Overview. Splay trees. Balanced binary search trees. Inge Li Gørtz. Self-adjusting BST (Sleator-Tarjan 1983).

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1 Ovrvw B r rh r: R-k r -3-4 r 00 Ig L Gør Amor Dm rogrmmg Nwork fow Srg mhg Srg g Comuo gomr Irouo o NP-om Rom gorhm B r rh r -3-4 r Aow,, or 3 k r o Prf Evr h from roo o f h m gh mr h E w E R E R rgr h R A A C H I N S S r Sf-jug BST (Sor-Trj 93) Mo frqu o r o o h roo Tr rorg f fr h oro Afr o o mov o h roo oro Wor m for ro, o rh O() Amor m r oro O(og ) R-k r Th roo w k A roo-o-f h hv h m umr of k o A A C E I R S R o o o hv r hr A v (NIL) r k H N 3

2 Sg Sg S(): o foowg roo u h roo L h r of rgh (or f): f h o grr S(): o foowg roo u h roo L () h r of rgh (or f): f h o grr g-g (or g-g): f o of,() f h h ohr rgh h rgh f w w w rgh roo ( f roo ) g-g Sg S(): o foowg roo u h roo L h r of rgh (or f): f h o grr g-g (or g-g): f o of, f h h ohr rgh h ror-or: f () r hr oh f hr or oh rgh hr Dm mmo Wor rug m ( r) Immo rh r mmum mmum uor ror k O() O() O() O() O() O() O() orr rr O(og ) O() O() O() O() O(og ) O(og ) BST O(h) O(h) O(h) O(h) O(h) O(h) O(h) -3-4 r O(og ) O(og ) O(og ) O(og ) O(og ) O(og ) O(og ) r-k r O(og ) O(og ) O(og ) O(og ) O(og ) O(og ) O(og ) r O(og ) O(og ) O(og ) O(og ) O(og ) O(og ) O(og ) : mor rug m rgh ror-or ( f ror-or )

3 Amor Amor Dm Progrmmg Gr gorhm hqu C u wh h rom hv om uruur : ouo oru from om ouo o urom Tm rqur o rform qu of oro vrg ovr h oro rform Em: m wh oug hvg If h fu o h m o w rr of ou If h qurr fu o h m o w rr of hf h Wor m for ro or o: O() Amor m for ro o: O() A qu of ro o k m O() Em Ro ug Log ommo uqu Squ gm A r hor h Mho Aggrg mho Aoug mho Po mho Log ommo uqu Log ommo uqu urom ror: X- urom ror: X- Yj- j Yj- j >< 0 f = 0 or j =0 LCS(X,Y j )= LCS(X,Y j )+ f = j >: m(lcs(x,y j ), LCS(X,Y j )) f = j >< 0 f = 0 or j =0 LCS(X,Y j )= LCS(X,Y j )+ f = j >: m(lcs(x,y j ), LCS(X,Y j )) f = j S A N D A L S D o LCS(X 5,Y 4 ) S A 0 N 0 D 0 A L S S A N Vu, o ouo D A L S

4 Log ommo uqu urom ror: X- Yj- >< 0 f = 0 or j =0 LCS(X,Y j )= LCS(X,Y j )+ f = j >: m(lcs(x,y j ), LCS(X,Y j )) f = j S A 0 N 0 D 0 A L S j S A N D A L S Nwork Fow Nwork fow: grh G=(V,E) S vr (our) (k) Evr g (u,v) h (u,v) 0 Fow: or: vr g h fow 0 f(u,v) (u,v) fow orvo: for u, : fow o u qu fow ou of u X f(v, u) = X f(u, v) u v:(v,u)e v:(u,v)e Vu of fow f h um of fow ou of mu um of fow o : f = X f(, v) X f(v, ) v:(,v)e v:(v,)e Mmum fow rom: f - fow of mmum vu Nwork fow: - Cu Cu: Pro of vr o S T, uh h S T S T Augmg h Augmg h (fo ffr h CLRS): - h whr forwr g hv fovr kwr g hv ov fow +δ -δ +δ +δ -δ -δ f < f > 0 f3 < 3 f4 < 4 f5 > 0 f > 0 Thr o ugmg h <=> f mmum fow C of u: o of g gog from S o T Fow ro u: fow from S o T mu fow from T o S Vu of fow fow f (S,T) for - u (S,T) Suo w hv fou fow f u (S,T) uh h f = (S,T) Th f mmum fow (S,T) mmum u For-Fukro gorhm: R f ugmg h, u, u o ugmg h Rug m: O( f* m) Emo-Kr gorhm: R f hor ugmg h, u, u o ugmg h U BFS o f hor ugmg h Rug m: O(m ) F mmum u A vr o whh hr ugmg h from go o S, r o T

