MiYu鍘熷垱, 杞笘璇鋒敞鏄?: 杞澆鑷?nbsp;______________鐧界櫧銇眿 
浼稿睍鏍戯紙Splay Tree錛夋槸涓縐嶄簩鍙夋帓搴忔爲錛屽畠鑳藉湪O(log n)鍐呭畬鎴愭彃鍏ャ佹煡鎵懼拰鍒犻櫎鎿嶄綔銆傚畠鐢盌aniel Sleator鍜孯obert Tarjan鍒涢犮傚畠鐨勪紭鍔垮湪浜庝笉闇瑕佽褰曠敤浜庡鉤琛℃爲鐨勫啑浣欎俊鎭傚湪浼稿睍鏍戜笂鐨勪竴鑸搷浣滈兘鍩轟簬浼稿睍鎿嶄綔銆? 鏌ユ壘鏍戠殑鐩稿叧鐭ヨ瘑 銆銆鍚勭鏌ユ壘鏍戝瓨鍦ㄤ笉瓚熾傛瘮濡傦細瀵逛簬涓涓湁n涓妭鐐圭殑騫寵 鏍戯紝铏界劧鏈鍧忔儏鍐典笅姣忔鏌ユ壘鐨勬椂闂村鏉傚害涓嶄細瓚呰繃O(logn)錛屼絾鏄鏋滆闂ā寮忎笉鍧囧寑錛屽鉤琛℃爲鐨勬晥鐜囧氨浼氬彈鍒板獎鍝嶃傛澶栵紝瀹冧滑榪橀渶瑕侀澶栫殑絀洪棿鏉ュ瓨鍌ㄥ鉤琛′俊鎭? 銆銆榪欎簺鏌ユ壘鏍戠殑璁捐鐩爣閮芥槸鍑忓皯鏈鍧忔儏鍐典笅鍗曟鎿嶄綔鏃墮棿錛屼絾鏄煡鎵炬爲鐨勫吀鍨嬪簲鐢ㄧ粡甯擱渶瑕佹墽琛屼竴緋誨垪鐨勬煡鎵炬搷浣滐紝姝ゆ椂鏇村叧蹇冪殑鎬ц兘鎸囨爣鏄墍鏈夎繖浜涙搷浣滄誨叡闇瑕佸灝戞椂闂淬傚浜庢綾誨簲鐢紝鏇村ソ鐨勭洰鏍囧氨鏄檷浣庢搷浣滅殑鎽婂鉤鏃墮棿錛屾澶勭殑鎽婂鉤鏃墮棿鏄寚鍦ㄤ竴緋誨垪鏈鍧忔儏鍐電殑鎿嶄綔搴忓垪涓崟嬈℃搷浣滅殑騫沖潎鏃墮棿銆傝幏寰楁憡騫蟲晥鐜囩殑涓縐嶆柟娉?灝辨槸浣跨敤“鑷皟鏁?#8221;鐨勬暟鎹粨鏋勩? 銆銆鍜屽鉤琛＄殑鎴栨槸鍏跺畠瀵圭粨鏋勬湁鏄庣‘闄愬埗鐨勬暟鎹粨鏋勬瘮璧鋒潵錛岃嚜璋冩暣鏁版嵁緇撴瀯鏈変互涓嬪嚑涓紭鐐癸細 銆銆1銆佷粠鎽婂鉤瑙掑害鑰岃█錛屽畠浠拷鐣ュ父閲忓洜瀛愶紝鍥犳緇濆涓嶄細姣旀湁鏄庣‘闄愬埗鐨勬暟鎹粨鏋勫樊銆傝屼笖鐢變簬瀹冧滑鍙互鏍規嵁浣跨敤鎯呭喌榪涜璋冩暣錛屼簬鏄湪浣跨敤妯″紡涓嶅潎鍖鐨勬儏鍐典笅鏇村姞鏈夋晥銆?font color="#CC3300"> 銆銆2銆佺敱浜庢棤闇瀛樺偍騫寵 鎴栬呭叾瀹冪殑闄愬埗淇℃伅錛屽畠浠墍闇鐨勭┖闂存洿灝忋?font color="#CC3300"> 銆銆3銆佸畠浠殑鏌ユ壘鍜屾洿鏂扮畻娉曟蹇電畝鍗曪紝鏄撲簬瀹炵幇銆? 銆銆褰撶劧錛岃嚜璋冩暣緇撴瀯涔熸湁娼滃湪鐨勭己鐐癸細 銆銆1銆佸畠浠渶瑕佹洿澶氱殑灞閮ㄨ皟鏁達紝灝ゅ叾鏄湪鏌ユ壘鏈熼棿銆傦紙閭ｄ簺鏈夋槑紜檺鍒剁殑鏁版嵁緇撴瀯浠呴渶鍦ㄦ洿鏂版湡闂磋繘琛岃皟鏁達紝鏌ユ壘鏈熼棿鍒欎笉鐢級 銆銆2銆佷竴緋誨垪鏌ユ壘鎿嶄綔涓殑鏌愪竴涓彲鑳戒細鑰楁椂杈冮暱錛岃繖鍦ㄥ疄鏃跺簲鐢ㄧ▼搴忎腑鍙兘鏄釜涓嶈凍涔嬪銆? 