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

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1 Ovrvw Bn nr rh r: R-k r n -- r 00 Ing L Gør Amor n Dnm rogrmmng Nwork fow Srng mhng Srng nng Comuon gomr Inrouon o NP-omn Rnom gorhm Bn nr rh r -- r. Aow,, or k r no Prf n. Evr h from roo o f h m ngh. mr hn E wn E n R E R rgr hn R A A C H I N S S r Sf-jung BST (Sor-Trjn 9). Mo frqun no r o o h roo. Tr rorgn f fr h oron. Afr o no mov o h roo oron. Wor m for nron, on n rh O(n). Amor m r oron O(og n). R-k r. Th roo w k A roo-o-f h hv h m numr of k no. A A C E I R S R no o no hv r hrn A v (NIL) r k H N

2 Sng Sng S(): o foowng roon un h roo. L h rn of. rgh (or f): f h no grnrn. S(): o foowng roon un h roo. L () h rn of. rgh (or f): f h no grnrn. g-g (or g-g): f on of,() f h n h ohr rgh h. rgh f w w w rgh roon (n f roon ) g-g Sng S(): o foowng roon un h roo. L h rn of. rgh (or f): f h no grnrn. g-g (or g-g): f on of, f h n h ohr rgh h. ror-or: f n () r hr oh f hrn or oh rgh hrn. Dnm mmnon Wor runnng m ( r) Immnon rh nr mnmum mmum uor ror nk O(n) O() O() O(n) O(n) O(n) O(n) orr rr O(og n) O(n) O(n) O() O() O(og n) O(og n) BST O(h) O(h) O(h) O(h) O(h) O(h) O(h) -- r O(og n) O(og n) O(og n) O(og n) O(og n) O(og n) O(og n) r-k r O(og n) O(og n) O(og n) O(og n) O(og n) O(og n) O(og n) r O(og n) O(og n) O(og n) O(og n) O(og n) O(og n) O(og n) : mor runnng m rgh ror-or (n f ror-or )

3 Amor n Amor n. Dnm Progrmmng Gnr gorhm hnqu Cn u whn h rom hv om uruur : ouon n onru from om ouon o urom. Tm rqur o rform qun of oron vrg ovr h oron rform. Em: nm wh oung n hvng If h fu o h mn o nw rr of ou. If h qurr fu o h mn o nw rr of hf h. Wor m for nron or on: O(n) Amor m for nron n on: O() An qun of n nron n on k m O(n). Em Wgh nrv hung Sgmn qur Squn gnmn Shor h (Bmn-For) Mho. Aggrg mho Aounng mho Pon mho Squn gnmn j f =0 >< f j =0 SA(X,Y j )= >< (, j )+SA(X,Y j ), mn + SA(X,Y j ), ohrw >: >: + SA(X,Y j )} Squn gnmn: Fnng h ouon j f =0 >< f j =0 SA(X,Y j)= >< (, j)+sa(x,y j ), mn + SA(X,Y j ), >: >: + SA(X,Y j)} Pn mr A C G T A 0 ohrw C 0 δ = G 0 T 0 A C A A G T C Pn mr A C A A G T C A C A A G T C C A T G T δ = SA(X 5, Y ) Dn on? A C G T A 0 C 0 G 0 T C 5 A 5 T G 5 T C A T G T

4 Bmn-For Em 0 f = 0 OPT(, v) = { mn{opt(,v), mn (v,w) E {OPT(,w) + vw }} ohrw Bmnn-For(G,,) Bmnn-For(G,,) for h no v V M[0,v] = M[0,] = 0. for = o n- for h no v V M[,v] = M[-,v] for h g (v,w) E M[,v] = mn(m[,v], M[-,w] + vw f 0 for h no v V M[0,v] = M[0,] = 0. for = o n- for h no v V M[,v] = M[-,v] for h g (v,w) E M[,v] = mn(m[,v], M[-,w] + vw f 9 0 Em Em Bmnn-For(G,,) for h no v V M[0,v] = M[0,] = 0. for = o n- for h no v V M[,v] = M[-,v] for h g (v,w) E M[,v] = mn(m[,v], M[-,w] + vw f Bmnn-For(G,,) for h no v V M[0,v] = M[0,] = 0. for = o n- for h no v V M[,v] = M[-,v] for h g (v,w) E M[,v] = mn(m[,v], M[-,w] + vw) f f f 9 5 0

