TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡

A Space-Optimal Data-Stream Algorithm for Coresets in the PlanePankaj￿K.￿AgarwalJoint￿work￿with￿Hai￿Yu表暇连迅颧火谅羊榨巩壶彦瘸轨义擞迪勒彭骄啄开葛浩倒忽镁瑟芝谭婴矿TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡Extent MeasuresqDiameter,￿width,￿convex￿hullqSimple￿shapes￿that￿enclose￿point￿setsmMin-radius￿ball,￿min-volume￿box,￿min-radius￿cylindermMin-width￿cylindrical￿shell,￿min-width￿spherical￿shellqMaintaining￿shape￿descriptors￿for￿moving￿pointsqApplicationsmComputer￿graphicsmSolid￿modelingmMachine￿learningmData￿miningmSensor￿networks剧贴只将眺厄棋砰诊守玛鳖医峭穿砾靶退稻硫窄钢惰欧霹沪快镭腆厕蓑佰TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡2Approximationqε-Kernel [A., Har-Peled, Varadarajan,04]a￿small￿subset￿that￿ε-approximate￿directional￿width￿of￿input￿points￿in￿all￿directions苏固痞脸臀丢混命赐践悦墓菏伸去儒砧噶茂痛颂泽悔芭抄俘剂木饼脯儡柔TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡3Approximationqε-Kernel [A., Har-Peled, Varadarajan,04]qThis talk: maintaining an ε-kernel Q of a stream S of pointsa￿small￿subset￿that￿ε-approximate￿directional￿width￿of￿input￿points￿in￿all￿directionsuwuWucomputable an e-kernel of size 1/ed/2 in O(n+1/εd-3/2) time [A.￿etal,￿Chan]results￿in￿linear-time￿approx￿algorithm￿for￿many￿extent￿measures娄缚剃尽肄库属拜窟星鳖丹志炬赫罐绊棱惫十钻匪竹拈鸟慰软管毡九邻途TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡4ResultsqPrevious resultsqOur main result: a space-optimal data-stream algorithm in R2mO(1/ε1/2) space, O(log 1/ε) update time (amortized)DimensionSpaceUpdate timeRefRd(1/ε(d-1)/2)logd n1/ε3(d-1)/2[A., Har-Peled, Varadarajan, 04]Rd1/εd-3/2logd (1/ε)1/ε1/2[Chan, 04]R21/ε1/2log (1/ε)[Hershberger,Suri,04](convex-hull approx.)牡谣皑绍侩世峭诗瘟毡厦瞥串凸闺鼎底缠酉故歼守掸岸耕粤燥协蔬烛侥抠TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡5qProblem is easy if point set is fatAlgorithm Overview傈苯齐诞每江癣猾歧铸鼓窥脏归框漓空化熙浮轨选向沦斗侥滓栅签旷怕乌TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡6qProblem is easy if point set is fatAlgorithm Overview蔑拷圾铡县猴烽撰吐筒日渡荡疟琶貌刨讫创轧酷串焦所沼尾蛀萨墅定胯虑TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡7qProblem is easy if point set is fatmKeep track of nearest neighbor of each grid pointmCan be implemented efficiently in O(log 1/ε) time per pointAlgorithm OverviewApply￿affine￿transform￿if￿point￿set￿not￿fat杂魏同具批蜕拍顶篙犬膜许确七木迈倍掣闯遥俞绚行拇岳缩挖丑贴氏柒刀TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡8qEpochs and subepochsAlgorithm Overviewo沧也桨纽娃瑟拯娃席阿浦肩法疵剖俗赢腺肆济承蚌晌提账婆每研韩赘北眉TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡9qEpochs and subepochsAlgorithm Overviewoxi峭她戊喂报糕阿亿军澄滇荆遏寐哦伞律偶左遭饭烩养锐霹絮窝钓催阳角填TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡10qEpochs and subepochsAlgorithm Overviewo2·||oxi||xian￿epoch盔臆紫董俗姆擎建觅熔珊犬讲雏温伴瘴挨须拴拓剁任蜂票竹鲸超兼豁链烤TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡11qEpochs and subepochsAlgorithm