浼稿睍鏍戝瓨鍦ㄧ殑鎰忎箟 銆銆鍋囪鎯寵瀵逛竴涓簩鍙夋煡鎵炬爲鎵ц涓緋誨垪鐨勬煡鎵炬搷浣溿備負浜嗕嬌鏁翠釜鏌ユ壘鏃墮棿鏇村皬錛岃鏌ラ鐜囬珮鐨勯偅浜涙潯鐩氨搴斿綋緇忓父澶勪簬闈犺繎鏍戞牴鐨勪綅緗備簬鏄兂鍒拌璁′竴涓畝鍗曟柟娉曪紝 鍦ㄦ瘡嬈℃煡鎵句箣鍚庡鏍戣繘琛岄噸鏋勶紝鎶婅鏌ユ壘鐨勬潯鐩惉縐誨埌紱繪爲鏍硅繎涓浜涚殑鍦版柟銆俿play tree搴旇繍鑰岀敓銆俿play tree鏄竴縐嶈嚜璋冩暣褰㈠紡鐨勪簩鍙夋煡鎵炬爲錛屽畠浼氭部鐫浠庢煇涓妭鐐瑰埌鏍戞牴涔嬮棿鐨勮礬寰勶紝閫氳繃涓緋誨垪鐨勬棆杞妸榪欎釜鑺傜偣鎼Щ鍒版爲鏍瑰幓銆? 宸茬煡閲嶆瀯鏂規硶涓庝幾灞曟爲鐨勯噸鏋勬柟娉? 銆銆鍏堝墠錛屽凡緇忓瓨鍦ㄤ袱縐嶉噸鏋勬柟娉曪細 銆銆1銆佸崟鏃嬶細鍦ㄦ煡鎵懼畬浣嶄簬鑺傜偣x涓殑鏉＄洰i涔嬪悗錛屾棆杞摼鎺鍜屽叾鐖惰妭鐐圭殑杈廣傦紙闄ら潪x灝辨槸鏍戞牴錛?font color="#CC3300"> 銆銆2銆佹惉縐昏嚦鏍戞牴錛氬湪鏌ユ壘瀹屼綅浜庤妭鐐箈涓殑鏉＄洰i涔嬪悗錛屾棆杞摼鎺鍜屽叾鐖惰妭鐐圭殑杈癸紝鐒跺悗閲嶅榪欎釜鎿嶄綔鐩磋嚦x鎴愪負鏍戞牴銆? 銆銆splay tree鐨勯噸鏋勬柟娉曞拰鎼Щ鑷蟲爲鏍圭殑鏂規硶鐩鎬技錛屽畠涔熶細娌跨潃鏌ユ壘璺緞鍋氳嚜搴曞悜涓婄殑鏃嬭漿錛屽皢琚煡鎵炬潯鐩Щ鑷蟲爲鏍廣備絾涓嶅悓鐨勬槸錛屽畠鐨勬棆杞槸鎴愬榪涜鐨勶紝欏哄簭鍙栧喅浜庢煡鎵捐礬寰勭殑緇撴瀯銆備負浜嗗湪鑺傜偣x澶勫鏍戣繘琛宻play鎿嶄綔錛屾垜浠渶瑕侀噸澶嶄笅闈㈢殑姝ラ錛岀洿鑷硏鎴愪負鏍戞牴涓烘錛?font color="#CC3300"> 銆銆1銆佺涓縐嶆儏鍐碉細濡傛灉x鐨勭埗鑺傜偣p(x)鏄爲鏍癸紝鍒欐棆杞繛鎺鍜宲(x)鐨勮竟銆傦紙榪欑鎯呭喌鏄渶鍚庝竴姝ワ級 銆銆2銆佺浜岀鎯呭喌錛氬鏋減(x)涓嶆槸鏍戞牴錛岃屼笖x鍜宲(x)鏈韓閮芥槸宸﹀瀛愭垨鑰呴兘鏄彸瀛╁瓙錛屽垯鍏堟棆杞繛鎺(x)鍜寈鐨勭鐖惰妭鐐筭(x)鐨勮竟錛岀劧鍚庡啀鏃嬭漿榪炴帴x鍜宲(x)鐨勮竟銆?font color="#CC3300"> 銆銆3銆佺涓夌鎯呭喌錛氬鏋減(x)涓嶆槸鏍戞牴錛岃屼笖x鏄乏瀛╁瓙錛宲(x)鏄彸瀛╁瓙錛屾垨鑰呯浉鍙嶏紝鍒欏厛鏃嬭漿榪炴帴x鍜宲(x)鐨勮竟錛屽啀鏃嬭漿榪炴帴x鍜屾柊鐨刾(x)鐨勮竟銆? 