CSC 263 Lecture 3. October 4, 2006

Transcription

1 S 263 Lece 3 Ocobe 4, Seach and Roaion on Red-lack Tee We ill no implemen he hee oine Seach, Ine and Delee fom he DITIONRY DT ing he Red-lack Tee daa ce. Since a Red-lack Tee i a ST, e can e he ame Seach oine a befoe o each he ee in o cae ime Θ(log n) (ince no he heigh of he ee i Θ(log n) in he o cae). Ine and Delee ill alo ake ime Θ(log n) b if e e he ame oine a befoe, he ill cae iolaion of one of he hee Red-lack popeie. Fo inance, if e e he egla ST Ine, hen e ll add he ne node a he boom of he ee (o boh i childen ae nll). Then e hae o decide hehe o make i ed o black. If e make i black, e ll ceainl iolae pope 3 of Red-lack ee. If e make i ed, e don hae o o abo pope 3, b e migh iolae pope 2 (if i paen i ed). The folloing o pocede ill be efl in bilding o Ine and Delee mehod: RoaeLef(Tee T, Node ) Node = ighhild(); ighhild() = lefhild(); lefhild() = ; RoaeRigh(Tee T, Node ) Node = lefhild(); lefhild() = ighhild(); ighhild() = ; Thee o mehod pefom ha i efeed o a a oaion on he ee T. The folloing i a gaphical epeenaion of hee o mehod. and ae node and,, and ae bee. RoaeRigh(T, ) RoaeLef(T, ) 12

2 6.3 Ineion We ll e he folloing pocede o ine a node ino a Red-lack ee: RedlackIne(Tee Roo R, Node ) Ine(R, ); colo() = ed; If pope 2 i iolaed hen Fi he ee Pope 2 can be iolaed onl if paen() i ed. If paen() = R, he oo of he ee, hen e can j ecolo R o be black. Thi on iolae pope 3 fo an node ince hee i nohing aboe R. If paen() R, hen e hae hee cae. We can ame paen(paen()) i coloed black ince oheie e old be iolaing pope 2 befoe e een ineed. We ma need o appl he fiing opeaion mliple ime befoe he ee i a pope Red-lack ee. Hence, een hogh a off ih boh childen nll, e migh hae moed i pad in he ee ing peio fiing opeaion, o migh in geneal hae non-nll childen. We ill no conide he hee cae. In each diagam, he objec hon ae bee of he enie Red-lack ee. Thee ma be node aboe i o node belo i, ecep hee oheie pecified. and ae bee and,, and ae ingle node. Sqae epeen black node and cicle epeen ed node. In ee cae, e ame ha he ee on he lef doe no iolae pope 3. aed on hi ampion, o hold check ha he final bee on he igh alo doe no iolae pope 3. ae 1 i he onl cae hich leae he ee on he igh ih a iolaion of Pope 3. ae 2 and 3 podce a pope Red-lack ee on he igh iho fhe ieaion. ae 1: ncle i ed. (i.e. node i ed) Recolo The poblem hee i ha paen migh be ed, o e ill hae a iolaion of pope 2. noice ha e hae moed he conflic pad in he ee. Eihe e can keep appling cae 1 nil e each he oo, o e can appl cae 2 o cae 3 and end he fi p poce. If e each he oo b appling cae 1 (in ohe od, paen() i he oo and paen() i ed, hen e j change paen() o black. ae 2: ncle i no ed (i black o doe no ei) and ke() ke() ke() (o ke() ke() ke()). 13

3 RoaeRigh Recolo No hee ae no iolaion of eihe pope 2 o pope 3, o e ae finihed. ae 3: ncle i no ed and ke() ke() ke() (o ke() ke() ke()). RoaeLef No e can appl cae 2 ing a and e e done nali We kno fom o peio anali ha Ine ake o cae ime Θ(log n). No conide he nning ime of fiing he ee. In he o cae, e migh hae o appl cae 1 nil e moe he ed-ed conflic all he a p o he oo aing fom he boom of he ee. Thi ake ime Θ(log n) ince he heigh of he ee i Θ(log n). ombined ih he Θ(log n) ime o do he Ine, e find ha RedlackIne ake ime Θ(log n) in he o cae. 6.4 Deleion I emain o be een ho Red-lack ee can ppo deleion in ime Θ(log n), hee n i he nmbe of node in he ee. Recall ha RedlackIne a eeniall he ame a he Ine opeaion fo bina each ee ecep ha i a folloed b a fi-p poce henee he ne node (hich e coloed ed) had a ed paen. Recall ha Delee(R, ) fo ST delee eihe node if ha 0 o 1 childen and cc() if ha 2 childen. RedlackDelee ill be a follo: RedlackDelee(Tee Roo R, Node ) Delee(R, ) b don acall delee, inead le be he 14

