ALG 2.2 Search Algorithms

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1 Algorithms Professor Joh Reif ALG 2.2 Search Algorithms (a Biary Search: average case (b Biary Search with Errors (homework (c Iterpolatio Search (d Ubouded Search Biary Search Trees (i sorted Table of keys k 0,..., k - Biary Search Tree property: at each ode x key (x > key(y y odes o left subtree of x key (x < key(z V z odes o right subtree of x 0 V Mai Readig Selectios: CLR, Chapter 3 Auxillary Readig Selectios: AHU-Desig, 4. ad 4.5 AHU-Data, Sectios 5. ad 5. BB, Sectios 4.3 ad Hadout: "A Almost Optimal Algorithm for Ubouded Searchig" left subtree 2 3 =7 I=0 E= right subtree 2

2 Assume ( keys iserted ito tree i radom order (2 Search with all keys equally likely legth = # of edges + = umber of leaves iteral path legth I = sum of legths of all iteral paths of legth > (from root to oleaves successful search: expected #comparisos C = - = Â i=0 +(I/ (C' i + / exteral path legth E = sum of legths of all exteral paths (from root to leaves = I+2 N=4= 3 I I=4= 2 E E=2= 8 usuccessful search: expected #comparisios C' = E/ ( + = (I+2/(+ = ( C +/ (+ = = Â i= - Â i=0 (C' i 2 (i / (+ ª 2 l( =.386 log 3 4

3 Model of Radom Iput over Reals iput Set S of keys each idepedetly radomly chose over real iterval [L,U] for 0 < L < U iput set of keys, S radomly chose over [L,U] algorithm BUCKET-SORT(S: operatios - compariso operatios - Î, È operatios begi for i= to do B[i] empty list for i = to do add x i to B (x -L i Î (U-L + results ( sort i 0( expected time (2 selectio i 0(loglog expected time ed for i= to do sort (B[i] output B[] B[2] B[] 5 6

4 Theorem The expected time T of BUCKET- SORT is 0( Radom Search Table = ( x 0 < x <... < x < x + where x,..., x radom reals chose i depedetly from real iterval (x 0, x + p r oof B[i ] i s upper bouded by a Biomial variable with parameters, p = Hece $ c> "i,j Prob { B[i] > j } < c - j So T Â c -j (jlogj = O( j=0 ote geeralizes to case keys have distributio F Selectio Problem iput key Y problem fid idex k * s.t. k * = Y ote k * has Bi omi al d i stributio wi th parameters,p = (Y- 0 / (

5 Algorithm INTERPOLATION-SEARCH (,Y [ ] iitialize k È p commet k = È E( k * [2 ] if k = Y the retur k [3 ] if k < Y the output INTERPOLA TION- SEA RCH (',Y where ' = ( k,..., + [4 ] else k > Y ad output INTERPOLA TION- SEA RCH ( ",Y where " = ( 0,..., k k = È p Tricky Aalysis! k k * + 9 Rad om Tab l e = ( 0,,...,, + Algorithm pseudo iterpolatio search (,Y [0 ] k È p where p = (Y- 0 / ( [] if Y = k the retur k [2 ] if Y > k the for k' = k, k+, k+2,... if Y < È the exit with k'+ output pseudo iterpolatio search (',Y [3 ] else if Y < k the where ' = ( k',..., k'+ for k' = k, k-, k-2,... if Y > È the exit with k'- output pseudo iterpolatio search ( ",Y where " = ( k'-,..., k' 0

6 Probabilistic Aalysis of Psuedo Iterpolatio Search Easy Aalysis! k * is Biomial with mea p variace s 2 = p(-p p -2 p - p p + p +2 so k * - È p s approximates ormal as Æ Hece k Prob( * - È p s Z Y (Z/ Z - Z 2 2 where Y (Z = e 2P 2

7 Lemma C 2.03 where C = expected umber of probes i give call So Prob( i probes used i give call < Pr ob ( k * - È p > (i - 2 proof C = Â i> i Prob (i probes used Y (Z i / Z i where Z i = (i-2 = s (i-2 p(-p 2(i-2 sice p(-p 4 = Â i> 2 + Â i 3 Prob ( i probes used Y (Z i / Z i 2.03 Theorem Pseudo Iterpolatio Search has expected time T C l oglog proof T( C + T( C l oglog 3 4

8 Probabilistic Aalysis of Iterpolatio Search Theorem The expected umber of comparisos of Iterpolatio Search is T( l oglog + c (logloglog 2 Lemma Prob( k * - È p 0( log a proof where a is costat Si ce k * is Bi omi al wi t h parameters p, Prob( k * 2 e -z2 /2 - p Zs Z 2 p a 2 for s = p(-p ad Z = 0( l og proof T( + ( - T ( O( log + +loglog( log a +log( log 2 + c (logloglog2 loglog + c (logloglog 2 sice log2= a 2 +c l ogloglog( log +o( 5 6

9 Ubouded Search iput table [], [2],... where for j =,2,... { 0 j< [j] = j Applicatios ( Table Look-up i a ordered, ifiite table (2 biary ecodig of itegers if S represets iteger, the S is ot a prefix of ay S, j π j {S, S 2,...} called a prefix set ubouded Search Problem fid such that [-] = 0 ad []= Cost for algorithm A: C A (=m if algorithm A uses m evaluatios to determie that is the solutio to the ubouded search problem idea: use S = ( b, b,..., b 2 C A ( where b m = if the m'th evaluatio of is i algorithm A for ubouded search 7 8

10 Biary Search Algorithm Uary Search Algorithm Algorithm B o try [], [2],..., util [] = Cos t C ( = Bo Algorithm B lst stage try [2 i -] for i=,2,..., m util [ 2 m - ] = ( cost m = Î l og + where 2m- 2 m - 2d stage biary search over 2 m- elemets cost l og(2 m- = m- = Î l og Tot al Cost C B ( = 2 Î l og

11 Double Biary Search Algorithm B 2 st stage try 2 (2 - -,..., where m = Î l og + m 2 (2 - = (cost is C B (m = 2 Î l ogm d stage same as 2d stage of B after m was foud Cos t C Bo ( = m- = Î l og Tot al Cost C B2 ( = C B (m + C Bo ( = 2 Î l og ( Î l og+ + + Î l og 2 22

12 ... B 0 B fid by uary search fid m ( = log Î + by uary search fid fid by biary search B k fid fid m ( m 0 (= m (= Î logm 0 + fid by biary search B 2 fid fid m ( k- m j (= Î log m j - + fid m ( fid m 2 ( fid m ( by uary by biary search search m 2 ( = logm + m ( = logm + Î Î 0 fid by biary search m 0 ( = 23 fid m k( by uary search fid m ( by k- biary search Cos t C Bk ( = C Bk- ( - m k- (+(2m k - k = Â i= L i ( + L k ( + (where L i ( = m i ( - cost of ew biary search 24

13 g(0 = 2 25

) n. ALG 1.3 Deterministic Selection and Sorting: Problem P size n. Examples: 1st lecture's mult M(n) = 3 M ( È

) n. ALG 1.3 Deterministic Selection and Sorting: Problem P size n. Examples: 1st lecture's mult M(n) = 3 M ( È Algorithms Professor Joh Reif ALG 1.3 Determiistic Selectio ad Sortig: (a) Selectio Algorithms ad Lower Bouds (b) Sortig Algorithms ad Lower Bouds Problem P size fi divide ito subproblems size 1,..., k