Balanced binary search trees

Transcription

1 02110 Inge Li Gørtz

2 Overview Blnced binry serch trees: Red-blck trees nd trees Amortized nlysis Dynmic progrmming Network flows String mtching String indexing Computtionl geometry Introduction to NP-completeness Rndomized lgorithms

3 Blnced binry serch trees trees. Allow 1, 2, or 3 keys per node smller thn E E R lrger thn R Perfect blnce. Every pth from root to lef hs sme length. A A C between E nd R H I N S Red-blck trees. The root is lwys blck All root-to-lef pths hve the sme number of blck nodes. A A C E I R S Red nodes do not hve red children All leves (NIL) re blck H N 3

4 Sply trees Self-djusting BST (Sletor-Trjn 1983). Most frequently ccessed nodes re close to the root. Tree reorgnizes itself fter ech opertion. After ccess to node it is moved to the root by sply opertion. Worst cse time for insertion, deletion nd serch is O(n). Amortized time per opertion O(log n).

5 Splying Sply(x): do following rottions until x is the root. Let y be the prent of x. right (or left): if x hs no grndprent. x y right x y c left b b c right rottion t x (nd left rottion t y)

6 Splying Sply(x): do following rottions until x is the root. Let p(x) be the prent of x. right (or left): if x hs no grndprent. zig-zg (or zg-zig): if one of x,p(x) is left child nd the other is right child. z z x w x w z x d w c d b c d b c b zig-zg t x

7 Splying Sply(x): do following rottions until x is the root. Let y be the prent of x. right (or left): if x hs no grndprent. zig-zg (or zg-zig): if one of x,y is left child nd the other is right child. roller-coster: if x nd p(x) re either both left children or both right children. z y x y x z y x c d b c d b z b c d right roller-coster t x (nd left roller-coster t z)

8 Dynmic set implementtions Worst cse running times (except sply trees) Implementtion serch insert delete minimum mximum successor predecessor linked lists O(n) O(1) O(1) O(n) O(n) O(n) O(n) ordered rry O(log n) O(n) O(n) O(1) O(1) O(log n) O(log n) BST O(h) O(h) O(h) O(h) O(h) O(h) O(h) tree O(log n) O(log n) O(log n) O(log n) O(log n) O(log n) O(log n) red-blck tree O(log n) O(log n) O(log n) O(log n) O(log n) O(log n) O(log n) sply tree O(log n) O(log n) O(log n) O(log n) O(log n) O(log n) O(log n) : mortized running time 8

9 Amortized nlysis Amortized nlysis. Time required to perform sequence of dt opertions is verged over ll the opertions performed. Exmple: dynmic tbles with doubling nd hlving Methods. If the tble is full copy the elements to new rry of double size. If the tble is qurter full copy the elements to new rry of hlf the size. Worst cse time for insertion or deletion: O(n) Amortized time for insertion nd deletion: O(1) Any sequence of n insertions nd deletions tkes time O(n). Aggregte method Accounting method Potentil method

10 Dynmic Progrmming Generl lgorithmic technique Cn be used when the problem hve optiml substructure : solution cn be constructed from optiml solutions to subproblems. Exmples Rod cutting Longest common subsequence Sequence lignment All pirs shortest pth

11 Longest common subsequence subproblem property: Xi-1 xi Yj-1 yj LCS(X i,y j )= 8 >< 0 if i = 0 or j =0 LCS(X i 1,Y j 1 )+1 ifx i = y j >: mx(lcs(x i,y j 1 ), LCS(X i 1,Y j )) if x i 6= y j S A N D A L S B A N A N A S Depends on LCS(X 5,Y 4 )

12 Longest common subsequence subproblem property: Xi-1 xi Yj-1 yj LCS(X i,y j )= 8 >< 0 if i = 0 or j =0 LCS(X i 1,Y j 1 )+1 ifx i = y j >: mx(lcs(x i,y j 1 ), LCS(X i 1,Y j )) if x i 6= y j B A N A N A S S A N D A L S B A N A N A S S A N Vlue, not solution D A L S

13 Longest common subsequence subproblem property: Xi-1 xi Yj-1 yj LCS(X i,y j )= 8 >< 0 if i = 0 or j =0 LCS(X i 1,Y j 1 )+1 ifx i = y j >: mx(lcs(x i,y j 1 ), LCS(X i 1,Y j )) if x i 6= y j B A N A N A S S A N D A L S B A N A N A S S A N D A L S

14 Network Flow Network flow: 1 2 grph G=(V,E). 2 Specil vertices s (source) nd t (sink). s Every edge (u,v) hs cpcity c(u,v) 0. Flow: 1 cpcity constrint: every edge e hs flow 0 f(u,v) c(u,v). flow conservtion: for ll u s, t: flow into u equls flow out of u. X f(v, u) = X f(u, v) v:(v,u)2e v:(u,v)2e Vlue of flow f is the sum of flows out of s minus sum of flows into s: f = X X f(s, v) f(v, s) v:(s,v)2e v:(v,s)2e Mximum flow problem: find s-t flow of mximum vlue u 2 2 t

15 Network flow: s-t Cuts Cut: Prtition of vertices into S nd T, such tht s S nd t T. S T s t Cpcity of cut: totl cpcity of edges going from S to T. Flow cross cut: flow from S to T minus flow from T to S. Vlue of flow ny flow f c(s,t) for ny s-t cut (S,T). Suppose we hve found flow f nd cut (S,T) such tht f = c(s,t). Then f is mximum flow nd (S,T) is minimum cut.