銆銆鍦ㄨ妭鐐箈澶勮繘琛宻play鎿嶄綔鐨勬椂闂存槸鍜屾煡鎵緓鎵闇鐨勬椂闂存垚姣斾緥鐨勩俿play鎿嶄綔涓嶅崟鏄妸x鎼Щ鍒頒簡鏍戞牴錛岃屼笖榪樻妸鏌ユ壘璺緞涓婄殑姣忎釜鑺傜偣鐨勬繁搴﹂兘澶ц嚧鍑忔帀浜嗕竴鍗娿? 浼稿睍鏍戯紙Splay Tree錛夋敮鎸佺殑鎿嶄綔 銆銆鍏蜂綋鎿嶄綔鍖呮嫭錛?font color="#CC3300"> 銆銆1銆乤ccess(i,t):濡傛灉i鍦ㄦ爲t涓紝鍒欒繑鍥炴寚鍚戝畠鐨勬寚閽堬紝鍚﹀垯榪斿洖絀烘寚閽堛備負浜嗗疄鐜癮ccess(i,t)錛屽彲浠ヤ粠鏍憈鐨勬牴閮ㄥ悜涓嬫煡鎵緄銆傚鏋?鏌ユ壘鎿嶄綔閬囧埌浜嗕竴涓惈鏈塱鐨勮妭鐐箈錛屽氨鍦▁澶勮繘琛宻play鎿嶄綔錛屽茍榪斿洖鎸囧悜x鐨勬寚閽堬紝璁塊棶緇撴潫銆傚鏋滈亣鍒頒簡絀烘寚閽堬紝琛ㄧずi涓嶅湪鏍戜腑錛屾鏃跺氨鍦ㄦ渶鍚庝竴涓潪 絀鴻妭鐐瑰榪涜splay鎿嶄綔錛岀劧鍚庤繑鍥炵┖鎸囬拡銆傚鏋滄爲鏄┖鐨勶紝灝嗗拷鐣ユ帀splay鎿嶄綔銆?font color="#CC3300"> 銆銆2銆乮nsert(i,t):灝嗘潯鐩甶鎻掑叆鏍憈涓紙鍋囪鍏跺皻涓嶅瓨鍦級銆備負浜嗗疄鐜癷nsert(i,t)錛岄鍏堟墽琛宻plit(i,t)錛岀劧鍚庢妸t鎹㈡垚涓涓敱鏂扮殑鍖呭惈鏈塱鐨勬牴鑺傜偣緇勬垚鐨勬爲錛岃繖涓牴鑺傜偣鐨勫乏鍙沖瓙鏍戝垎鍒槸split榪斿洖鐨勬爲t1鍜宼2銆?font color="#CC3300"> 銆銆3銆?strong>delete(i,t):浠庢爲t涓垹闄ゆ潯鐩甶錛堝亣璁懼叾宸茬粡瀛樺湪錛夈備負浜嗗疄鐜?strong>delete(i,t)錛岄鍏堟墽琛宎ccess(i,t)錛岀劧鍚庢妸t鎹㈡垚鍏跺乏瀛愭爲鍜屽彸瀛愭爲join涔嬪悗鐨勬柊鏍戙?font color="#CC3300"> 銆銆4銆乯oin(t1,t2):灝嗘爲t1鍜宼2鍚堝茍鎴愪竴媯墊爲錛屽叾涓寘鍚箣鍓嶄袱媯墊爲鐨勬墍鏈夋潯鐩紝騫惰繑鍥炲悎騫朵箣鍚庣殑鏍戙傝繖涓搷浣滃亣璁総1涓殑鎵鏈夋潯鐩兘灝忎簬t2 涓殑鏉＄洰錛屾搷浣滃畬鎴愪箣鍚庝細閿姣乼1鍜宼2銆備負浜嗗疄鐜癹oin(t1,t2)錛岄鍏堣闂畉1涓渶澶х殑鏉＄洰i銆傝闂粨鏉熶箣鍚庯紝t1鐨勬牴鑺傜偣涓寘鍚殑灝辨槸i錛屽畠 