CSE Algorithms

Transcription

1 CSE 0 - Algoithms R-Black Ts Augmnting Sach Ts Intval Ts 0//00 CSE 0 - R Black Ts Binay Sach Ts... A binay t in which fo ach no y: If x is in y s lft subt, thn x y. If z is in y s ight subt, thn y z. Binay Sach Ts can o: Inst(S,x), Dlt(S,x) Sach(S,k), Minimum(S), Maximum(S) Succsso(S,x), Pcsso(S,x) Combin Min- an Max-Pioity Quu & Dictionay Can you o Incas-Ky(S,x,k) in a BST? Obvious implmntation BST can b unbalanc. O(lg n) xpct, O(n) wost cas (n anom qusts). x y z CSE 0 - R Black Ts

2 Balanc Sach Ts Basic ia: O(lg n) ynamic st opations t by nsuing t hight is O(lg n) n = siz of th st Vaious implmntations: R-Black ts, B ts (fatu in CLRS) AVL ts, - ts, Splay ts (oth mthos) Binomial haps, Fibonacci haps (allow mging) Paticia ts ( O(k) fo lngth-k kys) CSE 0 - R Black Ts R-Black Ts BST, guaant to b naly balanc.. Nos hav ky, point, & two (possibly NIL) chiln. NIL is ally a black laf w ll gloss ov this.. Evy nos is colo o black.. Root is black.. R no can t hav pant.. Paths fom oot to NIL s all hav th sam numb of black nos (th black-hight ). =black = CSE 0 - R Black Ts

3 No pth is O(lg n) in R-B t Black Subt: igno nos Evy non-laf has,, o chiln. Evy laf has th sam pth, k. Thus # black nos k+ ( = fo full binay t). In full -black t, n k+ (n = # nos) So lg n lg ( k+ ) lg k = k; that is, k < lg n. Longst path k (obvious fom pictu). So longst path lg n. CSE 0 - R Black Ts Instions into R-B t Inst as nw laf in coct spot. May cat mnac ( no with pant) Eliminat, pomot, o staightn mnac. Dtails on nxt pag. bnt staight bnt staight Aft at most x no pth movs, mnac will b gon If oot is, colo it black CSE 0 - R Black Ts

4 Moving R Mnacs Cas : Uncl (ganpants oth chil) is. Rcolo pant, uncl & ganpant. Ganpant may b a mnac now. Cas : Uncl is black, mnac is bnt. Rotat mnac with pant. R mnac is now staight. Poc to Cas Cas : Uncl black, mnac staight. Rotat & colo pant & ganpant. R mnac isappas. uncl mnac CSE 0 - R Black Ts You tun Inst,,,... into mpty t. Sach Oth Opations Compa to oot, go obvious iction, cusivly. Succsso Dlt If no has non-mpty ight subt, fin its MIN. Els, tun closst ancsto that s bigg (if any). Complicat (lik inst). 8 CSE 0 - R Black Ts

5 Augmnting Sach Ts (chapt ) Basic ia: Suppos w want to maintain aitional infomation about ata in a ynamic st. suppos this infomation can b comput at ach no using only th ata at that no an its immiat chiln. Such infomation is call a synthsiz attibut. W can o so without incasing th big-o complxity of Inst o Dlt. Why? Bcaus th only infomation that is moifi is on o th (oiginal) path fom laf to oot. th t may shift a bit whn w Inst o Dlt CSE 0 - R Black Ts (Tivial) Exampl Suppos satllit ata inclus a siz fil At ach no n w can maintain:. maximum siz of any no in n s subt, an/o. total siz of all nos in n s subt. ky siz total ( ++ = ) 0... CSE 0 - R Black Ts

6 CSE 0 - R Black Ts Total Siz Exampl 0 - Inst 0 0 total n upating sval stps lat CSE 0 - R Black Ts Exampl (continu) Pvious sli. Still has mnac.

7 Is ank a synthsiz attibut? ( Rank mans plac in sot list) In both ths ts, oot has ank- an ank- chil. But it s ank is iffnt. So ank can t b comput just fom chiln s anks. But... CSE 0 - R Black Ts Sach T & Rank in O(lg n) tim Numb of nos in subt is synthsiz. Rank can b comput by aing + nos in lft subt + # of lft ancstos + # nos in thi lft subts. whil going fom oot to th no. 8 ky p no nos in subt 8 Rank of no is th numb of nos insi ott lin. CSE 0 - R Black Ts

8 Intval Ts (Clos) intval [a,b] is { x Ñ a x b }. Intval t is a sach t with: ky = lft-han npoint of intval satllit ata = ight-han npoint of intval synthsiz attibut = max -h npoint of subt [,] [,8] 8 [-] [,] [,] [-] [-0] 0 CSE 0 - R Black Ts Ovlapping Intval Poblm (Two intvals ovlap if thy hav any points in common.) Rtun an intval in t that [x,y] ovlaps, o non. Cas : [x,y] ovlaps oot: you on. Cas : [x,y] is ntily lft of oot: sach lft subt. It can t possibly ovlap oot o anything in ight subt. (Ty [,0] in pvious xampl.) Cas : [x,y] is ntily ight of oot & max ight-han npoint in oot s lft subt: sach ight subt. It can t ovlap oot o anything in lft subt. (Ty [,8].) Cas : [x,y] is ight of oot, but not of max ight-han npoint in oot s lft subt: ovlap in lft subt. Continu with [,8] xampl this cas hols at scon stp. CSE 0 - R Black Ts