16 Augmenting pths Augmenting pth (definition different thn in CLRS): s-t pth where forwrd edges hve leftover cpcity bckwrds edges hve positive flow s +δ -δ +δ +δ -δ -δ f1 < c1 f2 > 0 f3 < c3 f4 < c4 f5 > 0 f6 > 0 t There is no ugmenting pth <=> f is mximum flow. Ford-Fulkerson lgorithm: Repetedly find ugmenting pth, use it, until no ugmenting pth exists Running time: O( f* m). Edmonds-Krp lgorithm: Repetedly find shortest ugmenting pth, use it, until no ugmenting pth exists Use BFS to find shortest ugmenting pth. Running time: O(nm 2 ) Find minimum cut. All vertices to which there is n ugmenting pth from s goes into S, rest into T.

17 Network flow Cn model nd solve mny problems vi mximum flow. Mximum biprtite mtching k edge-disjoint pths cpcities on vertices Mny sources/sinks ssignment problems: Exmple. X doctors, Y holidys, ech doctor should work t t most c holidys, ech doctor is vilble t some of the holidys. 1 c 1 s c c t

18 String Mtching String mtching problem: string T (text) nd string P (pttern) over n lphbet Σ. T = n, P = m. Report ll strting positions of occurrences of P in T. String mtching utomton. Running time: O(n + m Σ ) Knuth-Morris-Prtt (KMP). Running time: O(m + n) Rbin-Krp (fingerprinting). Running time: Expected O(m + n)

19 Finite Automton Finite utomton: lphbet Σ = {,b,c}. P= bbc. strting stte b b c ccepting stte b b

20 Finite Automton Finite utomton: lphbet Σ = {,b,c}. P= bbc. strting stte b b c ccepting stte b b longest prefix of P tht is suffix of b'

21 Knuth-Morris-Prtt (KMP) Mtched P[1 q]: Find longest block P[1..k] tht mtches end of P[2..q]. b b b b b Find longest prefix P[1...k] of P tht is proper suffix of P[1...q] Arry π[1 m]: π[q] = mx k < q such tht P[1...k] is suffix of P[1 q]. Cn be seen s finite utomton with filure links: i b b c π[i]

22 String Indexing String indexing problem. Given string S of chrcters from n lphbet Σ. Preprocess S into dt structure to support Serch(P): Return strting position of ll occurrences of P in S. Tries. Compressed trie. Chins of nodes with single child merged into single node Suffix tree. Compressed trie over ll the suffixes of string.

23 Suffix tree Suffix tree. Compressed trie over ll suffixes of string. bnn n $ 6 n n $ $ 0 5 n 2 $ Suffix trees cn be used to solve the String indexing problem in: Spce: O(n) Serch time: O(m+occ) Preprocessing: O(sort(n, Σ )) time

24 Suffix tree Suffix tree. Compressed trie over ll suffixes of string. [4,6] [2,3] [6] 1 3 Suffix trees cn be used to solve the String indexing problem in: Spce: O(n) [1] [6] Serch time: O(m+occ) Preprocessing: O(sort(n, Σ )) time 5 [0,6] [2,3] 0 [4,6] [6] 2 4 [6] b n n $

25 Longest common substring Find longest common suffix of strings S1 nd S2. Construct the suffix tree over S1$1S2$2 (remove ll prts of suffixes crossing $1). Exmple: Find longest common substring of nns nd bnn: Construct suffix tree of nns$1bnn$2. bnn$2 n s$1 $1 $2 n s$1 $2 n s$1 $2 n s$1 $2 s$1 $2 nn is the longest common substring s$1 $2 Mrk bottom up ll nodes which hs both $1 nd $2 in descendnt. The deepest one (the one representing the longest string) is the longest common substring.

26 Computtionl Geometry Geometric problems (this course Eucliden plne). Convex hull Jrvis s mrch O(nh) Grhm s scn O(n log n) Counterclockwise test in O(1) time

27 Convex Hull 3 equivlent definitions of convex hull: Given set of points Q, the convex hull CH(Q) is Def 1. The smllest convex polygon contining Q. Def 2. The lrgest convex polygon, whose vertices ll re points in Q. Def 3. The convex polygon contining Q nd whose vertices ll re points in Q. p 6 p 5 p 4 p1 p2 p3 Lst time Grhm s scn O(n log n) Jrvis s mrch O(nh)

28 Grhm s scn Grhm s scn. Pick lowest point p0 s strting point Sort remining points in counterclockwise order round p0. Use liner time scn to build hull: Push p0, p1 nd p2 onto the stck. Next point p: If dding p gives left turn push p onto stck If dding p gives right turn pop top element from stck nd check gin. Continue checking until we get right turn or only 3 vertices left on stck. p8 p9 p6 p4 p 7 p 5 p 3 p 2 p1 p 0

29 Jrvis s mrch Strt with dding lowest point p0 to CH(Q). Next point fter p: point ppering to be furthest to the right to someone stnding t p nd looking t the other points (smllest if sorted in counterclockwise order). If q point following p then for ny other point r in Q then p,q,r re in counterclockwise order. Cn find next vertex by performing n-1 counterclockwise tests.

30 Rndomized lgorithms Medin/Select. Quick-sort

31 P nd NP P solvble in deterministic polynomil time. NP solvble in non-deterministic (with guessing) polynomil time. Only the time for the right guess is counted. P NP (every problem T which is in P is lso in NP). It is not known (but strongly believed) whether the inclusion is proper, tht is whether there is problem in NP which is not in P. There is subclss of NP which contins the hrdest problems, NP-complete problems. Reductions.

32 Thnk you