5 Nwork Fow Nwork fow: grh G=(V,E). S vr (our) n (nk). Evr g (u,v) h (u,v) 0. Fow: onrn: vr g h fow 0 f(u,v) (u,v). fow onrvon: for u, : fow no 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 mnu um of fow no : f = X f(, v) X f(v, ) v:(,v)e v:(v,)e Mmum fow rom: fn - fow of mmum vu Nwork fow: - Cu Cu: Pron of vr no S n T, uh h S n T. S T C of u: o of g gong from S o T. Fow ro u: fow from S o T mnu fow from T o S. Vu of fow n fow f (S,T) for n - u (S,T). Suo w hv foun fow f n u (S,T) uh h f = (S,T). Thn f mmum fow n (S,T) mnmum u. Augmnng h Sng gorhm Augmnng h (fnon ffrn hn n CLRS): - h whr forwr g hv fovr kwr g hv ov fow +δ -δ +δ +δ -δ -δ f < f > 0 f < f < f5 > 0 f > 0 Thr no ugmnng h <=> f mmum fow. For-Fukron gorhm: R fn ugmnng h, u, un no ugmnng h Runnng m: O( f* m). Emon-Kr gorhm: R fn hor ugmnng h, u, un no ugmnng h U BFS o fn hor ugmnng h. Runnng m: O(nm ) Fn mnmum u. A vr o whh hr n ugmnng h from go no S, r no T. Sng rmr Δ On onr g wh Δ n ru grh Gf(Δ). Em: Δ = Gf 5 Gf() 5 0

6 Nwork fow Cn mo n ov mn rom v mmum fow. Mmum r mhng k g-jon h on vr Mn our/nk owr oun on fow on g. gnmn rom: Em. X oor, Y ho, h oor hou work mo ho, h oor v om of h ho. Srng Mhng Srng mhng rom: rng T () n rng P (rn) ovr n h Σ. T = n, P = m. Ror rng oon of ourrn of P n T. Srng mhng uomon. Runnng m: O(n + m Σ ) Knuh-Morr-Pr (KMP). Runnng m: O(m + n) Fn Auomon Fn uomon: h Σ = {,,}. P=. Fn Auomon Fn uomon: h Σ = {,,}. P=. rng ng rng ng ong rf of P h uff of '

7 Fn Auomon Fn uomon: h Σ = {,,}. P =. Knuh-Morr-Pr (KMP) Mh P[ q]: Fn ong ok P[..k] h mh n of P[..q]. Fn ong 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]. Cn n fn uomon wh fur nk: T = π[] KMP mhng KMP: Cn n fn uomon wh fur nk: ong rf of P h uff of wh w hv mh un now. T = 5 Srng Inng Srng nng rom. Gvn rng S of hrr from n h Σ. Prro S no ruur o uor Srh(P): Rurn rng oon of ourrn of P n S. Tr. S Com r. Chn of no wh ng h mrg no ng no Suff r. Com r ovr h uff of rng. S S S S 7 S h S5 h

8 Srng Inng Srng nng rom. Gvn rng S of hrr from n h Σ. Prro S no ruur o uor Srh(P): Rurn rng oon of ourrn of P n S. Tr. Com r. Chn of no wh ng h mrg no ng no S S S Suff r. Com r ovr h uff of rng. h S 5 S 7 h S S Suff r Suff r. Com r ovr uff of rng. Suff r n u o ov h Srng nng rom n: S: O(n) n Srh m: O(m+o) n Prrong: O(or(n, Σ )) m n n n n 5 7 Suff r Suff r. Com r ovr uff of rng. Long ommon urng Em: Fn ong ommon urng of n : [,] [,] [7,7] [,7] [,] [7,7] [5,7] [7,7] [5,7] 5 [7,7] Suff r n u o ov h Srng nng rom n: S: O(n) Srh m: O(m+o) Prrong: O(or(n, Σ )) m 0 5 n n h ong ommon urng

9 Rnom gorhm Mn/S. Quk-or Co r Co Pr of Pon: Rnom gorhm U hh o or whh qur on n. On or on r ook (r on). 7 5 δ 5 9 δ 0 Ponom-m ruon Ruon. Prom X onom ru o rom Y f rrr nn of rom X n ov ung: Ponom numr of nr omuon, u Ponom numr of o or h ov rom Y. Noon. X P Y. W for m o wr own nn n o k o nn of Y mu of onom. Ponom-m ruon Puro. Cf rom orng o rv ffu. Dgn gorhm. If X P Y n Y n ov n onom-m, hn X n o ov n onom m. Eh nr. If X P Y n X nno ov n onom-m, hn Y nno ov n onom m. Eh quvn. If X P Y n Y P X, w u noon X =P Y. u o onom for 5

10 P v NP P ov n rmn onom m. NP ov n non-rmn onom m/ h n ffn (onom m) rfr. P NP (vr rom T whh n P o n NP). I no known (u rong v) whhr h nuon ror, h whhr hr rom n NP whh no n P. Thr u of NP whh onn h hr rom, NP-om rom: X NP-Com f X NP Y P X for Y NP How o rov rom NP-om. Prov X NP (h n vrf n onom m).. S known NP-om rom Y.. Gv onom m ruon from Y o X: En how o urn n nn of Y no on or mor nn of X En how o u onom numr of o h k o gorhm/ or for X o ov Y. Prov/rgu h h ruon orr. 7 How o u gorhm DTU 0 Agorhm for mv 09 Comuon Hr Prom 07 Grh hor 007 Comuon Too for D Sn Th Thnk ou

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

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