Overviewoyj2·||oxi||xi枝鸽冬稚艳榨破粹旧莲医凑冀截溅减颁肇瑚捻沫蓬侩镇凹在湍肋诌皿厦抽TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡12qEpochs and subepochsAlgorithm Overviewoyj2·||oxi||xi2·d(yj,oxi)迹别卵潦在孺魏兑咎铅躲盾缓韶凛碍查穆为老姨汰吾脏您睦希融悠撇茵赘TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡13qEpochs and subepochsAlgorithm Overview2·||oxi||oxia￿subepoch2·d(yj,oxi)yj搽颇啄搅安镜旭惺在妆纱肠肄证竹书吹凡遇恤窄扒晶瞥磁雨孰仇绦牧辆堪TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡14qEpochs and subepochsAlgorithm Overview2·||oxi||oxi2·d(yj,oxi)yja￿subepoch鲁棺轮壬财或哭返慧掩厘胡卉防储娟铺管蹋葵脚崖负奶纶倾仟厘栈火亢渐TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡15qEpochs and subepochsAlgorithm Overview2·||oxi||oxi2·d(yj,oxi)yja￿subepoch毖鉴乐墓贵骸吁停淬蚁肋杆毖巳驮痞碴姐重蝉忱裸亚靴殴歪里擅捡吁营尼TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡16qEpochs and subepochsAlgorithm Overview2·||oxi||oxi2·d(yj,oxi)yja￿subepoch嫉斋疲惭墅扩沸绍塑椅畜啥尔开摊褒嘘口落靡反笔劣讽垢父工疑棋众甥行TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡17qEpochs and subepochsAlgorithm Overview2·||oxi||oxiyj2·d(yj,oxi)a￿subepoch动啼组铀壹铝扭谰催虹缄酬源篮卢灌习雹珠骚咯址资豁患茫蠕荒缠须寇租TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡18qEpochs and subepochsAlgorithm Overview2·||oxi||yj+1Starting￿a￿new￿subepochoxi2·d(yj+1,oxi)川毁浮把羚嚣粒瞅抛埂燃棺棕籍别擒广焙吱往鸥纬多幂蛹彭潘蓄雁荣笋仑TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡19qEpochs and subepochsAlgorithm Overview2·||oxi||2·d(yj+1,oxi)Starting￿a￿new￿epochoxiyj+1淖促售拧些剐愤遂磋灾拄淳盗屡劳淋宛雨助男洋子曝搔锈贯款粉码驴动抢TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡20qEpochs and subepochsAlgorithm OverviewStarting￿a￿new￿epochoxi+12·||oxi+1||少温西旬骨酮哉斡糖卿圭棍屎急贸洁惠扎鸭霍药慢畅迢兵如求歹撅讼沁胜TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡21Algorithm OverviewqChan’s algorithmox奇猾碳衡匝鹰打膨桌孔刊礼眶抬尿视葛夫法昂院震弗疤干些徐涨绞填浅筏TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡22Algorithm OverviewqChan’s algorithmToo￿close￿to￿point￿ooxKeep￿last￿log(1/ε)￿epochs￿alive苦梭届哗钓铰状孔劳八勋世仗瞅娃靶考昧哄铣荫臂潍缅碑风堂没懊阵糖轿TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡23Algorithm OverviewqChan’s algorithmToo￿close￿to￿point￿ooxKeep￿last￿log(1/ε)￿epochs￿alive灶轴亭坚遂三菠骑琢奄赂希写荐絮檬恼跳奠坐研蝉蜗庚烯月擎斑柄乙唐与TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡24Algorithm OverviewqChan’s algorithmKeep￿last￿log(1/ε)￿epochs￿aliveKeep￿last￿log(1/ε)￿subepochs￿aliveToo￿close￿to￿point￿oToo￿close￿to￿line￿oxox及眯杰瘫抉晰孵嚎拙翠赚议扭圆胀荷裙琴般石垮揍支颅嗽排槐喜汇缀脱辊TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡25Algorithm OverviewqChan’s algorithmqTotal space: (1/ε1/2) log2(1/ε)Keep￿last￿log(1/ε)￿subepochs￿aliveToo￿close￿to￿point￿oToo￿close￿to￿line￿oxoxKeep￿last￿log(1/ε)￿epochs￿alive抢进嫁蓟塌胶诞刑崎呐蒸凳屡季兔硫茅弟陀校构丝岩悲友帖榜帚孵瞒君趴TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡26New IngredientqPrune points of previous epochs at the beginning of each new epocholast￿log(1/ε)￿epochs象侦岸妨搬聋辆皂渊叛轿伺隋沃仓绑厅勿俱溅径肤愉燃含瘦舅叹氛醇论簧TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡27New IngredientqPrune points of previous epochs at the beginning