鐨勫彸瀛╁瓙鏄劇劧涓虹┖銆備簬鏄妸t2浣滀負榪欎釜鏍硅妭鐐圭殑鍙沖瓙鏍戝茍榪斿洖瀹屾垚涔嬪悗鐨勬柊鏍戝嵆鍙疄鐜癹oin鎿嶄綔銆?font color="#CC3300"> 銆銆5銆乻plit(i,t):鏋勫緩騫惰繑鍥炰袱媯墊爲t1鍜宼2錛屽叾涓璽1鍖呭惈t涓墍鏈夊皬浜庣瓑浜巌鐨勬潯鐩紝t2鍖呭惈t涓墍鏈夊ぇ浜巌鐨勬潯鐩傛搷浣滃畬鎴愪箣鍚庨攢姣乼銆備負 浜嗗疄鐜皊plit(i,t)錛岄鍏堟墽琛宎ccess(i,t)錛岀劧鍚庢牴鎹柊鏍硅妭鐐逛腑鐨勫兼槸澶т簬榪樻槸灝忎簬絳変簬i鏉ュ垏鏂繖涓牴鑺傜偣鐨勫乏閾炬帴鎴栧彸閾炬帴錛屽茍榪斿洖褰?鎴愮殑涓ゆ５鏍戙? 銆銆鍙﹀insert鍜?strong>delete鏂規硶鏈夋洿濂界殑瀹炵幇錛屾椂闂村鏉傚害鏇村皬錛?font color="#CC3300"> 銆銆1銆乮nsert(i, t):鏌ユ壘i錛屾妸閬囧埌鐨勭┖鎸囬拡鏇挎崲鎴愪竴涓惈鏈塱鐨勬柊鑺傜偣錛岀劧鍚庡啀鍦ㄦ柊鑺傜偣澶勫鏍戣繘琛宻play鎿嶄綔銆?font color="#CC3300"> 銆銆2銆?strong>delete(i, t):鏌ユ壘鍚湁i鐨勮妭鐐癸紝璁炬鑺傜偣涓簒錛屽叾鐖惰妭鐐逛負y銆傛妸x鐨勫乏鍙沖瓙鏍戝悎騫朵箣鍚庢浛鎹㈡帀x錛岀劧鍚庡啀浠巠澶勮繘琛宻play鎿嶄綔銆? 浼稿睍鏍戠殑浼樺娍 銆銆鐢變簬Splay Tree浠呬粎鏄笉鏂皟鏁達紝騫舵病鏈夊紩鍏ラ澶栫殑鏍囪錛屽洜鑰屾爲緇撴瀯涓庢爣鍑咮ST娌℃湁浠諱綍涓嶅悓錛屼粠絀洪棿瑙掑害鏉ョ湅錛屽畠姣擳reap銆丷ed-Black Tree銆丄VL瑕侀珮鏁堝緱澶氥傚洜涓虹粨鏋勪笉鍙橈紝鍥犳鍙鏄氳繃宸︽棆鍜屽彸鏃嬭繘琛岀殑鎿嶄綔瀵筍play Tree鎬ц川閮芥病鏈変笣姣獎鍝嶏紝鍥犺屽畠涔熸彁渚涗簡BST涓渶涓板瘜鐨勫姛鑳斤紝鍖呮嫭蹇熺殑鎷嗗垎鍜屽悎騫訛紙榪欓噷鎸囩殑鏄皢鍘熸爲鎷嗗垎鎴愪袱媯靛瓙鏍戯紝鍏朵腑涓媯靛瓙鏍戞墍鏈夎妭鐐歸兘姣斿彟涓瀛愭爲灝忥紝浠ュ強瀹冪殑閫嗚繃紼嬶級錛屽茍涓斿疄鐜版瀬涓轟究鎹楓傝繖涓鐐規槸鍏跺畠緇撴瀯杈冮毦瀹炵幇鐨勩傚叾鏃墮棿鏁堢巼涔熺浉褰撶ǔ瀹氾紝鍜孴reap鍩烘湰鐩稿綋
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