Algorithms

Algorithms - C SC I H om ew ork#So lu t ion P ro f.M ing-D...

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Unformatted text preview: C SC I H om ew ork#So lu t ion P ro f.M ing-D ehH uang TA s :Ift ikharBu rhanudd in ,C han sookL im // G au ss ,K ar lF r ied r ich(- ) Itisno tknow ledge ,bu ttheac to f learn ing ,no tpossess ionbu ttheac to fge ttingthere ,wh ichgran tsthegrea test en joym en t.W henIhavec lar iedandexhaustedasub jec t,thenIturnaway from it,inordertogoin todarknessaga in ;thenever-sa tisedm an isso strangeifhehascom p le tedastruc ture ,then itisno tinordertodw e llin itpeace fu lly ,bu tinordertobeg inano ther .Iim ag inethewor ldconqueror m ustfee lthus ,who ,a fteronek ingdom isscarce lyconquered ,stre tchesou t h isarm sforo thers . C orrect ion s G ramm at ica ltyposhavebeenxed . Q.- P k + i = >W hasbeen in serted . Q-c m akechangefor hasbeenchangedto m akechangefor . Q.-thew ord ca ll hasbeenrep lacedby opera tion Q-bthew ord list hasbeenrep lacedby array ).- A lgor ithm : Sorttheitem sby increas ingw e igh t .C hoosetherst k item s suchthat P k i = w i W and P k + i = >W ,w here W isthea llow edw e igh t . G reedy-cho icep roperty : Suppose S theop t im a lso lu t iondoesn 'tcon- ta in a theitemw iththeleastw e igh t ,then ,byrem ov inganyo ftheitem s from thelisto fitem sin S andrep lac ing itw ith a w eob ta in S anew so- lu t ionw h ich ism orep rotab lethantheop t im a lso lu t ion S (s incetheva lue o ftheligh testitem isa lsothegreatest) .T h isisacon trad ict ion !H encethe op t im a lso lu t ionm u stcon ta inthegreedycho ice(theligh testitem ) . O p t im a lsub structu re : L et f a ;a ;:::;a k g betheitem sintheop t im a l so lu t ion( in increas ingordero fw e igh t)and T betheseto fa llthe item s .O u r a lgor ithmw ou ldchoose a asitsrstcho ice .W ec la im that O = f a ;:::;a k g form sanop t im a lso lu t iontothesubp rob lemw herethesetis T ? f a g and therequ iredw e igh tis W ? w ( a ) .Ifnot ,bysuppos ing O istheop t im a l so lu t ion , O [f a g betterstheop t im a lso lu t ion O [f a g .A con trad ict ion ! H enceu s ingas im p leR ep lacem en tstrategyargum en tw ehavep rovedthat anop t im a lso lu t iontotheor ig ina lp rob lem con ta in sanop t im a lso lu t ionto thesubp rob lem . T im eC om p lex ity : Sort ingtakes O ( n log n )t im eandatm ost n check s havetobem adeto k .H encew orstcaserunn ingt im eis O ( n log n ) . ) .- A lgor ithm : L et S = f x ;x ;:::;x n g .Sort S .C hoosetherstun it-length c losed in terva las[ x ;x + ].R em ovea llthepo in tsw h icharecoveredw ith inth isin terva ltoget S andrepeat . G reedy-cho icep roperty : L ettheun it-length in terva l[ y...
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This note was uploaded on 10/02/2011 for the course CS 7300 taught by Professor R during the Spring '11 term at LSU.

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Algorithms - C SC I H om ew ork#So lu t ion P ro f.M ing-D...

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