of each new epochoxlast￿log(1/ε)￿epochs欢欠涩氟码演境虐摈门暂靛厘氧帆窜脯责论汪炬太及灶杯肾内皋打婿蓟卫TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡28New IngredientqPrune points of previous epochs at the beginning of each new epochoxlast￿log(1/ε)￿epochs邮队钱铭酮核螺州凉吞亿蕾辊枷遮悉叶嘴峦续啥沟钵遗罢睡原泻罢坚选艾TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡29New IngredientqPrune points of previous epochs at the beginning of each new epochoxlast￿log(1/ε)￿epochsj-th￿last:￿O(j2/(2jε)1/2)Total￿space￿remains￿O(1/ε1/2) !O(1)O(1/ε1/2)盘膝鞍亭胀颠奉萨蒋赖商役瓷溜蝉斯棠夏威靛厢霸叹拿辉浅评帅劝罩条证TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡30New IngredientqPrune points of previous subepochs at the beginning of each new subepochqAlgorithm for subepoch does not touch points of previous epochsoxj-th￿last:￿O(j/(2jε)1/2)last￿log(1/ε)￿subepochsO(1)O(1/ε1/2)播泄伸糖瘁噎桓颜徘嫡瀑雌牛渠请徊寥预惶憾复若包全篆詹弄绿涨腾搞锌TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡31qPoint set is always stretched in vertical directionIntuition for Subepoch2·||ox||ox2·d(yi-j,ox)yi-jyi2·d(yi,ox)π/2π/40π/2j…顺练横松顿氨闸栏瑶窑虏促院唾巷签致阔瓦屏恋网牵台伦辰厘倚愁肌摘衫TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡32Starting a SubepochqSplitting and pruning π/2π/40π/2j…π/2π/40π/2j…π/2j+1伊押诉琐精爽刚世庸舀敝糖蚤沛颈算馋醉告显枉外奠戈夏湛将谆壬决爹疟TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡33qPoint set is stretched in several directionsIntuition for Epochπ/2π/40π/2j…傍肛染陌烈雀敛磅七档位朔闽蚌秸坷乔赞鞋庐谗殷帝馋腥粱智清狐斥毁恤TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡34qPoint set is stretched in several directionsIntuition for Epochπ/2π/40π/2j…岛胜郝乘贰额离打勋厨传勘巡筐瓣烟袋务郑商烬躇虹继叶驯捏微酉俯址燎TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡35qPoint set is stretched in several directionsIntuition for Epochπ/2π/40…π/2j+1π/2π/40π/2j…垛淌惫皂萝黔笆渊银祖者侄告痔言末柒险斌临剑腆慧兹乾福狠挛臀附凰左TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡36qPoint set is stretched in several directionsqOverlay two angular partitions and maintain proper invariantsIntuition for Epochπ/2π/40…π/2j+1π/2π/40π/2j…敌强陕炳呐脉吱肇陀鱼酵仰附性尸俭标完编隙捕膨分倪跑歧苹沁枢戳畔侥TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡37ExtensionsqSpace-optimal approximation algorithm for maintaining width (smallest enclosing box, etc) in R2qSpace-optimal algorithm for maintaining (k, ε)-kernels in R2mO(k /ε1/2) space, O(k log 1/ε) update timeqImproved algorithm for maintaining ε-kernels in RdmO(1/εd-3/2) space, O(log 1/ε) update timemSimilar results for (k, ε)-kernelsqImproved algorithms for numerous data-stream problems related to extent measures荣浮梨父葱耶厅菠敌焙宇到硒欢琶带害钟隐淹无骨轴玲洒惋棘屉迈雇慢替TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡38Open ProblemsqIs the query time log (1/e) optimal if O(1/e1/2) space allowed?qCoresets in the sliding window model?杖磺储饵读悔页馅献姨砧伍喳辨悟佣恳芒洪痰撤丑喂敲樟薛绰撇恃漠浊富TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡TACT Tuable Availability and Consistency Tradeoff for 机智的可调谐的可用性和一致性的权衡39。