5 Nwork fow C mo ov m rom v mmum fow Mmum r mhg k g-jo h o vr M our/k gm rom: Em X oor, Y ho, h oor hou work mo ho, h oor v om of h ho Srg Mhg Srg mhg rom: rg T () rg P (r) ovr h Σ T =, P = m Ror rg oo of ourr of P T Srg mhg uomo Rug m: O( + m Σ ) Kuh-Morr-Pr (KMP) Rug m: O(m + ) F Auomo F uomo: h Σ = {,,} P= F Auomo F uomo: h Σ = {,,} P= rg g rg g og rf of P h uff of '

6 F Auomo F uomo: h Σ = {,,} P = Kuh-Morr-Pr (KMP) Mh P[ q]: F og ok P[k] h mh of P[q] F og rf P[k] of P h ror uff of P[q] Arr π[ m]: π[q] = m k < q uh h P[k] uff of P[ q] C f uomo wh fur k: T = π[] KMP mhg KMP: C f uomo wh fur k: og rf of P h uff of wh w hv mh u ow T = Srg Ig Srg g rom Gv rg S of hrr from h Σ Prro S o ruur o uor Srh(P): Rur rg oo of ourr of P S Tr S Com r Ch of o wh g h mrg o g o Suff r Com r ovr h uff of rg S S 4 S S 7 S 3 h S5 h

7 Srg Ig Srg g rom Gv rg S of hrr from h Σ Prro S o ruur o uor Srh(P): Rur rg oo of ourr of P S Tr Com r Ch of o wh g h mrg o g o S 4 S S Suff r Com r ovr h uff of rg h S 5 S 7 h S S 3 Suff r Suff r Com r ovr uff of rg Suff r u o ov h Srg g rom : S: O() Srh m: O(m+o) Prrog: O(or(, Σ )) m Suff r Log ommo urg Suff r Com r ovr uff of rg 4 Suff r u o ov h Srg g rom : S: O() [5,7] Srh m: O(m+o) [3,4] [7,7] [7,7] [,] Prrog: O(or(, Σ )) m [,7] [3,4] [7,7] [5,7] [7,7] F og ommo uff of rg S S Coru h uff r ovr SS Mrk f wh f uff r S ohrw A rgh Mrk oom u o whh h oh Th o (h o rrg h og rg) h og ommo urg 9 5 Em: F og ommo urg of : h og ommo urg

8 Co r: Dv--Coqur Dv: rw vr L o h rough / o o h Coqur: f o r h rurv Com: F o r wh o o h Rur of 3 ouo F o r wh o o h F o r wh o o h, umg h < δ Orvo: o o or o wh δ of L Sor o δ-r hr oor O hk w h o wh 7 oo or! L 7 L δ = m(, ) Thk ou o Kv W for ro o Thk ou o Kv W for ro o δ Co Pr of Po: Rom gorhm U hh o or whh qur o O or o r ook (r o) Rom gorhm M/S Quk-or Co r δ δ 0

9 P NP P ov rm oom m NP ov o-rm (wh gug) oom m O h m for h rgh gu ou P NP (vr rom T whh P o NP) I o kow (u rog v) whhr h uo ror, h whhr hr rom NP whh o P Thr u of NP whh o h hr rom, NP-om rom Ruo How o u gorhm DTU 0 Agorhm for mv 049 Comuo Hr Prom 07 Grh hor Th Thk ou