4 node ha old hae been deleed If i ed hen Delee If i black hen Fi he ee If e pefom Delee(R,) on a Red-lack ee R, hen e emoe a node (hich i eihe o cc()). If i ed hen e cold no hae poibl inodced an iolaion. Eecie. Wh cold e hae no inodced an iolaion hen deleing a ed node? If happen o be coloed black, he black-heigh balance of he ee ill almo ceainl be pe and pope 3 of Red-lack ee ill be iolaed. So again, e ill need a fi-p poce. Recall alo ha Delee ala emoe a node ha ha a mo one child (if ha o childen, hen e emoe cc(), hich nee ha o childen). Thefoe, e hae o o abo onl hoe cae hee i black and ha a mo one child. ae : ha one child: child, call i, m be ed ince oheie pope 3 old be iolaed in he bee ooed a. So e can j emoe and make black o peee he black-heigh balance fo an node aboe. ae : ha no childen: We can appl he aboe ick if ha no childen. Recall ha he nll ale a he boom of he ee ae conideed black leae. To peee he black-heigh fo anceo, e ll emoe and eplace i ih a nll node ha i no j black, b doble-black (denoed b a doble cicle). hile hi phold pope 3 of Red-lack ee, i iolae pope 1 (ha ee node m be eihe ed o black). No, e conide he poblem of emoing a doble-black node fom an abia poiion in he ee. Thee ae fie cae fo hi; in all, (hich migh be nll) ill be he doble-black node ha e an o emoe. Yo hold check ha he anfomaion peee pope 3 and do no inodce an pope 2 iolaion. 15

5 ae 1: ibling i ed: In hi cae, e modif he ee o ha neighbo i black and hen appl one of he ohe cae. Thi i o ha in geneal e can el on he fac ha neighbo ill be black: RoaeLef Recolo Noice ha hi anfomaion ha moed he doble-black node donad in he ee. ae 2: paen, ibling and nephe ae all black: Recolo Noice ha hi anfomaion ha moed he doble-black node pad in he ee. If hi keep happening, hen eenall he oo ill become he doble-black node and e can j change i o black iho iolaing an popeie. Oheie e ill be able o appl one of he ohe cae. Eecie. I i poible ha ae 1 and ae 2 can conflic ih each ohe b moing he doble-black node donad and hen pad in an infinie loop? ae 3: ibling and nephe ae black, paen i ed: Recolo We can op hee becae e hae eliminaed he doble-black. 16

6 ae 4: fa nephe i ed: paen can a off a eihe colo hee (e ll denoe hi b a ecangla node). fe he anfomaion, ake haee colo had befoe. RoaeLef Recolo gain, ince hee ae no doble-black node on he igh, e can op. ae 5: fa nephe i black, nea nephe i ed: Hee e e going o pefom a anfomaion o ha fa nephe become ed. Then e can appl ae 4. RoaeRigh Recolo nali gain, e kno he Delee ake Θ(log n) ime in he o cae. No, conide he fi-p poce. ae can be pefomed in conan ime. ae fhe beak don ino fie cae. ae 1 moe he doble black node don one poiion in he ee. ae 2 moe he doble black node p one poiion in he ee. The ohe hee poiion eliminae he doble black node in conan ime. If ae 1 i eqied, hen he paen of he doble black qae i ed. Theefoe, one of ae 3-5 ae applied and he enie opeaion i pefomed in conan ime. Finall, ae 2 moe he doble black node p one poiion in he ee and hi ill need o be pefomed Θ(log n) ime in he o cae. Theefoe, he o cae nning ime of he fi-p opeaion i Θ(log n). Theefoe, RedlackDelee ha a o cae nning ime of Θ(log n). 17