Self-Adjusting Top Trees

Transcription

1 Th Polm Sl-jsting Top Ts ynmi ts: ol: mintin n n-tx ost tht hngs o tim. link(,w): ts n g twn tis n w. t(,w): lts g (,w). pplition-spii t ssoit with gs n/o tis. ont xmpls: in minimm-wight g in th pth twn ny two tis. l to ll gs in th pth twn two tis. in totl wight o ll tis in st. Rot Tjn (Pinton Unisity/P) Rnto nk (Pinton Unisity) O(log n) tim p option. mp ity t onto ln t. t Stts t Stts ST-ts [ST83] ST-ts [ST85] Topology [85] R-ts Top Ts [03] [LT97] T-ts [K95] ST-ts [ST83] ST-ts [ST85] Topology [85] R-ts Top Ts [03] [LT97] T-ts [K95] ity st qis? ity st qis? ity pth qis? ity pth qis? Simpl to implmnt? int only Simpl to implmnt? ni int? ni Int? O(log n) wost s? motiz nomiz int only O(log n) wost s? motiz nomiz motiz Pinipl pth omp. pth omp. t onttion t onttion t onttion liniztion (l to) Pinipl pth omp. pth omp. t onttion t onttion t ont./ pth omp. liniztion (l to) onttions: Rk n ompss Popos y Mill n Ri [1985] (plll stting). Rk: limints -on tx. ollpss g onto ssso. ssms il o o gs. ompss: limints -two tx. omins two gs into on. Oiginl gs n slting g lsts. onttions: Rk n ompss onttion: Sis o ks n ompsss; Rs t to singl lst (g). Top t mois onttion: it ss only to oot lst. Us ins wht inomtion to sto in pnt. ny o o ks n ompsss is ight : oot will h th ot inomtion. ln: pts in O(log n) tim. lstp t l. [1997] s topology ts: high oh. show it implmnttion.

2 onsi som noot t: Pik -on tx s oot, it ll gs tows it. ll this nit t (oot t with -on oot). Pik oot pth: stts t som l; ns t th oot. Rpsnt th oot pth s iny t: Ls: s lsts (oiginl gs). Intnl nos: ompss lsts. N N N N ht i th o tx is not two? Rsily psnt h st oot t th tx. t most two s o il o. ht i th o tx is not two? Rsily psnt h st oot t th tx. o tx is ompss, k st onto jnt lst. N N N N N N N N

3 : Up to o hiln p no (p to two ost hiln). Mning: p to two ks ollow y ompss. ow os th si psnttion wok? Mst psnt sts oot t th oot pth. N N N N xmpl: N = ompss(k(, ), k(, )) = ow os th si psnttion wok? Mst psnt sts oot t th oot pth. h st is sqn o nit ts. ow os th si psnttion wok? Mst psnt sts oot t th oot pth. h st is sqn o nit ts. Rpsnt h nit t sily. ow os th si psnttion wok? Mst psnt sts oot t th oot pth. h st is sqn o nit ts. Rpsnt h nit t sily. il iny t o ks. = Intpttions: Us int: t onttion. sqn o ks n ompsss; singl t; simil to topology ts n R-ts. Implmnttion: pth omposition. mximl g-isjoint pths; hihy o iny ts (k ts/ompss ts). simil to ST-ts. (h il is k lst.)

4 Sl-jsting Top Ts Topmost ompss t psnts th oot pth. Top t int llows th s to ss th oot pth only. xpos mks no pt o th oot pth (n/o hngs oot). Min tools: sply n spli. Sl-jsting Top Ts Splying: sis o ottions within k/ompss st: kps st ln (in th motiz sns); ings tx to th oot o th st. N y N x ott ight z N x yz ott lt wx N y xpos() y wx xy xy yz x t stt w tl t Sl-jsting Top Ts Sl-jsting Top Ts Spli: hngs th ptition o th oiginl t into pths. xpos() in 3 psss: 1. Sply within h iny t twn n th oot; N N 2. pom sis o splis; 3. sply within th inl t. tl t spli() Min slt: O(log n) motiz tim. xpos() spli() t stt Links in tils link(,w): ist xpos n w, thn ng ppopitly. w x xposing th tx is slightly int om hnging th oot. Top t nos psnt gs; mst lso ssoit with tis. g o tis xpos mtts (spil ss). Lt-ight ltion mst lx in ompss ts. Mst ll s-in ntions in th ppopit o. N w N w xw w link(,w) xw N N w w

5 Ptil onsitions ompss no: tlly psnts p to 3 lsts. ol implmnt s on lst = on no. Splying n spliing gt slightly mo omplit. Spil ss (pplition-pnnt): No il o: ompss nos h t most 3 (not 4) hiln. Simpl splis. Tiil ks: ssntilly ST-Ts. No k ts. No points to mil hiln (sh gs). th ok ost-s int? l xpimntl sty: Top ts tn to slow thn T-ts n ST-ts, t: Mo gni: /n s; st/pth options; il o on tis. Mh si to pt to int pplitions; si to son ot. ow os it omp to R-ts?