒØi¬caØiÓÒ Óf di«eÖeÒØÛaÝ× ØÓ iÑÔÐeÑeÒØiØ× ×Ùb¹ ×Ý×ØeÑ׸ iÒcÐÙdiÒg BaÝe×iaÒ ÒeØÛÓÖk׸ cÐÙ×ØeÖiÒg¸ aÒd ÀÓÖع Øa×k׺ iÒgº BaÝe×iaÒ ÒeØÛÓÖk× cÖeaØe a ÑÓ deÐ ba×ed ÓÒ a ØÖaiÒiÒg ¾º FÓÖÑÙÐaØiÓÒ Óf a ÔÖecÓÑÔÙØed ÑÓ deÐ Óf iØeÑ ×iÑiÐaÖiØÝ ×eØ ÛiØh a deci×iÓÒ ØÖee aØ each ÒÓ de aÒd edge× ÖeÔÖe×eÒع ØÓ iÒcÖea×e Øhe ÓÒÐiÒe ×caÐabiÐiØÝ Óf iØeѹba×ed ÖecÓѹ iÒg Ù×eÖ iÒfÓÖÑaØiÓÒº Ìhe ÑÓ deÐ caÒ b e bÙiÐØ Ó«¹ÐiÒe ÓÚeÖ a ÑeÒdaØiÓÒ׺ ÑaØØeÖ Óf hÓÙÖ× ÓÖ daÝ׺ Ìhe Öe×ÙÐØiÒg ÑÓ deÐ i× ÚeÖÝ ×ÑaÐи ÚeÖÝ fa×ظ aÒd e××eÒØiaÐÐÝ a× accÙÖaØe a× ÒeaÖe×Ø ÒeighbÓÖ ¿º AÒ eÜÔ eÖiÑeÒØaÐ cÓÑÔaÖi×ÓÒ Óf Øhe ÕÙaÐiØÝÓf×eÚeÖaÐ ÑeØhÓ d× [6]º BaÝe×iaÒ ÒeØÛÓÖk× ÑaÝÔÖÓÚe ÔÖacØicaÐ fÓÖ eÒ¹ di«eÖeÒØ 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